Foundations of Physics manuscript No.
(will be inserted by the editor)
The Mechanics of Spacetime – A Solid Mechanics
Perspective on the Theory of General Relativity
T G Tenev · M F Horstemeyer
arXiv:1603.07655v6 [gr-qc] 1 Jun 2017
Received: date / Accepted: date
Abstract We present an elastic constitutive model of gravity where we identify
physical space with the mid-hypersurface of an elastic hyperplate called the “cosmic fabric” and spacetime with the fabric’s world volume. Using a Lagrangian
formulation, we show that the fabric’s behavior as derived from Hooke’s Law is
analogous to that of spacetime per the Field Equations of General Relativity. The
study is conducted in the limit of small strains, or analogously, in the limit of
weak and nearly static gravitational fields. The Fabric’s Lagrangian outside of inclusions is shown to have the same form as the Einstein-Hilbert Lagrangian for
free space. Properties of the fabric such as strain, stress, vibrations, and elastic
moduli are related to properties of gravity and space, such as the gravitational potential, gravitational acceleration, gravitational waves, and the energy density of
free space. By introducing a mechanical analogy of General Relativity, we enable
the application of Solid Mechanics tools to address problems in Cosmology.
Keywords modified gravity · constitutive model · spacetime · cosmic fabric
PACS 04.50.Kd · 46.90.+s
Mathematics Subject Classification (2000) 83D05 · 74L99
1 Introduction
In 1678, Robert Hooke, a contemporary of Isaac Newton, published what later
became known as Hooke’s Law [13]. In 1827, Cauchy [2] advanced Hooke’s Law by
defining the tensorial formulation of stress. For an isotropic linear elastic material,
T G Tenev
Mississippi State University, Starkville, MS 39759, USA
E-mail: [email protected]
M F Horstemeyer
Mississippi State University, Starkville, MS 39759, USA
E-mail: [email protected]
2
T G Tenev, M F Horstemeyer
Hooke’s Law states in tensorial form that,
ν
Y
ij kl
ik jl
kl
g g +g g
εij
σ =
1 + ν 1 − 2ν
(1.1)
where σ kl , εij , and g ij are the stress, strain, and the metric tensors, respectively,
Y is Young’s elastic modulus, and ν is the Poisson’s ratio. Latin indexes, i, j, k, l =
1 . . . 3, run over the three spatial dimensions, and Einstein summation convention
is employed.
In 1916 Einstein published the field equations of General Relativity [8], which
can be written as,
1
1
Tµν =
Rµν − Rgµν
(1.2)
κ
2
where Tµν , Rµν , and gµν are the stress-energy tensor, Ricci curvature tensor, and
µ
spacetime metric tensor, respectively; R ≡ Rµ
is the Ricci scalar, κ ≡ 8πG/c4
is the Einstein constant as c and G are the speed of light and gravitational constant, respectively. Greek indexes, µ, ν = 0 . . . 3, run over the four dimensions of
spacetime with the 0th dimension representing time. For the purposes of this paper, we have omitted the Cosmological Constant, which is sometimes included in
Equation (1.2), because its value is negligible for length-scales below the size of
the observable universe.
Einstein’s Gravitational Law (1.2) suggests a material-like constitutive relation, similar to Hooke’s Law (1.1), because it relates stress, on the left-hand side,
to deformation on the right-hand side. At first glance, the similarity appears imperfect because the right-hand sides differ in dimensionality: whereas the strain
term, εij , is dimensionless, the curvature terms, Rµν and R, have dimensions of
Length−2 . However, this problem is resolved once bending deformation is considered instead of purely longitudinal deformation (stretch or contraction). In the
equations for bending, stress is proportional to the second spatial derivative of
strain.
In this paper, we develop a formal analogy between Solid Mechanics and General Relativity by identifying physical space with the mid-hypersurface of a four
dimensional hyperplate, called the “cosmic fabric,” which has a small thickness
along a fourth spatial dimension and exhibits a constitutive stress-strain behavior. Matter-energy fields act as inclusions within the fabric causing it to expand
longitudinally and consequently to bend. The effect, illustrated on Fig. 1, is analogous to the result from General Relativity in which matter causes space to bend
resulting in gravity.
We conduct our study in the limit of weak and nearly static gravitational
fields, and demonstrate that outside of inclusions, the fabric’s action SF , assumes
the form of the Einstein-Hilbert action SEH ,
Z p
Z p
YL
1
4
SF =
R |g| dx
vs. SEH =
R |g| dx4
(1.3)
48
2κ
where L is the reference thickness of the fabric, g ≡ det gµν , and the integral is
taken over a large enough volume of spacetime sufficient to ensure convergence.
The action integral of any physical system fully determines its dynamics, because
the system’s equations of motion can be derived from the variation of the action
with respect to the metric. Therefore, once we recognize SF as analogous to SEH ,
The Mechanics of Spacetime
3
Fig. 1 A plate bending from flat geometry (a) into a curved geometry (b) because of an
inclusion that prescribes uneven strain field, as indicated by the concentric dashed lines and
the diverging arrows. The strain is larger near the center and tapers off with the distance from
it. For the geometry of the plate to accommodate the prescribed strain, the plate must bend
into the transverse dimension.
we can interpret various attributes of the cosmic fabric, such as its shape, strain,
vibrations, and elastic moduli as analogous to properties of gravity and space,
such as curvature, gravitational potential, gravitational waves, and the zero point
energy density of space.
In some aspects, our approach resembles the Arnowitt-Deser-Misner (ADM)
formulation of gravity. Under the ADM approach, spacetime is foliated into spacelike hypersurfaces related to each other via shift and lapse functions. ADM has
been used in computational models of gravity, because it reformulates the equations of General Relativity (1.2) as an initial value problem (IVP). Likewise, the
Cosmic Fabric model discussed here could also be used in the future to construct
an IVP. Nevertheless, the Cosmic Fabric model differs from the ADM formulation
in that it associates constitutive behavior with the geometric description of gravity
and derives its governing equations from a material-like constitutive relation.
Since as early as Isaac Newton [16] there have been various scientific theories about an all pervasive cosmic medium, also known as “ether,” through which
light and matter-matter interactions propagate. These theories culminated with
the Lorentz Ether Theory (LET) [20, 21], which postulated length contractions
and time dilations for objects moving through ether [10, 20] in order to explain the
negative outcome of the Michelson and Morleys ether detection experiment [23].
Although the theory of Special Relativity (SR) [7] appeared to obviate the need
for an ether, in reality SR and LET are mathematically equivalent and experimentally indistinguishable from one another. After the development of General Relativity [8], which attributed measurable intrinsic curvature to spacetime, Einstein
conceded [9] that some notion of an ether must remain. More recently, theoretical
predictions from Quantum Field Theory [30] include zero-point energy density of
space, which further supports the material view of space.
After Einstein’s publication of General Relativity [8], a number of researchers
have investigated the relationship between Mechanics notions and General Relativity. One category of publications dealt with generalizing the equations of Solid
Mechanics to account for relativistic effects. Synge [31] formulated a constitutive relationship in relativistic settings. Rayner [28] extended Hooke’s Law to a
relativistic context. Maugin [22] generalized the special relativistic continuum mechanics theory developed by Grot and Eringen [12] to a general relativistic context.
More recently, Kijowski and Magli [18] presented the relativistic elasticity theory as
a gauge theory. A detailed review of relativistic elasticity can be found in Karlovini
and Samuelsson [17].
4
T G Tenev, M F Horstemeyer
Another category of publications interprets General Relativity in Solid Mechanical terms. Kondo [19] mentions an analogy between the variation formalism
of his theory of global plasticity and General Relativity. Gerlach and Scott [11]
introduce a “metric elasticity” tensor in addition to the elasticity of matter itself
and “stresses due to geometry.” However, these stress and strain terms are not
a constitutive model of gravity, because they are not expected to apply in the
absence of ordinary matter. Tartaglia [32] attempted to describe spacetime as a
four-dimensional elastic medium in which one of the spatial dimensions has been
converted into a time dimension by assuming a uniaxial strain. However, many
of the ideas in Tartaglia’s paper appear to be incomplete. Antoci and Mihich [3]
explored the physical meaning of the straightforward formal extension of Hooke’s
Law to spacetime but did not consider the possibility, which is explored in this
paper, that Einstein’s Gravitational Law may be related to Hooke’s Law. Beau [5]
pushed the material analogy further by interpreting the cosmological constant Λ
as related to a kind of a spacetime bulk modulus, but the analogy is to a fluid-like
material and not a solid. A set of recent publications, for which Rangamani [27]
presents a literature review, explore the applicability of the Navier-Stokes equations of Fluid Dynamics to gravity. While a fluid analogy is useful for some applications, it does not account for shear waves in space, such as gravity waves, because
fluids are only capable of propagating pressure waves and not shear waves. In contrast to the prior literature recounted above, the work presented here begins with
the premise that space exhibits material-like behavior subject to a constitutive
relationship.
The Cosmic Fabric model of gravity allows General Relativity problems to be
formulated as Solid Mechanics problems, solved within the Solid Mechanics domain, and the solution interpreted back in General Relativity terms. The reverse
is also true. Thus, ideas, methodologies and tools from each field become available
to the other field. Over the past century, Solid Mechanics and General Relativity
have advanced independently from each other with few researchers having expertise in both. Consequently, significant terminology and focus gaps exist between
these two fields, which obscure their underlying physical similarities. Our research
attempts to bridge these gaps.
The remainder of this paper is organized as follows: In Section 2 we develop the
Solid Mechanics analogy of gravity by specifying a material body whose behavior,
determined solely based on Hooke’s Law (1.1), is demonstrably analogous to the
behavior of spacetime. In Section 3 we discuss the implications of the resulting
model, and summarize and conclude in Section 4.
2 Formulation of the Cosmic Fabric Model of Gravity
Consider a four dimensional hyperplate, called here the “cosmic fabric,” which is
thin in the fourth spatial dimension, x4 . We show that, for a suitably chosen initial
geometry and Poisson’s ratio
p of the fabric, its Lagrangian density outside of inclusions is LF = (Y L/48)R |g|, where LF is the integrand in Equation (1.3). This
result enables us to subsequently analyze how the remaining kinematic properties
of the cosmic fabric correspond to properties of gravity.
For the remainder of this paper, we will use the following notation and conventions: Lower case Latin indexes, i, j, k, l = 1 . . . 3 run over the three ordinary spatial
The Mechanics of Spacetime
5
Fig. 2 Multi-axial stress state (a), and a uniaxial deformation of an object (b) from the
transparent to the opaque shape. Each component σij represents the stress through the ith
surface in the j th direction. The Poisson’s ratio ν measures the effect of the longitudinal stress
along the ith direction on the longitudinal strain along the j th direction, for j 6= i. In the case
of uniaxial stress state, εjj = (−ν/Y )σii = −νεii .
dimensions. Upper case Latin indexes, I, J, K, L = 1 . . . 4 run over the four hyperspace dimensions, while Greek indexes, µ, ν, α = 0 . . . 3 run over the four spacetime
dimensions, where indexes 0 and 4 represent, respectively, the time dimension and
the extra spatial dimensions. Indexes appearing after a comma represent differentiation with respect to the indexed dimension. For example, ui,j ≡ ∂ui /∂xj .
For spacetime, we adopt the space-like metric signature (−, +, +, +) and denote
the flat metric tensor as ηµν , where [ηµν ] ≡ diag[−1, 1, 1, 1]. Furthermore, we will
use geometric units where the gravitational constant and speed of light are set to
unity: G = 1 and c = 1, in which case Einstein’s constant κ = 8π.
2.1 Deformation Basics
Let xI be material coordinates assigned to the Cosmic Fabric, and let gIJ be the
metric tensor of the fabric. Material coordinates are those that remain attached to
the fabric during deformation and displace along with it. The metric tensor defines
how coordinate differences relate to distances. Thus, the distance ds between two
nearby material points is given by,
ds2 = gIJ dxI dxJ
(2.1)
The distance ds between the same two material points prior to deformation is
given by,
ds2 = g IJ dxI dxJ
(2.2)
where g IJ is the metric tensor of the undeformed fabric.
The strain tensor εIJ describes the amount of relative length change during
deformation. By definition, εIJ is such that,
2εIJ dxI dxJ = ds2 − ds2 = (gIJ − g IJ )dxI dxJ
1
∴ εIJ = (gIJ − g IJ )
2
(2.3)
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T G Tenev, M F Horstemeyer
The tensors in the above equations can be related to familiar Solid Mechanics
tensors. If we used Cartesian coordinates for the undeformed configuration, then
the xI would be the so called “reference coordinates.” The elements of the undeformed metric would be g IJ = δIJ , where δIJ is the Kronecker delta while the
elements of the deformed metric would be gIJ = CIJ , where CIJ is the CauchyGreen deformation tensor. Consequently, the definition of the strain tensor εIJ ,
per the above equation, would correspond to the definition of the Green-Saint
Venant strain tensor from Solid Mechanics, EIJ = (CIJ − δIJ )/2.
The Young’s elastic modulus Y , which figures in Hooke’s Law (1.1), is the
amount of longitudinal stress (force per unit of cross section area) σii in the ith
direction needed to produce a unit amount of longitudinal strain εii in the same
direction under a uniaxial stress condition (no summation intended over the index
i). The effect of longitudinal stress along a given orientation on the longitudinal
strains in the transverse orientations is known as the Poisson effect and is measured
by the Poisson’s ratio ν (see Fig. 2).
2.2 Postulates
We postulate the cosmic fabric to be a (1) thin hyperplate, (2) exhibiting isotropic
hyperelastic constitutive behavior, with (3) matter-energy fields as inclusions, and
(4) advancing in time at a rate inversely proportional to its thickness. Each of
these postulates is described and motivated in the sections below.
2.2.1 Hyperplate
Cosmic space is identified with the mid-hypersurface of a hyperplate called the
Cosmic Fabric that is thin along the fourth spatial dimension. Because of its
correspondence to cosmic space, the intrinsic curvature, R3D , of the fabric’s midhypersurface corresponds to that of three-dimensional (3D) space. Likewise, the
intrinsic curvature R of the fabric’s world volume, corresponds to that of fourdimensional (4D) spacetime. The term “world volume” refers to the four-dimensional
shape traced out by an object in spacetime as it advances in time.
From a Solid Mechanics perspective, the fabric must have thickness so it can
resist bending. At the same time, 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 to or comparable to
Planck’s length lp 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.
We imagine the cosmic fabric as immersed in a 4D hyperspace and able to
bend along the x4 dimension. The 4D hyperspace outside of the cosmic fabric is not
directly observable, but the bending of the fabric into x4 can be observed indirectly
by measuring the intrinsic curvature R3D of the fabric’s mid-hypersurface.
F
GR
Let gIJ
be the fabric’s metric and let gµν
be the spacetime metric of General
Relativity. We can identify the 3D spatial components of the two metrics with
each other as follows,
F
GR
gij
= gij
≡ gij
(2.4)
The Mechanics of Spacetime
7
so from here on, we can drop the superscripts F and GR , and understand from the
context and from the use of subscripts which metric tensor components are being
referenced.
2.2.2 Isotropicity and Hyperelasticity
The cosmic fabric exhibits an isotropic hyperelastic constitutive behavior subject
to Hooke’s Law (1.1) and the Poisson effect (see Fig. 2b). The meaning of stress,
strain, and the Poisson effect is extended from 3D to 4D as follows. First, we
observe that both strain and Poisson effect are geometric in nature and naturally
extend to the 4D hyperspace. Next, we define stress σIJ to be the thermodynamic
conjugate of the strain εIJ ,
1 ∂U
σ IJ = p
|g| ∂εIJ
(2.5)
where U is the elastic energy density, which is an intensive scalar quantity and
thus also readily generalizable to 4D. Note the use of script symbols, such as
p U,
to denote
tensor
or
scalar
densities.
A
density
quantity
includes
the
factor
|g|
p
3D
3D
(or |g | for 3D) as part of its definition, where g (or g ) is the determinant
of the metric. This factor represents the ratio between the coordinate volume
element d4 x, which depends on the choice of coordinates, and the proper volume
element, which is invariant under coordinate transformations. The 3D components
σij coincide with the conventional definition of stress as a traction force per unit
of cross-section area, while the additional components σI0 = σ0I are stress-like
quantities.
2.2.3 Inclusions
Matter-energy fields behave as inclusions in the fabric inducing membrane strains
leading to transverse displacements and hence bending (Fig. 1). Thus, matter is a
source of volumetric strain, so:
(
ε3D
,kk
ε3D
,kk
>0
=0
within a matter-energy inclusion
outside of inclusions
(2.6)
2 3D
where ε3D ≡ εii is the volumetric strain, and ε3D
is its Laplacian.
,kk ≡ ∇ ε
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 midhypersurface. Note that it is not the spatial extent of matter that causes it to
displace the fabric material, but rather it is the mass content of matter. In Section 3.5 we attempt to quantify more precisely the effect of inclusions on the
fabric.
8
T G Tenev, M F Horstemeyer
2.2.4 Lapse Rate
The lapse rate of proper time is postulated to be the inverse of the fabric’s transverse stretch. Therefore, in orthonormal coordinates as we have adopted here,
dτ
= (−g00 )1/2 = (g44 )−1/2
dx0
∴ g00 = −(g44 )−1
(2.7)
where τ designates proper time. Under the weak field condition, which is analogous
to a small strain condition, the metric tensors can be approximated as,
gIJ = δIJ + 2εIJ ,
|εIJ | 1
gµν = ηµν + 2εµν ,
|εµν | 1
(2.8)
where we have identified εij with the spatial components of a gravitational gauge
as per [26, Ch.18], and we have also applied the notation to the space-time components 2ε0µ and 2εi0 of the gauge. Thus, we have fixed the gauge to be 2εµν . This
gauge has the physical meaning of being the fabric’s material strain and for that
reason will be used throughout the paper. Note that, except under special conditions, this gauge does not necessarily comply with the harmonic gauge condition,
which is often used in Linearized Gravity.
Under weak field conditions, the postulated equation (2.7) can be approximated
as follows,
dτ
= −1 + 2ε00 = −(1 + 2ε44 )−1 ≈ −1 + 2ε44
dx0
∴ε00 ≈ ε44
(2.9)
2.3 Weak and Nearly Static Fields Condition
To keep the math tractable, we conduct our study under the assumption of weak
and nearly static fields. We believe that this assumption is not fundamental to the
model, and that it could be relaxed or removed in the future. As will be shown
in Section 3.1, the fabric strain is analogous to the gravitational potential, so the
weak field condition, which is the subject of Linearized Gravity (see [26, Ch. 18])
is analogous to the small strain condition, which is the subject of Solid Mechanic’s
Infinitesimal Strain Theory.
We consider a gravitational potential Φ to be weak if Φ/c2 1. By this
definition, most gravitational fields that we experience on an everyday basis are
weak. For example, the values for Φ/c2 at the Earth’s surface due to the gravitational fields of the Earth, Sun, and Milky way are 6.7 × 10−10 , 1.0 × 10−8 , and
1.4 × 10−6 [33], respectively. As such, we consider these gravity fields to be weak.
Except in regards to gravity waves (Section 3.3), we will assume nearly static
(or slow moving) fields in addition to weak fields. A field is considered slow moving
if its velocity v satisfies, v 2 /c2 1, which is the case for most gravitational or
strain fields that we experience. The nearly static field assumption means that
differentiation with respect to time results in negligibly small values.
The Mechanics of Spacetime
9
Fig. 3 The cosmic fabric is treated as a stack of three-dimensional hypersurfaces Σξ each
parameterized by ξ ≡ x4 = const. The reference thickness L ≡ ∆ξ is the difference between the
ξ coordinates of the two faces of the fabric. The actual thickness is N L, where the contraction
N = N (xi ) varies from point to point xi along the mid-hypersurface Σ0 .
2.4 Bending Energy Density
Rather than treating the fabric as a 4D hyperplate, it is convenient to approximate
it as a 3D hypersurface. This can be done once we have averaged the fabric’s elastic
energy density across its thickness and assigned it to its mid-hypersurface. At that
point, we can use the fabric’s mid-hypersurface as a proxy instead of the fabric in
future calculations.
To compute UB , we adapt the work of Efrati et al. [6] concerning the bending
of conventional thin plates. For ease of notation, let ξ ≡ x4 denote the coordinate offset from the mid-hypersurface of the fabric. The fabric, having a reference
thickness L, is regarded as foliated into infinitely many hypersurfaces Σξ each
parameterized by ξ = const. (Fig. 3). We carry over the simplifying assumption from Kirchoff-Love thin plate theory [1] to thin hyperplates and assume that
the set of material points along any given hypersurface that were along a normal
prior to bending remain along the normal after bending. Consequently, the fabric’s
four-dimensional metric gIJ can be expressed in terms of the three-dimensional
hypersurface metrics gij = gij (ξ) as follows,
gIJ
2
N
0
≈
0 gij
(2.10)
where gij = gij (ξ) is the metric of a given hypersurface Σξ that is offset by ξ from
the mid-hypersurface, and N is the fabric’s contraction along the x4 dimension
(Fig. 3).
It can be shown [6] that the metric gij of each Σξ takes the form,
gij = aij − 2N bij ξ + N 2 cij ξ 2
(2.11)
where aij = aij (xi ) and bij = bij (xi ) are, respectively, the first and second fundamental forms of the mid-hypersurface, and cij = akl bik bjl . In the same way, let
g IJ represent the undeformed metric of the fabric, and g ij (ξ) be the undeformed
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T G Tenev, M F Horstemeyer
metric of each hypersurface Σξ .
g µν ≈
2
N
0
0 g ij
where
(2.12)
2
g ij = aij − 2N bij ξ + N cij ξ
2
The total elastic energy density of any solid is half of the inner product of
its stress and strain tensors. Applying Hooke’s Law (1.1), the total elastic energy
density Uξ of each hypersurface Σξ is given by,
Uξ =
1 ij
σ εij
2
q
C ijkl
such that
q
1 ijkl
C
εij εkl |g 3D |
2
Y
ν
ij kl
ik jl
≡
g g +g g
1 + ν 1 − 2ν
|g 3D | =
(2.13)
where σij = σij (ξ) and εij = p
εij (ξ) are,p
respectively, the stress and strain at each
hypersurface Σξ . The factor |g 3D | ≡ det gij converts coordinate unit volume
into proper volume, which is required for Uξ to be a density.
Next, we compute the total elastic energy density U averaged across the fabric’s
thickness, and we separate it into a bending term UB and a membrane stretch term
UM . For this purpose, we split the strain at each surface, εij into a membrane strain
B
εM
ij and a bending strain εij as follows:
1
B
(gij − g ij ) = εM
ij + εij
2
1
= (aij − aij )
2
= − N bij − N bij ξ + O(ξ 2 )
εij =
εM
ij
εB
ij
U=
1
L
L
2
Z
1
=
2L
(2.14)
Uξ N dξ
−L
2
Z
L
2
−L
2
M
B B
M B
B M
C ijkl (εM
ij εkl + εij εkl + [εij εkl + εij εkl ]) dξ
= UM + UB
Z L
2
1
M
C ijkl εM
UM =
ij εkl dξ
2L − L
2
Z L
2
1
B
UB =
C ijkl εB
ij εkl dξ
2L − L
(2.15)
2
The mixed terms inside the square brackets in Equation (2.15) vanish under integration because the bending strain reverses sign across the mid-hypersurface;
B
M
M
hence εB
ij = εij (ξ) is an odd function, while εij = εij (ξ) is an even function.
For the remainder of this subsection, we focus on evaluating the term UB . The
term UM will be addressed in the following subsection where we show that it vanishes under appropriately chosen material properties and deformation kinematics.
The Mechanics of Spacetime
11
Evaluating UB from Equation (2.15), we obtain,
q
UB = N bij − N bij N bkl − N bkl N |g 3D | + O(L3 )
(2.16)
3D
3D
Using the identity, Rlijk
= bik bjl − bij bkl , where Rlijk
is the Riemann curvature
3
tensor of the mid-hypersurface, and setting O(L ) = 0, we can express UB in terms
of the intrinsic three-dimensional spatial curvature R3D as follows,
q
L2 Y
1−ν i k
UB = −
N 2 R3D +
bi bk N |g 3D |
24(1 + ν)
1 − 2ν
(2.17)
q
2
L ijkl
2
3D
+
C
−N N bij bkl − N bij N bkl + N bij bkl N |g |
24
The undeformed geometry and Poisson’s ratio of the cosmic fabric had remained unspecified as freedoms to be fixed at a later time such as now. Requiring
that the cosmic fabric’s mid-hypersurface has a flat undeformed geometry, bij = 0,
causes the last term in Equation (2.17) to vanish. The bii bkk term would also vanish
if we chose Poisson’s ratio ν = 1. In this case, the bending energy becomes the
following,
q
Y L2 3 3D
UB = −
N R
|g 3D |
(2.18)
48
subject to the conditions,
bij = 0,
and
ν=1
(2.19)
2.5 Membrane Energy Density
We now show that for any given small-strain deformed configuration, we can identify a material displacement field that results in no membrane energy. Consequently, we conclude that the bending energy UB is the only contribution to the
total elastic energy of the fabric for the case of slow moving fields. Since General
Relativity (GR) is only concerned with the curvature of the deformed body, in
developing the material analogy of GR we have freedom to prescribe a specific
material displacement field for the deformation.
Let us consider a displacement field where each point of the mid-hypersurface,
x4 = 0, of the fabric is displaced by the amount w = w(xi ) along a geodesic normal
to the hypersurface. It should be evident that using such a displacement field, one
can deform a flat body into any given shape that represents a small deviation from
flatness and does not contain folds. Flamm’s paraboloid, which is used to illustrate
the geometry of space around a static gravitating body, is an example of such a
shape. Let y I be the coordinate of the position to which the material point at
xi was displaced. Thus, y i = xi and y 4 = w. The metric tensor of the deformed
hypersurface can be computed from the dot product of the position differentials
as follows,
K K
gij = y,i
y,j = xk,i xk,j + w,i w,j = δij + w,i w,j
1
1
∴ εij = (gij − δij ) = w,i w,j
2
2
(2.20)
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T G Tenev, M F Horstemeyer
Using the formula for elastic energy density, UM = σ kl εkl /2 and applying Hooke’s
Law (1.1) to Equation (2.20) with ν = 1, we find,
UM ∝ σ kl εkl ∝ (g ik g jl − g ij g kl )εij εkl =
= εkj εjk − εjj εkk ∝ w,k w,j w,j w,k − w,j w,j w,k w,k = 0
(2.21)
∴ UM = 0
Fixing the fabric’s deformation to material displacements only along the hypersurface normals is a valid approximation under the assumption of nearly static
fields. In such cases, the reason for the deformation would have been to geometrically accommodate inclusions by bending into the x4 dimension. Once bending
has taken place, the material points of the fabric can shift within the plane of
the fabric to minimize its membrane energy without affecting the geometrical constraints imposed by the inclusion. For nearly static situations, we have shown that
the membrane energy can be minimized to where it vanishes. In such cases, the
net displacement would have taken the form described in this subsection.
2.6 Lagrangian Density
Ignoring the kinetic energy component, under the simplifying assumption of slow
moving fields, the Lagrangian density is LF = −UB ∝ R3D . We now derive an
expression for LF in terms of the Ricci curvature R of the fabric’s world volume.
Let us adopt the conditions (2.19), and choose orthonormal coordinates, such
that g ij = δij , where δij is the Kronecker delta. Then, gij = δij + 2εij , and so 2εij
plays the role of a gauge for the purposes of the linearized curvature equations per
[26, Ch. 18]. We extend this gauge to spacetime by defining additional strain-like
components ε0µ and εk0 such that gµν = ηµν +2εµν , where gµν is the metric of the
fabric’s world volume, and ηµν is the Minkowski metric. Using the gauge-invariant
linearized expression for R,
R = 2 −εµµ,αα + εαµ,αµ
= 2 −εii,kk + εik,ik − ε00,kk − εkk,00 + 2ε0k,0k
(2.22)
≈ R3D + 2ε00,kk
In the last step of the above derivation, we have recognized that the purely spatial
terms add up to the gauge-independent linearized expression for R3D . Furthermore, the terms differentiated with respect to x0 are negligible because of the slow
moving fields assumption, and lowering or raising a single 0 index, which is done
using ηµν , changes the sign of the term.
Next, we show that ε00,kk = 0 in free space. Because the fabric is very thin outside of inclusions, there are only in-plane stresses and hence plane-stress conditions
apply. Therefore, from Hooke’s Law (1.1) and considering that ν = 1,
(
Y ε3D = (1 − 2ν)σ 3D
Y ε44 = −νσ 3D
(2.23)
ν
3D
3D
∴ ε44 = −
ε =ε
1 − 2ν
The Mechanics of Spacetime
13
where σ 3D ≡ σkk and ε3D ≡ εkk are the 3D volumetric stress and strain, respectively,
and Y is the Young’s modulus of elasticity. Combining the above result with the
Inclusions and Lapse Rate postulates of the Cosmic Fabric Model, we conclude
that,
ε00 ≈ ε44 = ε3D
∴ ε00,kk = ε3D
,kk = 0
(2.24)
Based on the results form the previous subsections, the Lagrangian density of
the fabric for the case of weak and slow moving fields is simply,
LF = −UB =
Y L2 3 3D
N R
48
q
|g 3D |
(2.25)
Let g ≡ det gµν . For the case of slow and weak fields,
g ≈ g 3D g00 = −g 3D (g44 )−1 = −g 3D N −2
q
p
∴ |g 3D | = N |g|
(2.26)
Combining Equations (2.25), (2.26), (2.24), and (2.22) while also letting N ≈ 1,
we finally arrive at,
LF =
Y L2 p
R |g|
48
(2.27)
which has the same form as the Einstein-Hilbert Lagrangian density.
3 Discussion
In the previous section, we postulated a material body, which we named the “cosmic fabric” whose constitutive behavior outside of inclusions is analogous to the
behavior of gravity. For the analogy to be useful, it should allow us to map between
notions in Solid Mechanics and General Relativity. Such a mapping is possible on
the basis of identifying the fabric Lagrangian density LF with the Lagrangian density from the Einstein-Hilbert action, LEH , as applying to free space. Specifically,
LF =
Y L2 p
1 p
R |g| = LEH =
R |g|
48
2κ
(3.1)
where κ is the Einstein constant.
In the subsections below, we discuss the correspondence between mechanical
properties of the cosmic fabric and known properties of gravity.
14
T G Tenev, M F Horstemeyer
3.1 Fabric Strain and Gravitational Potential
A General Relativity result is that the ratio by which clocks slow down in gravitational field, also known as “time dilation,” is given by,
p
∆tΦ
= 1 + 2Φ/c2 ≈ 1 + Φ/c2
∆t0
(3.2)
where Φ is the gravitational potential, ∆t0 is a time interval measured away from
gravitational fields, while ∆tΦ is a corresponding time interval measured within a
gravitational field of potential Φ. In the coordinate system we have adopted, the
values of x0 correspond to the clock measurements away from gravitational fields,
so by the Lapse Rate postulate (2.7) and applying Equation (2.23) we conclude
that,
∆tΦ
dτ
=
∆t0
dx0
1 + Φ/c2 = 1 − ε00
(3.3)
∴ −Φ/c2 = ε00 = ε3D
Thus, the gravitational potential corresponds to the volumetric expansion of the
fabric. The above result is consistent with the Schwarzchild metric in the exterior
of a non-rotating body, because the space metric coefficients are the inverses of
the time coefficient.
3.2 Poisson’s Ratio and the Substructure of Space
Known materials with a Poisson’s ratio of ν = 1 have a fibrous substructure, which
suggests that the cosmic fabric is, in fact, a fabric! For ν = 1, the bulk modulus is
K = Y /[3(1−2ν)] < 0. A negative bulk modulus means that compressing the fabric
results in an overall increase of the material volume and vice versa. Although such
behavior is unusual for most conventional materials, there are recently discovered
compressive dilatant [29] and stretch densifying [4] materials, for which ν = 1
in either compression or tension, respectively. Compressive dilatant materials are
artificially manufactured and their substructure consists of entangled stiff wires.
Stretch densifying materials, have textile-like substructure comprised of woven
threads each consisting of twisted fibers.
3.3 Fabric Vibrations and Gravitational Waves
Having Poisson’s ratio ν = 1 also implies that there can only be transverse (shear)
waves in the fabric but no longitudinal (pressure) waves. The shear modulus µ
and the p-wave modulus M are as follows,
Y
Y
=
2(1 + ν)
4
1−ν
M =Y
=0
(1 − 2ν)(1 + ν)
µ=
(3.4)
The Mechanics of Spacetime
15
p
implying that the transverse (shear) wavepvelocity vs = µ/ρ 6= 0, while the
longitudinal (pressure) wave velocity vp = M/ρ = 0. This result shows why the
speed of light is the fastest entity of the universe, given that a longitudinal wave is
typically faster than a shear wave. For a shear wave to be the fastest, the Poisson’s
Ratio must be 1.0. This conclusion is consistent with observations, because all
known waves that propagate in free space, such as gravity or electromagnetic
waves, are transverse.
Let us consider the analogy between shear waves in the fabric and gravitational
waves. Such an analogy depends on demonstrating that the fabric’s behavior parallels that of spacetime for high velocity fields as well. We leave the rigorous proof
for a future article, and for the rest of this subsection we assume that the fabric’s
behavior implied by the Lagrangian (2.27) also holds for high velocities. Based
on this assumption, we proceed to investigate in-plane shear waves propagating
through the fabric and their correspondence to gravitational waves.
First, we show that if static fields are negligible and in the absence of torsion,
then the gravitational gauge defined by the strain 2εµν satisfies the harmonic gauge
condition, εµα,α = (1/2)εαα,µ . For shear waves, ε3D = 0, and by Equation (2.24),
e00 = 0, implying that εαα = 0, so proving the condition reduces to demonstrating
that, εµα
,α = 0. Furthermore, the shear time-space components must vanish, ε4j =
εj4 = 0 = ε0j = εj0 , because we are assuming negligible static fields and in-plane
shear waves. Therefore, in order to prove that the harmonic gauge condition holds,
we just need to show that εik,k = 0. Let ui be the material displacement field. In
terms of the displacement field, the strain is 2εij = ui,j + uj,i , and so,
2εij = 2uj,i + [ui,j − uj,i ]
2εik,k = 2uk,ki + [ui,k − uk,i ],k
(3.5)
But, uk,ki = 0 since εkk = uk,k = 0. The difference in the square brackets corresponds to material torsion and must vanish too, thus concluding,
eik,k = 0
(3.6)
Since εµν satisfies the harmonic gauge condition, we can apply the linearized
α
approximation for the Ricci tensor, Rµν ≈ −εµν,α
. After substituting into the
Einstein Field Equations (1.2), and taking into account that R ≈ εαα,µµ = 0, and
that in empty space Tµν = 0, we arrive at,
α
εµν,α
= εij,kk − εij,00 = 0
∴εij,00 = εij,kk
(3.7)
which is a wave equation with solutions that are traveling waves at the speed of
light c. To see this clearly, let us re-write Equation (3.7) in terms of the time
variable t, where x0 ≡ ct, and using the canonical form derivative operators ∂,
and ∇,
1 ∂2
εij = ∇2 εij
(3.8)
c2 ∂t2
The above equation can be related to the Solid Mechanics equation for the
propagation of a shear wave in elastic medium with density ρ and shear modulus
µ. In the absence of body forces, the equation of motion is the following,
ρ
∂2
ui = σij,j
∂t2
(3.9)
16
T G Tenev, M F Horstemeyer
Applying Hooke’s Law (1.1) and recognizing that, εij = (ui,j + uj,i )/2, µ =
Y /[2(ν + 1)], and uk,k = εkk = 0, we arrive at,
ρ
ρ
∂2
ui = µ∇2 ui
∂t2
∂2
(ui,j + uj,i ) = µ∇2 (ui,j + uj,i )
∂t2
∂2
∴ ρ 2 εij = µ∇2 εij
∂t
(3.10)
The parallel between Equations (3.8) and (3.10) confirms that gravitational waves
are analogous to shear waves propagating in a solid material and that furthermore
the speed of propagation, which is the speed of light c, is related to the shear
modulus and density of the medium per c2 = µ/ρ.
Although Equation (3.7) suggests that there are ostensibly ten strain components, εαβ , oscillating independently, in reality only two are independent and the
rest are coupled to the two. To show this, consider a traveling wave, which corresponds to a gravity wave, propagating along the x3 direction. Then ε3α = εα3 = 0
for the wave to be a shear wave. Furthermore, as shown earlier, ε00 = ε3D = 0
and εj0 = ε0j = 0. Finally, we have ε3D = ε11 + ε22 = 0, because ε33 = 0 already.
Therefore,
ε11 = −ε22
ε12 = ε21
(3.11)
are the only two independent degrees of freedom left implying just two types of
wave polarizations. The fact that we are using the material strain, which has concrete physical meaning, ensures that the waves must also be physical as opposed to
being mere coordinate displacements. This result, derived from a Solid Mechanic’s
perspective, is consistent with the analogous result from General Relativity about
the polarization of gravitational waves [26, Ch. 35].
3.4 Elastic Modulus and Density of Free Space
From the result in Equation (3.1), the fabric’s elastic modulus Y could be computed given an estimate
p for the fabric’s thickness L. As reasoned in Section 2.2.1,
Planck’s length lp ≡ ~G/c3 is a suitable estimate for L, where ~ is the reduced
Planck’s constant. So, assuming L ∼ lp , we can estimate Y to be,
Y ∼
24
= 4.4 × 10113 N m−2
lp2 κ
(3.12)
The density of the fabric ρ is related to the wave speed and shear modulus, as
shown in Section 3.3, and can now be computed,
ρ=
Y
µ
= 2 ∼ 1.3 × 1096 kg m−3
c2
4c
(3.13)
In accordance with the Cosmic Fabric analogy, the density of the fabric corresponds to the density of free space, which is also known as the zero-point energy
The Mechanics of Spacetime
17
density. The computed value for ρ agrees to an order of magnitude with the predictions of Quantum Field Theory (∼ 1096 kg m−3 ) for the energy density of free
space [30]. Note that the predictions of Quantum Field Theory are also based on
using Planck’s length lp as a length-scale parameter.
3.5 Generalizing the Cosmic Fabric Model
The Cosmic Fabric analogy to cosmic space was demonstrated subject to certain
simplifying conditions such as small strain (weak gravity), nearly static equilibrium
(slow fields) and outside of inclusions (free space). In this subsection, we consider
how each of these conditions might be removed or relaxed in a future generalization
of the model. We also sketch how we could quantify the effect of inclusions on the
fabric, and investigate the significance of non-flat unstrained geometry.
The small strain (weak gravity) condition was imposed so we could use the
linearized equations for strain and, analogously, the linearized equations for gravity. To relax this condition, we will need to account for the higher order terms in
the equations of strain and also use covariant derivatives instead of conventional
differentiation for all field variables. Some materials exhibit nonlinear elastic behavior under large strains. By requiring that the fabric behaves analogously to
cosmic space at large strains (strong gravity), we would determine whether the
fabric is one such material. Nonlinear elastic behavior is related to the substructure of the material, and discovering such behavior would suggest clues about the
substructure of cosmic space.
Imposing the nearly static (slow fields) condition allowed us to ignore the
kinetic energy term in the fabric’s Lagrangian. It also let us assume specific bending
kinematics that minimizes the membrane energy of the fabric and, in Section 2.5,
we showed that such kinematics result in zero membrane energy. Without the
condition of nearly static equilibrium (slow fields), we will need to take into account
the kinetic and membrane energies of the fabric and also consider a more complex
deformation state. One possible simplification would be to concentrate on the
condition without bending (away from static gravitational fields), and derive a
closed-form result for the fabric’s Lagrangian. This is the condition under which
we would study gravitational waves as discussed in Section 3.3. The mathematical
complexities resulting in the general non-static (fast fields) case will probably
require that it be studied using numerical techniques.
The reason we focused on deriving the fabric’s Lagrangian outside of inclusions
was so that we could eliminate the ε00,kk term in Equation (2.22), which we showed
equaled the Laplacian of the volumetric strain per Equation (2.24). The latter
had to vanish because of the Inclusions Postulate (Section 2.2.3). Consequently
we could conclude that R ≈ R3D and show that the Fabric Lagrangian outside
of inclusions (2.27) takes a form analogous to the Einstein-Hilbert Lagrangian for
free space (3.1). It maybe possible to show that in the case of inclusions, the term
ε00,kk in Equation (2.22) yields an additional Lagrangian term that is analogous to
the energy-matter
Lagrangian
term LM in the generalized Einstein-Hilbert action
p
R
SEH = (R/2κ + LM ) |g| dx4 . We propose the idea below and leave the rigorous
derivation for future research.
18
T G Tenev, M F Horstemeyer
First, we quantify the effect of inclusions as a source of strain by restating the
Inclusions Postulate (2.6) as follows,
ε3D
,kk = O(1)κρ
(3.14)
where ρ is the matter-energy density of the inclusion, κ is the Einstein constant,
and O(1) stands for some dimensionless coefficient of order unity. 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. This is in fact the mass-geometry relationship implied when
using geometric units, where the speed of light c and the gravitational constant
G are set to unity. In the same way, the geometric significance of a matter-energy
field, represented by the right hand side of Equation (3.14), can be understood
as the Schwarzchild radius density and κ as a units conversion factor (in geometric units κ = 8π). Thus, the meaning of Equation (3.14) is to postulate that the
Schwarzchild radius density of a matter-energy field is a source of volumetric strain
in the cosmic fabric.
Next, we relate ε3D
,kk to ε00,kk via the Poisson effect and the Lapse Rate postulate. Outside of inclusions, we had shown in Equations (2.23) and (2.24) that
ε3D
,kk = ε00,kk , but with an inclusion we will have to account for the transverse
stress, so in general,
ε00,kk = O(1)ε3D
,kk = O(1)κρ
(3.15)
Substituting the above results into Equation (2.22) and retracing the steps in the
derivation of the fabric’s Lagrangian LF in Section 2.6, we conclude that,
p
Y L2
(R − 2O(1)κρ) |g|
48
p
κY L2 R
=
|g|
− O(1)ρ
24
2κ
LF =
(3.16)
We note that, within General Relativity, the matter
p Lagrangian density LM for
a nearly static fluid of density ρ is LM = −O(1)ρ |g| [25]. Therefore, by identifying Y L2 /24 = 1/κ as was suggested already by Equation (3.1) and choosing
appropriately the various O(1) factors mentioned above, we can identify the full
fabric Lagrangian LF that accounts for inclusions with the full Einstein-Hilbert
Lagrangian which in turn accounts for matter energy fields as follows,
LF = LEH = LB + LM , where
Rp
LB =
|g| is due to bending
2κ
p
LM = −O(1)ρ |g| is due to matter-energy fields
(3.17)
Assuming that the above result can be demonstrated rigorously, it would imply
that matter-energy is analogous to the strain energy associated with introducing
an inclusion into the cosmic fabric.
The Mechanics of Spacetime
19
3.6 Possible Explanation for Dark Energy and Dark Matter
An initial analysis suggests that if we accounted for non-flat unstrained geometry
of the fabric (inherent curvature of cosmic space not due to matter-energy fields),
then we may be able to explain the observational results that are currently attributed to dark energy and dark matter. For a non-flat unstrained geometry, the
last term in Equation (2.17) yields an additional Lagrangian density term, LB ,
that must be included in the fabric’s Lagrangian density (3.16) as follows,
LF = LB + LB + L M
(3.18)
By the Cosmic Fabric analogy, the LB term is due to the inherent curvature of
cosmic space. Suppose that cosmic space had a nearly constant inherent global
curvature of radius comparable to the Hubble radius, that is c/H0 (about 13.7
billion light years), where H0 is the Hubble parameter. We may be able to show
that such curvature can explain the presence of a Cosmological Constant as well
as phenomena currently attributed to dark matter. The main distinction between
these two explanations depends on whether the effect of the global curvature is
considered away from or near matter-energy fields. Below is a brief sketch of the
idea, whose full development is left for future research.
Away from inclusions (matter-energy fields), LB would remain constant and,
upon variation of the fabric’s action, would yield a Cosmological Constant Λ term
as part of the resulting field equations, which is associated with dark energy.
The value of Λ would be such that Λ−1/2 ∼ c/H0 , which is consistent with the
observationally determined value of the known Cosmological Constant.
Near inclusions (matter-energy fields), the inherent curvature of the fabric
would interact with the bending caused by the matter-energy field as represented
by the mixed bij and bij terms in Equation (2.17). The overall effect would be
to boost the gravitational acceleration due to matter, which is observationally
equivalent to the presence of fictitious invisible mass, also known as dark matter.
The boost in gravitational acceleration due to the inherent curvature of cosmic
space, would only be appreciable for exceedingly small accelerations a where a ∼
cH0 . Such conclusion would be consistent with the calculations of the Modified
Newtonian Dynamics theory [24], which introduces an acceleration scale parameter
a0 whose empirically determined value happens to be such that a0 ∼ cH0 .
4 Summary and Conclusion
In this paper, we showed that the behavior of spacetime per Einstein’s Field Equations (1.2) is analogous to that of an appropriately chosen material body, termed
the “cosmic fabric” that is governed by a simple constitutive relation per Hooke’s
Law (1.1). In Section 2, we postulated several basic properties of the fabric and
how they correspond to physical space and matter in space. Other properties, such
as the elastic moduli and undeformed geometry, were left unconstrained as model
parameters. These were fixed once we computed the Lagrangian of the fabric and
required that it be identified with the Einstein-Hilbert Lagrangian of General Relativity. After the Cosmic Fabric model was calibrated in this way, in Section 3,
it was applied to interpret various properties of gravity in terms of the fabric’s
mechanics and vice versa. To a great extent, the interpretations seemed physically
20
T G Tenev, M F Horstemeyer
Table 1 Comparison between the General Relativity and Solid Mechanics Perspectives.
General Relativity Perspective
Solid Mechanics Perspective
Physical space
Mid-hypersurface of a hyperplate called
“cosmic fabric”.
Spacetime
The world volume of the cosmic fabric’s
mid-hypersurface
Intrinsic curvature of space
Intrinsic curvature of the fabric’s midhypersurface
Intrinsic curvature of spacetime
Intrinsic curvature of the fabric’s world
volume
Gravitational potential Φ in free space
Volumetric strain ε3D outside of inclusions, such that ε3D = −Φ/c2
Gravitational waves
Shear waves traveling at the speed of light
Matter curves spacetime
Matter induces prescribed strain causing
the fabric to bend
Action integral in free space,
Z p
1
R |g| d4 x
S=
2κ
Action integral outside of inclusions,
Constants of Nature:
Elastic constants:
G, ~, c
S=
L2 Y
24
Z
R
p
|g| d4 x
Y = 6c7 /2π~G2 ,
ν=1
meaningful from both perspectives of General Relativity and Solid Mechanics. Table 1 summarizes the correspondence between concepts from one field to analogous
concepts in the other.
The research presented in this paper suggests an equivalence between postulating the field equations of General Relativity and postulating a cosmic fabric
having material-like properties as described here. We believe that these are two
different approaches for studying the same underlying reality. The Cosmic Fabric model introduces a new paradigm for interpreting cosmological observations
based on well-established ideas from Solid Mechanics. In recent decades, Solid
Mechanics has made significant advancements in describing the structure of materials at various length and time scales ranging from electrons to large scale
engineering structures [14, 15]. Such advances, in conjunction with the advances
in high-performance computing, have made possible the construction of multiscale
models that accurately simulate the behavior of metals, ceramics, polymers, and
biomaterials. The Cosmic Fabric model should enable the application of these
techniques to simulate both the fine and large scale structures of the cosmos, and
consequently, to address some of the outstanding problems in Cosmology, such as
those pertaining to the density of free space, dark energy, and dark matter.
Our research in developing and applying the Cosmic Fabric model is still ongoing, and the results we have shared are subject to some simplifying assumptions.
In the future, we hope to relax these assumptions if not eliminate them completely.
Nevertheless, even at the current stage of the work the results appear promising
and useful.
The Mechanics of Spacetime
21
Acknowledgements MFH would like to acknowledge the Center for Advanced Vehicular
Systems (CAVS) at Mississippi State University for support of this work. The authors would
also like to thank Russ Humphreys and Anzhong Wang for their insight with respect to Cosmology, and Shantia Yarahmadian for his feedback with respect to Mathematics.
Conflict of Interest The authors declare that they have no conflict of interest
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