# Category:Astronomical Scale

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− | In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term | + | 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, |

:<math> | :<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. | 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. | ||

+ | |||

+ | 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. | ||

+ | |||

+ | 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 | ||

+ | <ref name=Horstemeyer2012> | ||

+ | </ref> | ||

+ | <ref name=Horstemeyer2018> | ||

+ | </ref> | ||

## Revision as of 11:54, 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.

In the context of the Lagrangian formulation of gravity, the inherent structure of space figures as the additional term in the following modification to the Einstein-Hilbert action integral,

where the integral is taken over all of spacetime and represents the proper volume element of spacetime with being its metric and the coordinate volume element. The various terms are Lagrangian densities, where is due to the curvature of spacetime, is a correction to due to the inherent (or undeformed) curvature of space, and accounts for energy-matter fields. The governing differential equations of spacetime (or the cosmic fabric) can be derived by variation of the action with respect to the metric tensor . Form GR's perspective, , where is the Ricci curvature scalar, is the Einstein constant, is the speed of light and is the gravitational constant. We show in \chref{ch:model} that from SM's perspective, , where and are, respectively, the average thickness and shear modulus of the cosmic fabric. The interpretation of is one and the same within both GR's and SM's paradigms. The term , 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.

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 (), continuum () , structure (), and cosmic () 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.

The Solid Mechanics perspective interprets the Lagrangian components , and 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
^{[3]}
^{[4]}

# Tutorials

# Research

## Cosmic Fabric Model of space

## Inherent Structure Hypothesis

## Numerical Simulations of Inherent Curvature

# References

- ↑
^{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) - ↑ Tenev, T. G., Horstemeyer, M. F., "Recovering the Principle of Relativity from the Cosmic Fabric Model of Space", http://arxiv.org/abs/1808.08804
- ↑
- ↑