https://icme.hpc.msstate.edu/mediawiki/api.php?action=feedcontributions&user=Tenev&feedformat=atomEVOCD - User contributions [en]2020-06-07T12:11:54ZUser contributionsMediaWiki 1.19.1https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:52:23Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
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<br />
= Tutorials = <br />
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* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
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= Research =<br />
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== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
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==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
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==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
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=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
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== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.''<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br />
One observation of anomalous gravity comes from the rotation curves of galaxies. <ref name=DeSwart2017/> A rotation curve is simply the plot of linear rotation velocities of galactic matter versus the distance from the galactic center. Since most of the visible mass of galaxy is concentrated within its stellar disk, one would expect that the rotation curve would increase and then begin to decrease with the distance from the center. However, the rotation curves of many galaxies do not show such decline with radial distance but instead appear to remain constant and sometimes increase monotonically. Such observations have suggested the presence of invisible mass, which is also known as "dark matter". <br />
<br />
The Inherent Structure Hypothesis is an alternative explanation for such gravitational anomalies. We have simulated its effect for the M33 galaxy (also known as the Triangulum galaxy) and the results are visualized. We could choose to study the effect of different kinds of inherent curvature. The result visualized here us based on a spherically symmetric inherent curvature whose profile is chosen to emulate the effect of the Modified Newtonian Dynamics (MOND) theory.<ref name=Milgrom1983><br />
Milgrom, M., "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis", The Astrophysical Journal, Vol. 270, 1983.<br />
</ref><br />
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= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:46:53Z<p>Tenev: /* Inherent Structure Hypothesis */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
<br clear=all/><br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.''<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br />
One observation of anomalous gravity comes from the rotation curves of galaxies. <ref name=DeSwart2017/> A rotation curve is simply the plot of linear rotation velocities of galactic matter versus the distance from the galactic center. Since most of the visible mass of galaxy is concentrated within its stellar disk, one would expect that the rotation curve would increase and then begin to decrease with the distance from the center. However, the rotation curves of many galaxies do not show such decline with radial distance but instead appear to remain constant and sometimes increase monotonically. Such observations have suggested the presence of invisible mass, which is also known as "dark matter". <br />
<br />
The Inherent Structure Hypothesis is an alternative explanation for such gravitational anomalies. We have simulated its effect for the M33 galaxy (also known as the Triangulum galaxy) and the results are visualized. We could choose to study the effect of different kinds of inherent curvature. The result visualized here us based on a spherically symmetric inherent curvature whose profile is chosen to emulate the effect of the Modified Newtonian Dynamics (MOND) theory.<ref name=Milgrom1983><br />
Milgrom, M., "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis", The Astrophysical Journal, Vol. 270, 1983.<br />
</ref><br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:46:05Z<p>Tenev: /* Numerical Simulations of Inherent Curvature */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.''<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br />
One observation of anomalous gravity comes from the rotation curves of galaxies. <ref name=DeSwart2017/> A rotation curve is simply the plot of linear rotation velocities of galactic matter versus the distance from the galactic center. Since most of the visible mass of galaxy is concentrated within its stellar disk, one would expect that the rotation curve would increase and then begin to decrease with the distance from the center. However, the rotation curves of many galaxies do not show such decline with radial distance but instead appear to remain constant and sometimes increase monotonically. Such observations have suggested the presence of invisible mass, which is also known as "dark matter". <br />
<br />
The Inherent Structure Hypothesis is an alternative explanation for such gravitational anomalies. We have simulated its effect for the M33 galaxy (also known as the Triangulum galaxy) and the results are visualized. We could choose to study the effect of different kinds of inherent curvature. The result visualized here us based on a spherically symmetric inherent curvature whose profile is chosen to emulate the effect of the Modified Newtonian Dynamics (MOND) theory.<ref name=Milgrom1983><br />
Milgrom, M., "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis", The Astrophysical Journal, Vol. 270, 1983.<br />
</ref><br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:34:05Z<p>Tenev: /* Numerical Simulations of Inherent Curvature */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.'' (2014)<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:32:30Z<p>Tenev: /* Numerical Simulations of Inherent Curvature */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.'' (2014)<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect per \eref{eq:dm-mond-profile1}. The displayed fit was achieved with a scale mass of <math>M = 10^{11} M_\odot</math>, which is about <math>10\times</math> the mass of M33. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl/</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:31:15Z<p>Tenev: /* Numerical Simulations of Inherent Curvature */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
[[image:m33-strain.jpg|thumb|420px|<br />
Fabric strain due to the M33 galaxy.}{Fabric strain due to the M33 galaxy without (a) and with (b) inherent curvature. The top panels visualize a quadrant of the strain field, while the bottom panels show the density and strain profiles. The density profile used here is from Figure 10 of Corbelli ''et. al.'' (2014)<ref name=Corbelli2014><br />
Corbelli, E; Thilker, D.; Zibetti, S.; Giovanardi, C.; Salucci, P., "Dynamical signatures of a <math>\Lambda</math>CDM-halo and the distribution of the baryons in M33", Astronomy & Astrophysics, Vol. 572, 2014<br />
</ref>.<br />
]]<br />
<br />
[[image:m33-rotation-curve.jpg|thumb|420px|<br />
The top panel shows M33's observed rotation curve<br />
<ref name=Corbelli2003>Corbelli, E., "Dark matter and visible baryons in M33", Monthly Notices of the Royal Astronomical Society, Vol. 342, 2003<br />
</ref><br />
along those computed with and without inherent curvatuture (solid line versus dashed line, respectively). The inherent curvature was specified by a radially symmetric hyperspatial displacement <math>w= w(x^i)</math> (bottom panel) that produces MOND-equivalent effect per \eref{eq:dm-mond-profile1}. The displayed fit was achieved with a scale mass of <math>M = 10^11 M_\odot</math>, which is about <math>10\times</math> the mass of M33. The bottom panel also shows <math>\sqrt{g_{11}}</math> which represents the ratio between the proper distance element <math>dl/</math> and the coordinate element <math>dx^1</math>.<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:M33-rotation-curve.jpgFile:M33-rotation-curve.jpg2018-10-12T19:22:03Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:M33-strain.jpgFile:M33-strain.jpg2018-10-12T19:21:38Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:18:17Z<p>Tenev: /* Inherent Structure Hypothesis */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis ==<br />
<br />
[[image:dark-matter-effect.jpg|thumb|320px|<br />
Inherent Structure Hypothesis: The Dark Matter Effect is the gravity of conventional matter amplified by the inherent curvature of space.<br />
]]<br />
<br />
We propose that the gravitational anomalies currently attributed to dark matter<br />
<ref name=DeSwart2017><br />
DeSwrat, J.; Bertone, G.; Van Dongen, J, "How dark matter came to matter", Nature Astronomy, Vol 1., 2017<br />
</ref><br />
(which we call herein "Dark Matter Effect") can be alternatively explained by the affects of the inherent structure of spapce, which we call the "Inherent Structure Hypothesis" (ISH). The ISH is the idea that physical space has inherent structure, such as inherent curvature, that exists apart from matter and leads to modified gravity effects. While the ISH refers to inherent structure in general, herein we focus specifically on inherent curvature being one of its quantifiable attributes.<br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Dark-matter-effect.jpgFile:Dark-matter-effect.jpg2018-10-12T19:17:10Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:08:15Z<p>Tenev: /* Analogies */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T19:07:02Z<p>Tenev: /* Analogies */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
{| class="wikitable" style="width:70%"<br />
|+ Comparison between the General Relativity and Solid Mechanics Perspectives<br />
|-<br />
!General Relativity Perspective !! Solid Mechanics Perspective <br />
|-<br />
|Physical space || Mid-hypersurface of a hyperplate called "cosmic fabric".<br />
|-<br />
|Spacetime || The world volume of the cosmic fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of physical space || Intrinsic curvature of the fabric's mid-hypersurface<br />
|-<br />
|Intrinsic curvature of spacetime || Intrinsic curvature of the fabric's world volume<br />
|-<br />
|Gravitational potential <math>\Phi</math> || Volumetric strain <math>\varepsilon</math>, such that <math>\varepsilon = -\Phi/c^2</math><br />
|-<br />
|Gravitational waves || Shear waves traveling at the speed of light<br />
|-<br />
|Matter curves spacetime || Matter induces prescribed strain causing the fabric to bend<br />
|-<br />
|Action integral in free space, <math> \mathcal{S} = \frac{1}{2\kappa} \int R\sqrt{|g|}\,d^4x </math>|| Action integral outside of inclusions, <math> \mathcal{S} = \frac{L^2 Y}{24} \int R\sqrt{|g|}\,d^4x </math><br />
|-<br />
|Constants of Nature: <math> G,\;\hbar,\;c </math> || Elastic constants: <math> Y = 6c^7/ 2\pi\hbar G^2,\quad \nu=1 </math><br />
|-<br />
|}<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:55:33Z<p>Tenev: /* Postulate: Lapse Rate */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Allison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:54:41Z<p>Tenev: /* Formulation */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
[[image:cosmic-fabric-postulate-one.jpg|thumb|320px|<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate (''cosmic fabric'') that is thin along its fourth spatial dimension.<br />
The hyperplate is immersed in a four dimensional absolute space wherein it is free to bend.<br />
]]<br />
<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Inclusions ====<br />
<br />
[[image:cosmic-fabric-postulate-two.jpg|thumb|320px|<br />
A plate bending from a flat (a) into a curved (b) geometry to accommodate the strain prescribed by an inclusion.<br />
]]<br />
<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
<br clear=all><br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
[[image:cosmic-fabric-postulate-three.jpg|thumb|320px|<br />
Matter-matter interaction is mediated by signals propagating in the fabric, whose speed depends on the fabric’s mechanical properties. Clock ticks and hence proper time <math>\tau</math> slows down relative to absolute time <math>t</math> per: <math>d\tau/dt = (1+\varepsilon)^{-1} </math><br />
]]<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in Alison ''et. al.''<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
<br clear=all/><br />
<br />
=== Analogies ===<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Cosmic-fabric-postulate-three.jpgFile:Cosmic-fabric-postulate-three.jpg2018-10-12T18:53:26Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Cosmic-fabric-postulate-two.jpgFile:Cosmic-fabric-postulate-two.jpg2018-10-12T18:49:04Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Cosmic-fabric-postulate-one.jpgFile:Cosmic-fabric-postulate-one.jpg2018-10-12T18:47:10Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:41:19Z<p>Tenev: /* Cosmic Fabric Model of space */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
=== Formulation ===<br />
<br />
The Cosmic Fabric Model of space is formulated as follows: We postulate the cosmic fabric to be (1) an elastic thin hyperplate, with (2) matter-energy fields as inclusions, and (3) lapse rate of proper time proportional to the shear wave speed <math>v_s</math>. Each of these postulates is described and motivated in the sections below.<br />
<br />
<br />
==== Postulate: Elastic Thin Hyperplate ====<br />
<br />
Cosmic space is identified with the mid-hypersurface of a hyperplate called the Cosmic Fabric that is thin along the fourth spatial dimension. We imagine the fabric as foliated into 3D hypersurfaces each of which is isotropic and elastic, and each is subject to Hooke's Law (see the figure). Thus, Hooke's Law together with concepts such as stress, strain and the Poisson effect apply as conventionally understood in Solid Mechanics, because they pertain to individual hypersurfaces, which are 3D bodies.<br />
<br />
Because of its correspondence to physical space, the intrinsic curvature, <math>R^\text{3D}</math>, of the fabric's mid-hypersurface corresponds to that of three-dimensional (3D) space. Likewise, the intrinsic curvature <math>R</math> of the fabric's world volume, corresponds to that of four-dimensional (4D) spacetime. The term "world volume" refers to the four-dimensional shape traced out by an object in spacetime as it advances in time.<br />
<br />
The small transverse thickness of the fabric is needed to create resistance to bending, but once such resistance is accounted for, we treat the fabric as essentially a 3D hypersurface that bends within the 4D reference hyperspace. The thickness must be very small so that the fabric can behave as an essentially 3D object at ordinary length scales and be an appropriate analogy of 3D physical space. The thickness itself defines a microscopic length scale at which the behavior of the physical world would have to differ significantly from our ordinary experience. A value equal or comparable to Planck's length <math>l_p</math> meets this criteria. However, the exact value of the thickness is not essential to the model as long as it is small but not vanishingly so.<br />
<br />
==== Postulate: Inclusions ====<br />
<br />
Matter-energy fields behave as inclusions in the fabric inducing \emph{membrane} strains leading to transverse displacements and hence bending (\fref{fig-plate-bending}). The following equation postulates that matter is a source of volumetric strain,<br />
<br />
:<math><br />
\varepsilon_{,kk} = -\frac{1}{2} c^2\kappa \rho<br />
</math><br />
<br />
where <math>\varepsilon_{,kk} \equiv \nabla^2\varepsilon</math> is the Laplacian of the volumetric strain, <math>c</math> is the speed of light, <math>\kappa</math> is the Einstein constant, and <math>\rho</math> is the density of matter-energy. The term "membrane" strain (or stress) refers to strains (or stresses) that change in-plane but are uniform across the thickness of the fabric as opposed to bending strains (or stresses) that switch sign through the thickness across the mid-hypersurface.<br />
<br />
The mass content of matter, rather than its spatial extent, is what causes the displacement of fabric material. In the context of General Relativity, mass can be related to geometry through its Schwarzchild radius. Thus, one meter of mass is the amount of mass whose Schwarzchild radius is two meters. In the same way, the geometric significance of a matter-energy field, represented by the right hand side of the above equation, can be understood as the Schwarzchild radius density and <math>c^2\kappa</math> as a units conversion factor. In other words, the above equation postulates that the Schwarzchild radius density of a matter-energy field is a source of volumetric strain in the cosmic fabric.<br />
<br />
The analogy between a body in empty space and an inclusion in the cosmic fabric raises the question of how such an inclusion can move freely through a stiff fabric in the same way as a body can move through empty space. The wave nature of matter, at the length-scale of the body's elementary particles suggests the answer. Just like waves can propagate through a very stiff material, in the same way, elementary particles, which have wave nature, could propagate through the fabric. A detailed treatment of the matter-fabric interaction requires extending our model to include a theory about the nature of matter, which is beyond the scope of this work. Instead, the details of the underlying matter-fabric interactions are abstracted and only the effect is considered, namely, that matter inclusions prescribe a strain field on the fabric. This strain field is then treated as the input to our model. Representing matter as a strain field within the fabric allows us to aggregate the effects of individual elementary particles over large length scales, and treat planets and stars as individual inclusions.<br />
<br />
==== Postulate: Lapse Rate ====<br />
<br />
The Lapse Rate postulate relates the flow of proper time to the geometry of the cosmic fabric. All matter-matter interactions are mediated by signals propagating in the fabric as shear waves. Therefore, the rate of such interactions varies proportionally to the shear wave speed. A clock placed where fabric waves propagate slower would tick proportionally slower compared to a clock placed where fabric waves propagate faster. Such effect is independent of the clock's design, because the speed of fabric waves affects all matter-matter interactions. In other words, the lapse rate at each point in the fabric, that is how fast clocks tick, is proportional to the speed of shear waves propagating in the fabric when measured in relation to the reference space.<br />
<br />
Notice that the shear wave speed will appear to have remained constant when measured by an observer within the fabric, because the reduction in lapse rate exactly compensates for the reduction in shear wave speed. This perceived invariance of the shear wave speed is analogous to the speed of light invariance in General Relativity. <br />
<br />
Stated quantitatively, we postulate that the shear wave speed <math>v_s</math> depends on the fabric's volumetric strain <math>\varepsilon</math> as follows,<br />
<br />
:<math><br />
v_s = (1 + \varepsilon)^{-1}c<br />
</math><br />
<br />
Consequently, the lapse rate, that is the relationship between proper time <math>\tau</math> and coordinate time <math>t</math>, is as follows, <br />
<br />
:<math><br />
\frac{d \tau}{d t} = (1 + \varepsilon)^{-1}<br />
</math><br />
<br />
We motivate the above postulate by connecting the shear wave speed <math>v_s</math> to the mechanical properties of the cosmic fabric. <br />
A well known result from Solid Mechanics is that <math>v_s = \sqrt{\mu/\rho}</math> where <math>\mu</math> and <math>\rho</math> are, respectively, the shear modulus and density of the material. When such material is stretched, its density decreases by a factor of <math>(1+\varepsilon)</math> because the same amount of material now occupies <math>(1+\varepsilon)</math> times more volume. The elastic modulus also changes when the fabric is stretched, but its relationship to strain depends on the internal structure of the material. The choice of modulus-strain relationship becomes a parameter in our model that controls the effect of time dilation. By fixing this relationship to be such that,<br />
<br />
:<math><br />
\mu = (1+\varepsilon)^{-3}\mu_0<br />
</math><br />
<br />
where <math>\mu_0</math> is the reference modulus of the undeformed fabric, we can recover the equation for <math>v_s</math> above. One reason why the modulus changes is that the internal structure of the material weakens under stretch. As discussed in<br />
<ref name=Allison2012><br />
Allison, P.; Horstemeyer, M; Brown H., "Modulus dependence on large scale porosity of powder metallurgy steel", Journal of Materials Engineering and Performance, vol 21, 2012 <br />
</ref>, there are materials which exhibit modulus-strain relationship similar to the one in the above equation.<br />
<br />
=== Analogies ===<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:28:03Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties, and that <br />
# Having material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:07:50Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: <br />
<br />
# Physical space (that is cosmic space) exhibits material-like properties and, that <br />
<br />
# Paving material nature, physical space also has inherent structure at multiple length scales, which affects its behavior. <br />
<br />
The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:05:20Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: 1) physical space (that is cosmic space) exhibits material-like properties and, that 2) having material nature, it also has inherent structure at multiple length scales, which affects its behavior. The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
Different governing equations characterize different length scales. Information bridges, indicated as arrows, convey aggregate information from one length scale to another in the form of state variables. These state variables figure as parameters into each of the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math>. For example, <math>\mathcal{L}</math> is parameterized by the Ricci scalar <math>R</math>, while <math>\mathcal{L}_\text{M}</math> by the energy and momentum of matter. The term <math>\bar{\mathcal{L}}</math> is parameterized by the inherent curvature of space. Note that inherent curvature can exist at multiple length scales, such as (3) and (4) as indicated in the figure. Information about inherent curvature that propagates to a higher length scale can be treated in aggregate as a kind of texture. At the Continuum Scale, we expect the effects of larger scale structures to be negligible, but at the Structure and Cosmic scales these would become significant and manifest as, for example, "dark matter" effect or as the organization of matter into walls, filaments, and sheets, which are too large to be explainable by conventional gravity.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T18:01:14Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: 1) physical space (that is cosmic space) exhibits material-like properties and, that 2) having material nature, it also has inherent structure at multiple length scales, which affects its behavior. The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science", Hoboken, NJ, USA, 2012<br />
</ref><br />
<ref name=Horstemeyer2018><br />
Horstemeyer, M.F., "Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies", Wiley, 2018<br />
</ref><br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T17:54:28Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: 1) physical space (that is cosmic space) exhibits material-like properties and, that 2) having material nature, it also has inherent structure at multiple length scales, which affects its behavior. The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term <math>\bar{\mathcal{L}}</math> in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
The notion of inherent structure must be clarified in the context of the notion of length scale, because there can be diverse kinds of structures depending on the length scale. By "length scale" we understand a specific range of distances for which certain physical parameters and laws dominate, while others are of lesser significance. For our purpose, we consider the following four length-scales: substructure (<math>10^{-36} - 10^{-10}\text{m}</math>), continuum (<math>10^{-10} - 10^{14} \text{m}</math>) , structure (<math>10^{14}-3\times 10^{24}\text{m} </math>), and cosmic (<math>3 \times 10^{24} - 10^{27} \text{m} </math>) length-scales (see \fref{fig-lengthscales}). The specific ranges are indicated for the sake of concreteness, but are not intended to be precise. By analogy, the substructure length scale in a conventional material corresponds to the discrete entities comprising the material. Chapters \ref{ch:relativity} and \ref{ch:metric} discuss briefly the ramification of physical space having substructure. A more extensive treatment is a subject of subatomic physics and is beyond the scope of this dissertation work. At continuum length scale, as the name suggests, physical space is treated as a differentiable manifold. General Relativity is strictly a continuum scale theory, and at this length scale, the Cosmic Fabric model (see \chref{ch:model}) yields equivalent results with it. The structure length scale in a conventional material describes the components of which a mechanical system is built, such as the trusses in a bridge, for example. The behavior of these components depends not only on the continuum properties of their material but also on their shape. Our investigation of the Inherent Structure Hypothesis focuses on this length scale, where we have supposed that the space medium forms certain structures whose intrinsic curvatures can be measured and which in fact manifest as the effects currently attributed to dark matter. Finally, the cosmic length scale pertains to the global geometry of the cosmos. To use an analogy: the relationship between the global geometry of the cosmos versus the geometry at its structure length scale is like the relationship between the Earth's global geometry, which is approximately spherical, versus that of the local terrain at various regions on the Earth's surface.<br />
<br />
The Solid Mechanics perspective interprets the Lagrangian components <math>\mathcal{L}</math>, <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{M}</math> from the above equation as due to features of the cosmic fabric at different length scales. The figure illustrates a breakdown of these features that is analogous to similar breakdowns used in the multi-scale modeling of materials<br />
<ref name=Horstemeyer2012><br />
</ref><br />
<ref name=Horstemeyer2018><br />
</ref><br />
<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T17:46:47Z<p>Tenev: /* Overview */</p>
<hr />
<div>= Overview =<br />
<br />
We propose that: 1) physical space (that is cosmic space) exhibits material-like properties and, that 2) having material nature, it also has inherent structure at multiple length scales, which affects its behavior. The first part of the proposition underlies what we call the "Cosmic Fabric Model" of gravity. <ref name=Tenev2018>Tenev, T. G., Horstemeyer, M. F., "Mechanics of spacetime — A Solid Mechanics perspective on the theory of General Relativity", International Journal of Modern Physics D, Vol. 27 (2018)</ref> <ref name=Tenev2018a>Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804</ref><br />
The second part of the proposition, which we call the "Inherent Structure Hypothesis," offers a new approach for solving conundrums of modern cosmology, such as explaining phenomena currently attributed to dark matter. We also show how the Cosmic Fabric model of gravity makes the computational tools of modern mechanics applicable to studying the inherent structure of cosmic space, which introduced the Cosmic Fabric Model of gravity and began to illustrate its application to studying the inherent structure of space. <br />
<br />
The Cosmic Fabric model<ref name=Tenev2018/> is a formal analogy between General Relativity (GR) and Solid Mechanics (SM) that interprets physical space as a solid body and the field equations of GR as the bending equations governing the dynamics of said body. The vacuum of three-dimensional space is identified with a solid hyperplate called "cosmic fabric" that is embedded in four-dimensional hyperspace and has a small thickness along the fourth hyperspatial dimension. The fabric deforms elastically due to matter inclusions, such that its intrinsic curvature corresponds to that of space, while its volumetric strain to the reduction in the rate of time lapse, which in the case of weak gravity, is the same as the gravitational potential.<br />
<br />
[[image:astro-length-scales.jpg|thumb|420px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br />
<br />
In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term $\bar{\mathcal{L}}$ in the following modification to the Einstein-Hilbert action integral,<br />
<br />
:<math><br />
\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega<br />
</math><br />
<br />
where the integral is taken over all of spacetime <math>\Omega</math> and <math>d\Omega \equiv \sqrt{|g|}dx^4</math> represents the proper volume element of spacetime with <math>g</math> being its metric and <math>dx^4</math> the coordinate volume element. The various <math>\mathcal{L}</math> terms are Lagrangian densities, where <math>\mathcal{L}</math> is due to the curvature of spacetime, <math>\bar{\mathcal{L}}</math> is a correction to <math>\mathcal{L}</math> due to the inherent (or undeformed) curvature of space, and <math>\mathcal{L}_\text{M}</math> accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action <math>\mathcal{S}</math> with respect to the metric tensor <math>g</math>. Form GR's perspective, <math>\mathcal{L} = R/(2c\kappa)</math>, where <math>R</math> is the Ricci curvature scalar, <math>\kappa \equiv 8\pi G/c^4</math> is the Einstein constant, <math>c</math> is the speed of light and <math>G</math> is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, <math>\mathcal{L} = RL^2\mu/(24c)</math>, where <math>L</math> and <math>\mu</math> are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of <math>\mathcal{L}_\text{M}</math> is one and the same within both GR's and SM's paradigms. The term <math>\bar{\mathcal{L}}</math>, which represents the curvature of spacetime that is not due to matter-energy fields, has not been considered until now, but the need for it becomes apparent in the context of the material analogy. The unstrained cosmic fabric need not be flat, but could, for example, have global curvature and local relief. <br />
<br />
<br />
<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Astro-length-scales.jpgFile:Astro-length-scales.jpg2018-10-12T17:28:02Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T17:25:51Z<p>Tenev: </p>
<hr />
<div>= Overview =<br />
<br />
[[image:astro-length-scales.gif|thumb|320px|<br />
Cosmic hierarchical length scales and the information bridges between them. The field equations of General Relativity and, analogously, the constitutive equations of the cosmic fabric dominate continuum length scale (2). The effects of dark matter are directly observed at the structure length scale (3). The structures at length scales (3) and (4) contribute the <math>\bar{\mathcal{L}}</math> terms to the action equation, while length scale (1) contributes the <math>\mathcal{L}_\text{M}</math> term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms <math>\bar{\mathcal{L}}</math> and <math>\mathcal{L}_\text{F}</math> within the Einstein-Hilbert action. <br />
]]<br />
<br />
<br clear=all/><br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= Research =<br />
<br />
== Cosmic Fabric Model of space ==<br />
<br />
<br />
<br />
== Inherent Structure Hypothesis == <br />
<br />
== Numerical Simulations of Inherent Curvature ==<br />
<br />
<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astronomical_Scale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Astro-length-scales.gifFile:Astro-length-scales.gif2018-10-12T17:24:52Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Astronomical_ScaleCategory:Astronomical Scale2018-10-12T17:12:39Z<p>Tenev: Created page with "= Overview = = Tutorials = * Preliminaries to Studying Non-Euclidean Geometry = References = <references /> Category: Astroscale"</p>
<hr />
<div>= Overview =<br />
<br />
= Tutorials = <br />
<br />
* [[Preliminaries to Studying Non-Euclidean Geometry]]<br />
<br />
= References =<br />
<references /><br />
<br />
[[Category: Astroscale]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T22:51:20Z<p>Tenev: /* Fitting Procedure */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized. The environment temperature and strain rate were not specified, on the diagrams, so they were assumed to be 293K and 0.001 1/s respectively.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Material and environmental parameters for Mg AZ61 alloy<br />
|-<br />
! Constant<br />
! Value<br />
|--<br />
| Temperature<br />
| 293K<br />
|--<br />
| Strain rate<br />
| 0.001 1/s<br />
|--<br />
| Young's modulus<br />
| 44.8 GPa<br />
|--<br />
| Poisson ratio<br />
| 0.35<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence, and also helped the algorithm converge faster.<br />
<br />
The following parameters were fitted manually in the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. The results are detailed in the section below.<br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Estimated constants and their values<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100 MPa<br />
| Rate independent yield stress<br />
|-<br />
| C5<br />
| 1e-05 1/sec<br />
| Transition strain rate from rate independent to dependent yield<br />
|-<br />
| C7 1/MPa<br />
| 9.16489<br />
| Kinematic recovery constant<br />
|-<br />
| C9<br />
| 19485.5 MPa<br />
| Kinematic anisotropic hardening modulus<br />
|-<br />
| C13<br />
| 282.43 1/(MPa sec)<br />
| Isotropic dynamic recovery<br />
|-<br />
| C15<br />
| 109944 MPa<br />
| Isotropic hardening modulus<br />
|-<br />
| Cb<br />
| 1.49397<br />
| Tension/compression asymmetry constant<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T22:49:29Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized. The environment temperature and strain rate were not specified, on the diagrams, so they were assumed to be 293K and 0.001 1/s respectively.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Material and environmental parameters for Mg AZ61 alloy<br />
|-<br />
! Constant<br />
! Value<br />
|--<br />
| Temperature<br />
| 293K<br />
|--<br />
| Strain rate<br />
| 0.001 1/s<br />
|--<br />
| Young's modulus<br />
| 44.8 GPa<br />
|--<br />
| Poisson ratio<br />
| 0.35<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Estimated constants and their values<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100 MPa<br />
| Rate independent yield stress<br />
|-<br />
| C5<br />
| 1e-05 1/sec<br />
| Transition strain rate from rate independent to dependent yield<br />
|-<br />
| C7 1/MPa<br />
| 9.16489<br />
| Kinematic recovery constant<br />
|-<br />
| C9<br />
| 19485.5 MPa<br />
| Kinematic anisotropic hardening modulus<br />
|-<br />
| C13<br />
| 282.43 1/(MPa sec)<br />
| Isotropic dynamic recovery<br />
|-<br />
| C15<br />
| 109944 MPa<br />
| Isotropic hardening modulus<br />
|-<br />
| Cb<br />
| 1.49397<br />
| Tension/compression asymmetry constant<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T22:48:56Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized. The environment temperature and strain rate were not specified, on the diagrams, so they were assumed to be 293K and 0.001 1/s respectively.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Material and environmental parameters for Mg AZ61 alloy<br />
|-<br />
! Constant<br />
! Value<br />
|--<br />
| Temperature<br />
| 293K<br />
|--<br />
| Strain rate<br />
| 0.001 1/s<br />
|--<br />
| Young's modulus<br />
| 44.8 GPa<br />
|--<br />
| Poisson ratio<br />
| 0.35<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Fitted constants and their values<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100 MPa<br />
| Rate independent yield stress<br />
|-<br />
| C5<br />
| 1e-05 1/sec<br />
| Transition strain rate from rate independent to dependent yield<br />
|-<br />
| C7 1/MPa<br />
| 9.16489<br />
| Kinematic recovery constant<br />
|-<br />
| C9<br />
| 19485.5 MPa<br />
| Kinematic anisotropic hardening modulus<br />
|-<br />
| C13<br />
| 282.43 1/(MPa sec)<br />
| Isotropic dynamic recovery<br />
|-<br />
| C15<br />
| 109944 MPa<br />
| Isotropic hardening modulus<br />
|-<br />
| Cb<br />
| 1.49397<br />
| Tension/compression asymmetry constant<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T22:43:54Z<p>Tenev: </p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized. The environment temperature and strain rate were not specified, on the diagrams, so they were assumed to be 293K and 0.001 1/s respectively.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Material and environmental parameters for Mg AZ61 alloy<br />
|-<br />
! Constant<br />
! Value<br />
|--<br />
| Temperature<br />
| 293K<br />
|--<br />
| Strain rate<br />
| 0.001 1/s<br />
|--<br />
| Young's modulus<br />
| 44.8 GPa<br />
|--<br />
| Poisson ratio<br />
| 0.35<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Fitted constants and their values<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100<br />
|<br />
|-<br />
| C5<br />
| 1e-05<br />
|<br />
|-<br />
| C7<br />
| 9.16489<br />
|<br />
|-<br />
| C9<br />
| 19485.5<br />
|<br />
|-<br />
| C13<br />
| 282.43<br />
|<br />
|-<br />
| C15<br />
| 109944<br />
|<br />
|-<br />
| Cb<br />
| 1.49397<br />
|<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T18:59:40Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|+ Fitted constants and their values<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100<br />
|<br />
|-<br />
| C5<br />
| 1e-05<br />
|<br />
|-<br />
| C7<br />
| 9.16489<br />
|<br />
|-<br />
| C9<br />
| 19485.5<br />
|<br />
|-<br />
| C13<br />
| 282.43<br />
|<br />
|-<br />
| C15<br />
| 109944<br />
|<br />
|-<br />
| Cb<br />
| 1.49397<br />
|<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T18:58:19Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br />
{| class="wikitable" style="width:480"<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100<br />
|<br />
|-<br />
| C5<br />
| 1e-05<br />
|<br />
|-<br />
| C7<br />
| 9.16489<br />
|<br />
|-<br />
| C9<br />
| 19485.5<br />
|<br />
|-<br />
| C13<br />
| 282.43<br />
|<br />
|-<br />
| C15<br />
| 109944<br />
|<br />
|-<br />
| Cb<br />
| 1.49397<br />
|<br />
|}<br />
<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T18:57:51Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
<br />
{| class="wikitable" style="width:300"<br />
|-<br />
! Constant<br />
! Value<br />
! Meaning<br />
|-<br />
| C3<br />
| 100<br />
|<br />
|-<br />
| C5<br />
| 1e-05<br />
|<br />
|-<br />
| C7<br />
| 9.16489<br />
|<br />
|-<br />
| C9<br />
| 19485.5<br />
|<br />
|-<br />
| C13<br />
| 282.43<br />
|<br />
|-<br />
| C15<br />
| 109944<br />
|<br />
|-<br />
| Cb<br />
| 1.49397<br />
|<br />
|}<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T18:51:35Z<p>Tenev: /* Results */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
Screenshot of the DMGfit UI after performing the fitting operations listed in [[AZ61A#Fitting_Procedure]]<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-25T18:49:45Z<p>Tenev: /* Fitting Procedure */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
# Fit C3, which is the yield stress.<br />
# Fit C9, C7<br />
# Fit C15, C13<br />
# Fit Cb <br />
# Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T18:28:49Z<p>Tenev: /* Fitting Procedure */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1]] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
* Fit C3, which is the yield stress.<br />
* Fit C9, C7<br />
* Fit C15, C13<br />
* Fit Cb <br />
* Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T18:28:36Z<p>Tenev: /* Fitting Procedure */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
We followed the general outline of the fitting procedure detailed on the [[DMGfit_55p_v1p1] page with some modifications to accommodate for the types of available data. For example, temperature dependence constants were not fitted because we did not have temperature dependent data. <br />
<br />
First we preformed a rough manual fit, by adjusting several of the Cn constants as detailed below, followed by an automatic optimization for the same set of constants. The initial manual fit helped the goal seeking algorithm move in the right direction toward convergence and converge faster.<br />
<br />
The following parameters were fitted manually the order listed below:<br />
<br />
* Fit C3, which is the yield stress.<br />
* Fit C9, C7<br />
* Fit C15, C13<br />
* Fit Cb <br />
* Readjust C9, C7, C15 and C13 after fitting Cb.<br />
<br />
Finally, we performed an optimization procedure. To do this we unchecked the parameters mentioned above, while leaving the rest of the parameters checked and hit the 'optimize' button. <br />
<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:AZ61A-Mg-Tension.txtFile:AZ61A-Mg-Tension.txt2016-04-22T18:02:09Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:AZ61A-Mg-Compresion.txtFile:AZ61A-Mg-Compresion.txt2016-04-22T18:01:55Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T18:00:49Z<p>Tenev: /* Source Data */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
Digitized data files:<br />
<br />
* [[File:AZ61A-Mg-Compresion.txt]]<br />
* [[File:AZ61A-Mg-Tension.txt]]<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
'' Explain the fitting procedure and rationalization ''<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:58:41Z<p>Tenev: </p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below.<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
<br />
<br />
<br />
For the source we digitized Mg stress-strain curves. <br />
<br />
Digitized data:<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
'' Explain the fitting procedure and rationalization ''<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:AZ61A_Mg_Tension.pngFile:AZ61A Mg Tension.png2016-04-22T17:57:31Z<p>Tenev: </p>
<hr />
<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:AZ61A_Mg_Compression.pngFile:AZ61A Mg Compression.png2016-04-22T17:57:04Z<p>Tenev: </p>
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<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:56:02Z<p>Tenev: /* Source Data */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
<br><br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
<br />
<br />
<br />
For the source we digitized Mg stress-strain curves. <br />
<br />
Digitized data:<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
'' Explain the fitting procedure and rationalization ''<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:54:26Z<p>Tenev: /* Source Data */</p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br/><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"/><br />
<br />
</br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
<br />
<br />
<br />
For the source we digitized Mg stress-strain curves. <br />
<br />
Digitized data:<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
'' Explain the fitting procedure and rationalization ''<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:53:59Z<p>Tenev: </p>
<hr />
<div>The damage model parameters for Magnesium AZ61A alloy were fitted using [[DMGfit_55p_v1p1]]. The original sources, the fitting procedure and results are detailed below<br />
<br />
= Source Data =<br />
<br />
The source data was based on the digitization of AZ61A stress-strain curves. A compression and a tension stress-strain curves, respectively, were digitized.<br />
<br />
[[image:AZ61A Mg Compression.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging compressive stress-strain curve.<br />
<ref name="Mg-design1957"><br />
"Magnesium Design", Dow Chemical Co., 1957. As published in ''Aerospace Structural Metals Handbook'', vol 3, Code 3603, CINDAS/USAF CRDA Handbooks Operation, Purdue University, 1995, p4<br />
</ref><br />
<br />
<br/><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
[[image:AZ61A Mg Tension.png|thumb|320px|<br />
Mg.009 AZ61A magnesium alloy forging tensile stress-strain curve.<br />
<ref name="Mg-design1957"><br />
<br />
</br><br />
Composition: Mg-6Al-1Zn. UNS M11610<br />
]]<br />
<br />
<br />
<br />
<br />
For the source we digitized Mg stress-strain curves. <br />
<br />
Digitized data:<br />
<br />
<br clear=all/><br />
<br />
= Fitting Procedure =<br />
<br />
'' Explain the fitting procedure and rationalization ''<br />
<br />
<br clear=all/><br />
<br />
= Results =<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|<br />
caption goes here<br />
]]<br />
<br />
<br clear=all/><br />
<br />
= References =<br />
<br />
<references /><br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/File:AZ61A-Fit.pngFile:AZ61A-Fit.png2016-04-22T17:27:24Z<p>Tenev: </p>
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<div></div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:27:05Z<p>Tenev: </p>
<hr />
<div>''TODO: Info about AZ61A goes here''<br />
<br />
[[image:AZ61A-Fit.png|thumb|320px|]]<br />
<br />
[[Category:Magnesium]]</div>Tenevhttps://icme.hpc.msstate.edu/mediawiki/index.php/AZ61AAZ61A2016-04-22T17:26:46Z<p>Tenev: Created page with "''TODO: Info about AZ61A goes here'' 320px| Category:Magnesium"</p>
<hr />
<div>''TODO: Info about AZ61A goes here''<br />
<br />
[[[[image:AZ61A-Fit.png|thumb|320px|]]<br />
<br />
[[Category:Magnesium]]</div>Tenev