Category:Astronomical Scale

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= Overview =
 
= Overview =
  
[[image:astro-length-scales.gif|thumb|320px|
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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>
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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.
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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.
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[[image:astro-length-scales.jpg|thumb|420px|
 
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.  
 
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.  
 
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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,
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\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega
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</math>
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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.
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Revision as of 12:46, 12 October 2018

Contents

Overview

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. [1] [2] 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.

The Cosmic Fabric model[1] 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.

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 \bar{\mathcal{L}} terms to the action equation, while length scale (1) contributes the \mathcal{L}_\text{M} term. The effects of structure at length scales below and above the continuum length scale, are accounted for by the Lagrangian terms \bar{\mathcal{L}} and \mathcal{L}_\text{F} within the Einstein-Hilbert action.


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,


\mathcal{S} = \int_{\Omega}\left(\mathcal{L} - \bar{\mathcal{L}} + \mathcal{L}_\text{M} \right) d\Omega

where the integral is taken over all of spacetime \Omega and d\Omega \equiv \sqrt{|g|}dx^4 represents the proper volume element of spacetime with g being its metric and dx^4 the coordinate volume element. The various \mathcal{L} terms are Lagrangian densities, where \mathcal{L} is due to the curvature of spacetime, \bar{\mathcal{L}} is a correction to \mathcal{L} due to the inherent (or undeformed) curvature of space, and \mathcal{L}_\text{M} accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action \mathcal{S} with respect to the metric tensor g. Form GR's perspective, \mathcal{L} = R/(2c\kappa), where R is the Ricci curvature scalar, \kappa \equiv 8\pi G/c^4 is the Einstein constant, c is the speed of light and G is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, \mathcal{L} = RL^2\mu/(24c), where L and \mu are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of \mathcal{L}_\text{M} is one and the same within both GR's and SM's paradigms. The term \bar{\mathcal{L}}, 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.




Tutorials

Research

Cosmic Fabric Model of space

Inherent Structure Hypothesis

Numerical Simulations of Inherent Curvature

References

  1. 1.0 1.1 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)
  2. Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804

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