# Moving Manifolds and General Relativity

David V. Svintradze<EMAIL_ADDRESS>School of Medicine, New Vision
University, Bokhua 11, 0159 Tbilisi, Georgia  Niels Bohr Institute, University
of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

We revise general relativity (GR) from the perspective of calculus for moving
surfaces (CMS). While GR is intrinsically constructed in pseudo-Riemannian
geometry, a complete understanding of moving manifolds requires embedding in a
higher dimension. It can only be defined by extrinsic Gaussian differential
geometry and its extension to moving surfaces, known as CMS. Following the
recent developments in CMS, we present a new derivation for the Einstein field
equation and demonstrate the fundamental limitations of GR. Explicitly, we
show that GR is an approximation of moving manifold equations and only stands
for dominantly compressible space-time. While GR, with a cosmological
constant, predicts an expanding universe, CMS shows fluctuation between
inflation and collapse. We also show that the specific solution to GR with
cosmological constant is constant mean curvature shapes. In the end, by
presenting calculations for incompressible but deforming two-dimensional
spheres, we indicate that material points moving with constant spherical
velocities move like waves, strongly suggesting a resolution of the wave-
corpuscular dualism problem.

††preprint: APS/123-QED

Introduction– General relativity (GR) is the most successful theory of
gravitation [1, 2]. Indeed, experimental tests (to name a few) performed in
the nonlinear and dynamical regimes produced a high degree of accuracy [3, 4].
Recent extensions of GR, among many others, are embedding the brane in higher
dimensional space [5] and modifications of Einstein-Hilbert (EH) action with
quadratic terms [6]. It is noteworthy that recent experiments constrain
largely deviating from GR new theories [2], but there is an experimental
indication of the necessity of extending GR fundamentally [7].

Here, we refrain from making any adjustments in the field and, adhering to
mathematically rigorous notations, we label actions as

$\displaystyle S_{\Omega}$
$\displaystyle=\int_{\Omega}\mathscr{L}_{\Omega}d\Omega$ (1) $\displaystyle
S_{\mathbb{S}}$
$\displaystyle=\int_{\mathbb{S}}\mathscr{L}_{\mathbb{S}}d\mathbb{S}$ (2)

where $\Omega$ stands for the bounded by $\mathbb{S}$ hyper-surface space for
embedded manifold so that

$d\Omega=\sqrt{\eta}d^{n+1}X=\sqrt{\eta}dX^{1}\cdots dX^{n+1}$ (3)

$\eta=\pm\det(\eta_{ij})$ is the determinant of the space metric tensor with
the sign dependently whether space is Riemannian or pseudo-Riemannian. The
hyper-surface area element is defined as

$d\mathbb{S}=\sqrt{g}d^{n}x$ (4)

sign of $g$ depends on whether the manifold is Riemannian or pseudo-
Riemannian. Later, we omit ”hyper” and use surface or manifold for all
dimensional hyper-surfaces. We designate the space metric tensor as $\eta$ and
the surface metric as $g$. As far as the surface and enclosed space are
evolving and continuously changing, we evaluate (1, 2) actions according to
the extension of differential geometry called calculus for moving surfaces
(CMS). We do not modify fields $\mathscr{L}_{\Omega},\mathscr{L}_{\mathbb{S}}$
and, as a primary example, work with the EH action:

$\mathscr{L}_{\mathbb{S}}=\frac{1}{2k}R+\sigma$ (5)

where $k=8\pi Gc^{-4}$ Einstein gravitational constant, $G$ is the
gravitational constant, $c$ is the speed of light, $R$ is the Ricci scalar,
and $\sigma$ is some surface energy density. Quadratic or any other
modifications can be implemented at any stage due to the generality of our
formalism. Because the manifold is moving (changing by some extra parameter),
one needs an extrinsic approach to encounter the motion fully, embedding it in
a higher dimension. Therefore, both (1, 2) space and surface actions stay
relevant, though here we only concentrate on the surface action (2) to
demonstrate alternative ways of taking variations and generalizing GR. We
implement embedding in higher dimension space-time to consider the manifold’s
shape motion properly. Therefore, the presented formalism applies to all
$\mathscr{L}_{\Omega},\mathscr{L}_{\mathbb{S}}$ space and surface energy
densities. Because of independence from causality, the manifold is assumed to
be moving, the actions are

$\displaystyle S_{\Omega}$ $\displaystyle=\int_{\mathbb{S}}\frac{\rho
V^{2}}{2}d\mathbb{S}-\int_{\Omega}\mathscr{L}_{\Omega}d\Omega$ (6)
$\displaystyle S_{\mathbb{S}}$ $\displaystyle=\int_{\mathbb{S}}\frac{\rho
V^{2}}{2}d\mathbb{S}-\int_{\mathbb{S}}\mathscr{L}_{\mathbb{S}}d\mathbb{S}$ (7)

where $\rho$ is the surface mass density (initially can be zero), $V$ is the
surface velocity, and $\mathbb{S}\in\Omega$, therefore the most generic action
is (6). The action is (6) if the surface is embedded in higher dimension space
and the $\mathscr{L}_{\Omega}$ is the space energy density111energy per unit
volume, in three dimensions is pressure.. While the action is (7) if the
surface dynamics can be described by the surface energy density
$\mathscr{L}_{\mathbb{S}}\in\mathscr{L}_{\Omega}$222energy per unit area, in
three dimensions is tension. only. Following the definitions (6, 7), we
consider the variation of kinetic terms to be a variation of Lagrangian terms
and all the modeling of $\mathscr{L}_{\mathbb{S}},\mathscr{L}_{\Omega}$ serves
to capture kinetic energies of the surface motion. Since here we deal with
mostly EH action where there is no direct higher dimensional embedding, we
mostly work with $S_{\mathbb{S}}$ (7) and invoke space terms when needed
explicitly. In this formalism, we show that the surface neither expands nor
collapses for dominantly compressible space-time but oscillates non-
harmonically. Therefore, all measurements done in the expansion regime will
stay consistent with GR in some approximations.

Our motivation is simple. The Riemannian geometry is intrinsically defined as
GR. More implicitly, one can introduce a metric tensor without invoking
tangent space and construct all differential geometry based on partial
derivatives of it (see below). For instance, Christoffel’s symbols become the
combination of metric tensor partial derivatives and are renamed by Levi-
Civita connections, while the combination of partial derivatives of the
connections gives Riemann tensor. Metric and Riemann tensors are only
intrinsic fundamental tensors, and all other intrinsic tensors are given by
the combination of the two. While Riemannian geometry is the perfect language
to describe the not-moving manifold (static geometry), it cannot describe
shape motion, which is the manifold that evolves with some extra parameters.
Throughout the paper, we refer to the additional parameter as time, bearing in
mind that the extra parametric time might have nothing to do with the real or
proper time we use in physics. To distinguish the two, we designate the
parametric time as $t$ and the real/proper time as $\tau$. In some
approximations, parametric and real times can be equated $t=\tau$ and,
therefore, consequent equations simplified, though it is not our goal in this
paper. In GR, $\tau$ proper time is considered a pseudo coordinate, and
consequently, the space-time manifold becomes a four-dimensional hyper-
surface. The static manifold is considered to be moving because of the
presence of $\tau$ proper time in the metric tensor. The causality of this
assumption is that a-priory statically formulated Riemannian geometry is used
to describe equations of motions for moving manifolds. In other words, it is
”conjectured/hypothesized” that static geometry can still describe moving
manifolds. We challenge this idea here and, quite contrary, show that to
understand all possible deformations of moving manifolds fully, one needs an
extension of differential geometry called the calculus of moving surfaces
(CMS) [8]. Using CMS, we generalize GR and show that space-time fluctuates
with inflation and collapses in the dominantly compressible universe, while
the surface mass density non-harmonically, with growing amplitudes, oscillates
locally. In a predominantly incompressible universe, it is possible to form
various shapes with different patterns and varying pressure and temperature
distributions. Calculations in CMS formalism show that material points moving
with constant spherical velocities behave like waves due to surface dynamics
in two-dimensional incompressible spheres. While calculations are performed
for simple two-dimensional surface cases, the generality of the method allows
it to be readily extended to any pseudo-Riemannian geometry of any dimensions.

Calculus for Moving Surfaces (CMS) – Before proceeding with analyses, we
briefly define the extension of differential geometry called the calculus for
moving surfaces. The extension was initiated by Hadamard [9] and developed by
a generation of mathematicians [10, 11, 8].

Using the CMS in its current form, P. Grinfeld proposed equations of motions
for massive thin fluid films [8, 12, 13, 14]. We have generalized P. Grinfeld
equations for any manifolds and derived surface dynamics equations [15, 16],
solved shape-dynamics problems for water drops [17], and extended Young-
Laplace, Kelvin, and Gibbs-Thomson equations to arbitrarily curved surfaces
[18, 19], see also meeting abstracts that preceded the papers [20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31].

Gaussian Differential Geometry – Here, we revive the concept of extrinsic
differential geometry, referred to as Gaussian geometry. The approach
developed by Gauss, in contrast to Riemannian geometry, is fundamentally
extrinsic. We shall soon see that while Riemannian geometry is a powerful tool
for describing static surfaces, it is not enough to encounter moving shapes
entirely. Embedding in higher dimensions is necessary to better understand
moving manifolds. Therefore, instead of describing Riemannian geometry, we
start from the Gaussian one, bearing in mind that the formalism is also
sufficient for higher than two-dimensional surfaces. In mathematics, higher
dimensional surfaces are called hyper-surfaces. To minimize terminology, we
refer to any dimensional manifold as a surface and explicitly indicate whether
the referred surface is two-dimensional.

Let
$S^{\alpha}\in\mathbb{S}^{n},\alpha=0,1,2,...,n,X^{i}\in\mathbb{M}^{n+1},i=0,1,2...,n+1$
be the generic coordinates of the point on the hyper-surface $\mathbb{S}$
(referred as surface) embedded on one higher dimensional Minkowski space
$\mathbb{M}$ so that $\bm{R}$ is a position vector
$\bm{R}\in\mathbb{M}^{(n+1)},\bm{R}=\bm{R}(X^{i},t)=\bm{R}(S^{\alpha},t)$. In
contrast to our previous publications, greek indexes here stand for surface
and Latin ones for ambient Minkowski space with the space-like signature
$(-1,1,...,1)$, bold letters indicate vectors. Partial derivatives of the
position vector define covariant base vectors of ambient space and embedded
surface, while the scalar product of base vectors are metric tensors

$\begin{matrix}[l]\bm{X}_{i}=\partial_{i}\bm{R}&\bm{S}_{\alpha}=\partial_{\alpha}\bm{R}\\\
\eta_{ij}=\bm{X}_{i}\cdot\bm{X}_{j}&g_{\alpha\beta}=\bm{S}_{\alpha}\cdot\bm{S}_{\beta}\end{matrix}$
(8)

where $\partial_{i}=\frac{\partial}{\partial
X^{i}},\partial_{\alpha}=\frac{\partial}{\partial S^{\alpha}}$. Note here that
since in Minkowski space $\eta_{00}=-1$ according to the definition (8)
$\bm{X}_{0}^{2}=-1$ and therefore space-time is defined in complex space, that
makes the embedded surface pseudo-Riemannian manifold, therefore space and
surface metrics are defined in complex space so that
$g^{2}=gg^{*}=-g,\eta^{2}=\eta\eta^{*}=-\eta$. Inverse to space $\eta_{ab}$
and surface $g_{\alpha\beta}$ tensors are contravariant metric tensors
$\eta_{ab}\eta^{bc}=\delta_{a}^{c},g_{\alpha\beta}g^{\beta\gamma}=\delta_{\alpha}^{\gamma}$,
where $\delta_{\alpha}^{\gamma}$ is the Kronecker delta matrix. The surface
normal $\bm{N}$ is defined as the normal vector to surface bases so that
$\bm{N}\cdot\bm{S}_{\alpha}=0,\bm{N}^{2}=1$.

The definition of base vectors allows the introduction of Christoffel symbols
for ambient and embedded spaces. For ambient Minkowski flat space, the symbols
$\Gamma_{ij}^{k}=\bm{X}^{k}\cdot\partial_{i}\bm{X}_{j}=0$ are vanishing, while
for embedded surface
$\Gamma_{\alpha\beta}^{\gamma}=\bm{S}^{\gamma}\cdot\partial_{\alpha}\bm{S}_{\beta}\neq
0$ in general. In Riemannian geometry, the Christoffel symbols can be
presented as a linear combination of partial derivatives of metric tensors.
Therefore, the symbols are intrinsic

$\displaystyle\Gamma_{ij}^{k}$
$\displaystyle=\frac{1}{2}\eta^{kn}(\partial_{j}\eta_{in}+\partial_{i}\eta_{nj}-\partial_{n}\eta_{ij})$
(9) $\displaystyle\Gamma_{\alpha\beta}^{\gamma}$
$\displaystyle=\frac{1}{2}g^{\gamma\mu}(\partial_{\beta}g_{\alpha\mu}+\partial_{\alpha}g_{\mu\beta}-\partial_{\mu}g_{\alpha\beta})$
(10)

While all symbols are unchanged in differential geometry for moving manifolds,
they are not the only ones and are enhanced by extrinsic parameters.
Christoffel symbols allow the definition of invariant to reference frame
covariant derivatives

$\displaystyle\nabla_{k}T_{i}^{j}=$
$\displaystyle\partial_{k}T_{i}^{j}-\Gamma^{m}_{ki}T_{m}^{j}+\Gamma_{mk}^{j}T_{i}^{m}$
(11) $\displaystyle\nabla_{\gamma}T_{\alpha i}^{\beta j}=$
$\displaystyle\partial_{\gamma}T_{\alpha i}^{\beta
j}+\eta_{\gamma}^{k}\Gamma_{km}^{j}T_{\alpha i}^{\beta
m}-\eta_{\gamma}^{k}\Gamma_{ki}^{m}T_{\alpha m}^{\beta j}$
$\displaystyle+\Gamma^{\beta}_{\gamma\nu}T_{\alpha i}^{\nu
j}-\Gamma^{\nu}_{\gamma\alpha}T_{\nu i}^{\beta j}$ (12)

where $\eta_{\gamma}^{k}$ is the shift tensor, reciprocally shifting
space/surface bases so that

$\bm{S}_{\alpha}=\eta_{\alpha}^{i}\bm{X}_{i},g_{\alpha\beta}=\bm{S}_{\alpha}\cdot\bm{S}_{\beta}=\eta_{\alpha}^{i}\bm{X}_{i}\eta_{\beta}^{j}\bm{X}_{j}=\eta_{\alpha}^{i}\eta_{\beta}^{j}\eta_{ij}$
(13)

Note here that in flat Minkowski space-time, Christoffel symbols with Latin
indexes vanish, therefore ${\nabla_{k}T_{i}^{j}=\partial_{k}T_{i}^{j}}$ and
equations (11, 12) simplify accordingly.

Definitions (11–13) provide metrilinic property of metric tensors
$\nabla_{k}\eta_{ij},\nabla_{\gamma}g_{\alpha\beta}=0$, consequently the
surface base vectors are orthogonal to their covariant derivatives
${\bm{S}_{\alpha}\cdot\nabla_{\gamma}\bm{S}_{\beta}=0}$, therefore

$\nabla_{\alpha}\bm{S}_{\beta}=\bm{N}B_{\alpha\beta}$ (14)

$\bm{N}$ is the surface normal and $B_{\alpha\beta}$ is the curvature tensor
and according to definition (14) the curvature tensor is symmetric and
extrinsic parameter. Trace of the mixed curvature tensor $B_{\alpha}^{\alpha}$
is the mean curvature, while the determinant is the Gaussian curvature
$K=\lvert B^{\cdot}_{\cdot}\rvert$.

Note that, according to covariant derivatives and definitions of Christoffel
symbols (9–12), covariant derivatives are intrinsic, and therefore Riemann
tensor $R^{\delta}_{\alpha\beta\gamma}$ defined as

$\displaystyle(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})T_{\gamma}=R^{\lambda}_{\alpha\beta\gamma}T_{\lambda}$
(15) $\displaystyle
R^{\rho}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\alpha}\Gamma^{\alpha}_{\nu\sigma}-\Gamma^{\rho}_{\nu\alpha}\Gamma^{\alpha}_{\mu\sigma}$

is also intrinsic. The Trace of the Riemann tensor is the Ricci tensor, and a
trace of the mixed Ricci tensor is the Ricci scalar.

$\displaystyle R^{\delta}_{\alpha\delta\gamma}$
$\displaystyle=R_{\alpha\beta}$ (16) $\displaystyle R_{\alpha}^{\alpha}$
$\displaystyle=R$ (17)

Extrinsic curvature tensor is related to intrinsic Riemann tensor according to
Gauss’s Theorema Egregium

$R_{\gamma\delta\alpha\beta}=B_{\alpha\gamma}B_{\beta\delta}-B_{\beta\gamma}B_{\alpha\delta}$
(18)

Therefore, intrinsic $\nabla,g,R$ tensors can describe all static surfaces.
Note here that from the Gauss theorem (15), Ricci scalar follows

$R=R_{\alpha}^{\alpha}=(B_{\alpha}^{\alpha})^{2}-B_{\alpha\beta}B^{\alpha\beta}$
(19)

The contraction of the curvature tensor is negative twice Gaussian curvature
for two-dimensional surfaces.

$2K=-B_{\alpha\beta}B^{\alpha\beta}$ (20)

Therefore, the EH action resembles the Helfrich Hamiltonian for two-
dimensional surfaces, which has been intensively studied in biomembrane
applications. Note that (20) can be readily generalized for higher dimensional
manifolds as definition.

Introducing the velocity field for moving manifolds clarifies the
insufficiency of describing surface dynamics by only intrinsic tensors (see
below).

Velocity Field and CMS– All definitions from Gaussian geometry (8 – 14) are
also applicable to moving manifolds. However, extensions are required to
parameterize the motion because a surface changes. We refer to the parameter
as time $t$ and assign $\tau$ to the proper time to distinguish the two.
Parametric time might have nothing to do with real/proper time, but it is
necessary to describe manifold motion. The question to be asked is whether or
not the parametric time could be analyzed as relativistic real-time. That
question we leave open here.

The introduction of parametric time immediately necessitates defining time
derivatives to preserve covariance. For that, we first introduce surface
velocities: $C$ the surface normal velocity, and $V^{\alpha}$ tangent ones as
components of ambient velocity

$\bm{V}=\partial_{t}\bm{R}=C\bm{N}+V^{\alpha}\bm{S}_{\alpha}$ (21)

Note here that $\bm{V},C,V_{\alpha}$ are only defined in ambient flat space
and, therefore, are tensors only in the space. Note that the interface
velocity normal component of the surface velocity is invariant

$C=\bm{N}\cdot\bm{V}=N^{i}\bm{X}_{i}V_{j}\bm{X}^{j}=N^{i}V_{i}$ (22)

Therefore, if the surface is assumed to be moving initially, it will remain in
all reference frames. The invariance of interface velocity straightforwardly
indicates that, in general, for moving manifolds, a reference frame cannot be
found where $\bm{V}$ vanishes.

Introduction of the velocity field (21), without further elaboration [15, 8],
opens the path toward defining the covariant time derivative

$\displaystyle\dot{\nabla}T_{\alpha}^{\beta}$
$\displaystyle=\partial_{t}T_{\alpha}^{\beta}-V^{\gamma}\nabla_{\gamma}T_{\alpha}^{\beta}+\dot{\Gamma}_{\gamma}^{\beta}T^{\gamma}_{\alpha}-\dot{\Gamma}^{\gamma}_{\alpha}T_{\gamma}^{\beta}$
(23) $\displaystyle\dot{\Gamma}_{\alpha}^{\beta}$
$\displaystyle=\nabla_{\alpha}V^{\beta}-CB_{\alpha}^{\beta}$ (24)

where (24) $\dot{\Gamma}_{\alpha}^{\beta}$ is the Christoffel symbols analog
for invariant time derivatives first introduced by P. Grinfeld [8].
$\dot{\nabla}$ is invariant and has all properties as covariant derivatives.
From definitions (11, 12, 23, 24) straightforwardly follows that

$\nabla_{\gamma}g_{\alpha\beta},\dot{\nabla}g_{\alpha\beta}=0$ (25)

(25) is known as metrilinic property. Taking into account (23–25) we have

$\displaystyle\dot{\nabla}g_{\alpha\beta}=\partial_{t}g_{\alpha\beta}-V^{\gamma}\nabla_{\gamma}g_{\alpha\beta}-\dot{\Gamma}^{\gamma}_{\alpha}g_{\gamma\beta}-\dot{\Gamma}^{\gamma}_{\beta}g_{\gamma\beta}$
$\displaystyle=\partial_{t}g_{\alpha\beta}-(\nabla^{\gamma}V_{\alpha}-CB^{\gamma}_{\alpha})g_{\gamma\beta}-(\nabla^{\gamma}V_{\beta}-CB^{\gamma}_{\beta})g_{\alpha\gamma}$
$\displaystyle=\partial_{t}g_{\alpha\beta}-(\nabla_{\beta}V_{\alpha}-CB_{\alpha\beta})-(\nabla_{\alpha}V_{\beta}-CB_{\beta\alpha})$
$\displaystyle=\partial_{t}g_{\alpha\beta}-\nabla_{\alpha}V_{\beta}-\nabla_{\beta}V_{\alpha}+2CB^{\alpha\beta}=0,\Rightarrow$
$\displaystyle\partial_{t}g_{\alpha\beta}=\nabla_{\alpha}V_{\beta}+\nabla_{\beta}V_{\alpha}-2CB_{\alpha\beta}$
(26)

For the contravariant metric tensor, (26) has reversed signs: a negative sign
applies for tangent velocities and a positive sign for normal velocity.

Even though the invariant time derivative (23) and covariant space derivatives
(11, 12) possess the same properties, they are fundamentally different. The
first is extrinsically defined because it includes ambient surface velocities
and the curvature tensor, whereas the latter is intrinsic. The Gauss’s
Theorema Egregium (18) essentially explains that static geometry can be
intrinsically constructed. In contrast, equations (21–24) reveal that moving
manifolds are essentially extrinsic phenomena unless a direct reasonable
connection is made between extrinsically defined velocities and the
intrinsically defined tensors. Additionally, it is worth noting that, in
addition to (26), the definitions (23, 24) lead to beautiful theorems [8]:

$\displaystyle\dot{\nabla}B_{\alpha}^{\beta}=\nabla_{\alpha}\nabla^{\beta}C+CB_{\alpha}^{\gamma}B_{\gamma}^{\beta}$
(27)
$\displaystyle\dot{\nabla}B_{\alpha}^{\alpha}=\nabla_{\alpha}\nabla^{\alpha}C+CB_{\alpha}^{\beta}B_{\beta}^{\alpha}$
(28)

(28) is the special case of (27). (27, 28) theorems indicate that invariant
time derivatives of the curvature tensor and the mean curvature are strongly
dominated by the surface normal velocity $C$ and, therefore, cannot be
intrinsically defined.

Definitions (11–14, 21, 23) lead to integration theorems

$\displaystyle\partial_{t}\int_{\Omega}fd\Omega$
$\displaystyle=\int_{\Omega}\partial_{t}fd\Omega+\int_{\mathbb{S}}fCd\mathbb{S}$
(29) $\displaystyle\partial_{t}\int_{\mathbb{S}}fd\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}\partial_{t}fd\mathbb{S}-\int_{\mathbb{S}}fCB_{\alpha}^{\alpha}d\mathbb{S}$
(30)

Equations (8–24) along with integration theorems (29, 30) form the foundation
of CMS and extend understanding of GR. Proof of (29) is known in mathematics.
We provide a short proof of the (30) theorem here for brevity.

Let $F\in\mathbb{S}$ be a smooth scalar function on the closed smooth manifold
$\mathbb{S}$. Then, across any closed stationary contour $\gamma$, the contour
integral vanishes because the contour normal $n_{\alpha}$ lying in the tangent
plane always stays perpendicular to the surface tangent velocity $\bm{V}_{T}$,
so that

$v=\bm{n}\cdot\bm{V}_{T}=n_{\alpha}V^{\alpha}=0$ (31)

Using this and that, the total time derivative of the integral has two parts.
The first part reflects that $F$ changes as time evolves, and the second part
reflects that the surface changes as time evolves. Therefore

$\displaystyle\frac{d}{dt}$
$\displaystyle\int_{\mathbb{S}}Fd\mathbb{S}=\int_{\mathbb{S}}\partial_{t}Fd\mathbb{S}+\int_{\mathbb{S}}F\partial_{t}d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}(\partial_{t}F+F\nabla_{\alpha}V^{\alpha}-F\nabla_{\alpha}V^{\alpha})d\mathbb{S}$
$\displaystyle+\int_{x_{1}}...\int_{x_{n}}F\partial_{t}\sqrt{g}dx_{1}...dx_{n}$
$\displaystyle=\int_{\mathbb{S}}(\partial_{t}F+F\nabla_{\alpha}V^{\alpha}-F\nabla_{\alpha}V^{\alpha})d\mathbb{S}$
$\displaystyle+\int_{x_{1}}...\int_{x_{n}}F(\nabla_{\alpha}V^{\alpha}-CB_{\alpha}^{\alpha})\sqrt{g}dx_{1}...dx_{n}$
$\displaystyle=\int_{\mathbb{S}}(\partial_{t}F+F\nabla_{\alpha}V^{\alpha}-F\nabla_{\alpha}V^{\alpha})d\mathbb{S}$
$\displaystyle+\int_{\mathbb{S}}F(\nabla_{\alpha}V^{\alpha}-CB_{\alpha}^{\alpha})d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}(\partial_{t}F+\nabla_{\alpha}(FV^{\alpha})-V^{\alpha}\nabla_{\alpha}F-CFB_{\alpha}^{\alpha})d\mathbb{S}$
(32)

By Gauss theorem, the integral from $\nabla_{\alpha}(FV^{\alpha})$ is
converted to the contour integral and, according to the (31) boundary
condition, vanishes. If one relaxes the condition that the surface is closed
and is bounded by static contour, then according to (32), the integration
theorem (30) updates as

$\partial_{t}\int_{\mathbb{S}}Fd\mathbb{S}=\int_{\mathbb{S}}\dot{\nabla}Fd\mathbb{S}-\int_{\mathbb{S}}FCB_{\alpha}^{\alpha}d\mathbb{S}+\int_{\gamma}vFd\gamma$
(33)

Since we discuss only closed surfaces here, the last term from (33) becomes
irrelevant.

Note that, for calculation of $\partial_{t}\sqrt{g}$ in (32), we used Jacobi’s
theorem about taking derivative for determinants ${\partial g/\partial
g^{\alpha\beta}=gg_{\alpha\beta}}$ and (26), so

$\displaystyle\partial_{t}g=\frac{\partial g}{\partial
g^{\alpha\beta}}\partial_{t}g^{\alpha\beta}$
$\displaystyle=gg_{\alpha\beta}(\nabla^{\alpha}V^{\beta}+\nabla^{\beta}V^{\alpha}-2CB^{\alpha\beta})$
$\displaystyle=2g(\nabla_{\alpha}V^{\alpha}-CB_{\alpha}^{\alpha})$ (34)

From (34) according to the complex function derivative rule, the identity

$\partial_{t}\sqrt{g}=\frac{1}{2\sqrt{g}}\partial_{t}g=\sqrt{g}(\nabla_{\alpha}V^{\alpha}-CB_{\alpha}^{\alpha})$
(35)

trivially follows. Next, using the (23) definition written for scalar
functional

$\dot{\nabla}F=\partial_{t}F-V^{\alpha}\nabla_{\alpha}F$ (36)

Applying (36) to the (32) with taking into account the Gauss theorem and the
boundary condition (31), the integration theorems (30, 33) trivially follows.
This proof we also reported in [32].

Shape Dynamics Equations– Based on the definitions of velocity field and
spatiotemporal covariant derivatives, along with integrations theorems, one
can derive surface dynamics equations, see [15, 16] and for extensions see
[18, 19]. Instead of repeating the derivation, we will outline the path
forward. According to (7, 6), the most generic action is

$S=\int_{\mathbb{S}}\frac{\rho
V^{2}}{2}d\mathbb{S}-\int_{\mathbb{S}}\mathscr{L}_{\mathbb{S}}d\mathbb{S}-\int_{\Omega}\mathscr{L}_{\Omega}d\Omega$
(37)

In three-dimensional space, energy per volume is pressure and per area is the
tension therefore to reduce the number of letters, we call
$\mathscr{L_{\mathbb{S}}}\equiv\sigma$ and $\mathscr{L}_{\Omega}\equiv P$ even
though densities are in higher than three dimensions and $\sigma,P$ is only
reminiscent of the surface tension and the volumetric pressure. Also (37)
simplifies to (6) considering that
$\mathscr{L}_{\mathbb{S}}\in\mathscr{L}_{\Omega}$.

Next, taking into account the minimum action principle, from (37) directly
follows that

$\delta\int_{\mathbb{S}}\frac{\rho
V^{2}}{2}d\mathbb{S}=\delta(\int_{\mathbb{S}}\sigma
d\mathbb{S}+\int_{\Omega}Pd\Omega)$ (38)

According to integration theorems (30, 29) and calculations of the covariant
time derivative (23) (for details see [15, 16, 18, 19, 32]) we have

$\displaystyle\delta_{\bot}\int_{\mathbb{S}}\frac{\rho V^{2}}{2}d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}\rho
C(\dot{\nabla}C+2V^{\alpha}\nabla_{\alpha}C+V^{\alpha}V^{\beta}B_{\alpha\beta})d\mathbb{S}$
$\displaystyle\delta_{\|}\int_{\mathbb{S}}\frac{\rho V^{2}}{2}d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}\rho
V_{\alpha}(\dot{\nabla}V^{\alpha}+V^{\beta}\nabla_{\beta}V^{\alpha}$
$\displaystyle-C\nabla^{\alpha}C-CV^{\beta}B_{\beta}^{\alpha})d\mathbb{S}$
$\displaystyle\delta\int_{\mathbb{S}}\sigma d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}(\dot{\nabla}\sigma-\sigma
CB_{\alpha}^{\alpha})d\mathbb{S}$
$\displaystyle\delta_{\bot}\int_{\mathbb{S}}\sigma d\mathbb{S}$
$\displaystyle=\int_{\mathbb{S}}(\partial_{t}\sigma-\sigma
CB_{\alpha}^{\alpha})d\mathbb{S}$
$\displaystyle\delta_{\|}\int_{\mathbb{S}}\sigma d\mathbb{S}$
$\displaystyle=-\int_{\mathbb{S}}V_{\alpha}\nabla^{\alpha}\sigma d\mathbb{S}$
$\displaystyle\delta\int_{\Omega}Pd\Omega$
$\displaystyle=\delta_{\bot}\int_{\Omega}Pd\Omega=\int_{\Omega}\partial_{t}Pd\Omega+\int_{\mathbb{S}}PCd\mathbb{S}$

$\delta_{\|},\delta_{\bot}$ designate variations in the tangent space and
normal direction correspondingly. Considering that normal deformations mainly
govern $\partial_{t}\sigma$ while $V_{\alpha}\nabla^{\alpha}\sigma$ is
tangential so that $N^{i}\partial_{t}\sigma=\partial_{t}f^{i}$, then applying
Gauss theorem to the normal component and taking into account conservation of
mass the surface dynamics equations follow:

$\displaystyle\dot{\nabla}\rho+\nabla_{\alpha}(\rho V^{\alpha})=\rho
CB_{\alpha}^{\alpha}$
$\displaystyle\partial_{i}[V^{i}(\rho(\dot{\nabla}C+2V^{\alpha}\nabla_{\alpha}C+V^{\alpha}V^{\beta}B_{\alpha\beta})$
$\displaystyle-P+\sigma
B_{\alpha}^{\alpha})]=\partial_{t}P+\partial_{i}\partial_{t}f^{i}$ (39)
$\displaystyle\rho(\dot{\nabla}V^{\alpha}+V^{\beta}\nabla_{\beta}V^{\alpha}-C\nabla^{\alpha}C-CV^{\beta}B_{\beta}^{\alpha})=-\nabla^{\alpha}\sigma$

Here, the first equation describes the conservation of mass, the second
equation shows manifold evolution in the normal direction, and the last
equations lead to manifold dynamics in tangent space. Since $\alpha$ is the
manifold’s dimension, the number of the last equation also indicates the
dimensions of the tangent space in the case of GR $\alpha=4$. Note that in GR,
according to Einstein’s field equations, we have ten independent equations,
while in shape dynamic equations, the number is reduced to six.

EH Action According to CMS – To demonstrate the effectiveness of CMS in
understanding GR, we begin with the kinetic term of the EH action (5) without
any modifications

$S=\frac{1}{2k}\int_{\mathbb{S}}Rd\mathbb{S},...\Rightarrow\mathscr{L}_{\mathbb{S}}=\frac{1}{2k}R$
(40)

where $\mathbb{S}$ stands for surface and $g=-\det g_{\alpha\beta}$ because
space-time is pseudo-Riemannian. Variation of (40) is widely known [33], so we
do not repeat it here. Instead, let’s take a variation by CMS. For that, we
take into account the integration theorem (30) then, the variation of (40) is

$\delta
S=\frac{1}{2k}\frac{d}{dt}\int_{\mathbb{S}}Rd\mathbb{S}=\frac{1}{2k}\int_{\mathbb{S}}(\dot{\nabla}R-RCB_{\alpha}^{\alpha})d\mathbb{S}$
(41)

By expanding (41) according to the invariant time derivative of Ricci scalar
by the (36) equation, we have

$\delta S=\frac{1}{2k}\int_{\mathbb{S}}(\frac{\partial R}{\partial
t}-V^{\alpha}\nabla_{\alpha}R-RCB_{\alpha}^{\alpha})d\mathbb{S}$ (42)

Of course, instead of the (42) path, we could take a directly invariant
derivative of the Ricci scalar by the (19). Then,

$\displaystyle\dot{\nabla}R$
$\displaystyle=2\dot{\nabla}B_{\alpha}^{\alpha}-(B^{\alpha\beta}\dot{\nabla}B_{\alpha\beta}+B_{\alpha\beta}\dot{\nabla}B^{\alpha\beta})$
$\displaystyle=2(\nabla_{\alpha}\nabla^{\alpha}C+CB_{\alpha}^{\beta}B_{\beta}^{\alpha})$
$\displaystyle-(\nabla^{\alpha}\nabla_{\beta}C+CB^{\alpha}_{\gamma}B^{\gamma}_{\beta})(\nabla_{\alpha}\nabla^{\beta}C+CB_{\alpha}^{\gamma}B_{\gamma}^{\beta})$
(43)

(43) can be significantly simplified, but even without simplifications, (43)
shows strong dominance by normal velocities. That is, the variance of the
Ricci scalar is strongly dominated by normal velocities, and therefore, it
cannot be modeled by intrinsic tensors only. Any other path to the variance is
largely model-dependent and, therefore, approximation. Next, as an
approximation assume the normal velocity greatly dominates over tangent
velocity $C>>V^{\alpha}$, then we have

$\displaystyle\partial_{t}R$ $\displaystyle=\frac{\partial R}{\partial
g^{\alpha\beta}}\partial_{t}g^{\alpha\beta}$ $\displaystyle\frac{\partial
R}{\partial g^{\alpha\beta}}$ $\displaystyle\approx R_{\alpha\beta}$ (44)
$\displaystyle\delta S$
$\displaystyle\approx\frac{1}{2k}\int_{\mathbb{S}}(R_{\alpha\beta}\frac{\partial
g^{\alpha\beta}}{\partial t}-RCB^{\alpha\beta}g_{\alpha\beta})d\mathbb{S}$
(45)

where (44) follows from Palatini identity with assumed boundary condition.
Taking into account (26) and proposing that normal velocity largely dominates
over tangent ones, then

$\partial_{t}g^{\alpha\beta}\approx 2CB^{\alpha\beta}$ (46)

Hadamard proposed the equation (46) to explain shape dynamics for water drops.
We refer to (46) as the Hadamard flow. Taking into account (46) we obtain

$\delta
S\approx\frac{1}{2k}\int_{\mathbb{S}}2CB^{\alpha\beta}(R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta})d\mathbb{S}$
(47)

Next, let’s add a time-invariable matter field that moves the surface. Since
the system is restricted by the parametric time-independent $\sigma$,
according to the integration theorem (30) the variation of the energy term is

$\delta\int_{\mathbb{S}}\sigma d\mathbb{S}=-\int_{\mathbb{S}}\sigma
CB^{\alpha\beta}g_{\alpha\beta}d\mathbb{S}$ (48)

Introducing the designation $T_{\alpha\beta}=\sigma g_{\alpha\beta}$ and
demanding that (47) has to be identical to (48) for every curvature tensor and
normal velocity, one obtains Einstein field equation

$G_{\alpha\beta}=kT_{\alpha\beta}$ (49)

However, during the (40–47) derivation, we make several approximations.
Therefore, the field equation is not exact and has to be corrected. Without
consecutive approximations (42) variation is

$\displaystyle\delta
S=\frac{1}{2k}\int_{\mathbb{S}}(R_{\alpha\beta}(-\nabla^{\alpha}V^{\beta}$
$\displaystyle-\nabla^{\beta}V^{\alpha}+2CB^{\alpha\beta})$
$\displaystyle-V^{\alpha}\nabla_{\alpha}R-RCB_{\alpha}^{\alpha})d\mathbb{S}$
(50)

and we can still demand that (50) is identical to (48) energy variation for
every curvature tensor and normal velocity, then the generic field equation is

$\displaystyle\kappa
R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta}-\frac{V^{\alpha}\nabla_{\alpha}R}{2CB^{\alpha\beta}}=kT_{\alpha\beta}$
(51)
$\displaystyle\kappa=1-\frac{\nabla^{\alpha}V^{\beta}+\nabla^{\beta}V^{\alpha}}{2CB^{\alpha\beta}}=\frac{\partial_{t}g^{\alpha\beta}}{2CB^{\alpha\beta}}$
(52)

The scenario most closely resembling Einstein’s field equations with a
cosmological constant is when $\kappa=1$ and

$-V^{\alpha}\nabla_{\alpha}R/(2CB^{\alpha\beta})=\Lambda g_{\alpha\beta}$ (53)

Then either $C>>V^{\alpha}$, $V^{\alpha}=const$ or
${\nabla^{\alpha}V^{\beta}+\nabla^{\beta}V^{\alpha}=0}\Rightarrow\nabla_{\alpha}V^{\alpha}=0$.
Assuming that cosmological constant is time invariable than from (53), follow
that

$\displaystyle B_{\alpha}^{\alpha}\approx 0;C>>V^{\alpha}$
$\displaystyle\Lambda=\lambda\frac{I^{\alpha}\nabla_{\alpha}R}{B_{\alpha}^{\alpha}};\nabla_{\alpha}V^{\alpha}=0$
(54)

where $\bm{I}$ is some constant tangent unit vector so that
$V^{\alpha}=VI^{\alpha}$ and $\lambda=V^{\cdot}/C$. The trivial solution to
(54) is constant mean curvature surfaces

$B_{\alpha}^{\alpha}=\gamma\nabla_{\alpha}R^{\alpha}=const$ (55)

where $\gamma=\frac{\lambda}{\Lambda}$ and $R^{\alpha}=RI^{\alpha}$.
Therefore, possible solutions to field equations with a cosmological constant
are constant mean curvature shapes. Note that (52) can also be approximately
solved directly if one takes into account the (53) condition.

Note also that without any algebraic manipulations and approximations, taking
into account (41) and (48), we can also deduce straightforward field equations
for EH action that reads

$\frac{1}{2CB_{\gamma}^{\gamma}}\dot{\nabla}R_{\alpha\beta}-R_{\alpha\beta}=kT_{\alpha\beta}$
(56)

(56) is the generalization of Einstein field equations that look completely
different from GR equations, though, due to the course of (44–47)
approximations simplifies to the (49) field equations.

Equations of Surface Motion – Because we try to understand EH action from the
viewpoint of phenomenology, we neglect space terms in the (37) and concentrate
only on surface energy density term $\sigma$. We recover space-related terms
when needed to understand the global picture explicitly. In previous sections,
we largely dealt with the EH action. Here, we assume that the space-time
curving is induced by the kinetic energy of the moving manifold that is
modeled as the EH action in GR. Without EH modeling, we have (37) that gives
(39) surface dynamics equations which, after neglecting ambient space terms,
simplify as

$\displaystyle\dot{\nabla}\rho+\nabla_{\alpha}(\rho V^{\alpha})=\rho
CB_{\alpha}^{\alpha}$ (57)
$\displaystyle\partial_{i}[V^{i}(\rho(\dot{\nabla}C+2V^{\alpha}\nabla_{\alpha}C+V^{\alpha}V^{\beta}B_{\alpha\beta})+\sigma
B_{\alpha}^{\alpha})]=\partial_{i}\partial_{t}f^{i}$
$\displaystyle\rho(\dot{\nabla}V^{\alpha}+V^{\beta}\nabla_{\beta}V^{\alpha}-C\nabla^{\alpha}C-CV^{\beta}B_{\beta}^{\alpha})=-\nabla^{\alpha}\sigma$

Following conservation of energy $\sigma=const$, in EH action
$f_{i},\sigma=const$ the equations (57) further simplify as

$\displaystyle\dot{\nabla}\rho+\nabla_{\alpha}(\rho V^{\alpha})=\rho
CB_{\alpha}^{\alpha}$
$\displaystyle\rho(\dot{\nabla}C+2V^{\alpha}\nabla_{\alpha}C+V^{\alpha}V^{\beta}B_{\alpha\beta})=-\sigma
B_{\alpha}^{\alpha}$ (58)
$\displaystyle\rho(\dot{\nabla}V^{\alpha}+V^{\beta}\nabla_{\beta}V^{\alpha}-C\nabla^{\alpha}C-CV^{\beta}B_{\beta}^{\alpha})=0$

(58) is known as the Grinfeld equation for dynamic fluid films, and it was
proposed to describe the shape dynamics of soap bubbles [13].

Next, let’s assume that (58) is the approximate solution of surface dynamics
equations (39) for quasi-constant $P,\sigma,f_{i}\neq const$ fields. Then for
homogenous surfaces $\nabla^{i}\sigma=0$ and dominantly compressible cases
$C>>V^{i}$, when tangential velocities are infinitesimally small $V^{i}\approx
0$, the dynamic manifolds equations (39) simplify as

$\displaystyle\partial_{t}\rho$ $\displaystyle=\rho CB_{i}^{i}$
$\displaystyle-\partial_{i}(PV^{i})$
$\displaystyle=\partial_{t}P+\partial_{i}\partial_{t}f^{i}$ (59)
$\displaystyle\rho\partial_{t}C$ $\displaystyle=-\sigma B_{\alpha}^{\alpha}$

Note here that for homogeneous surfaces ${\partial_{i}P,\partial_{i}f^{i}=0}$
and the immediate solution to the second equation, therefore to the system
(59) is

$\displaystyle\partial_{i}V^{i}$ $\displaystyle=0$ (60)
$\displaystyle\partial^{2}_{t}C$ $\displaystyle=-\gamma(\Delta C-2KC)$ (61)

(61) provides the wave equation for near-planar surfaces where Gaussian
curvature $K\approx 0$ is infinitesimally small. It is noteworthy that, in
this particular case, even though the system is dominantly compressible, it
has an approximate solution (60) characteristic of dominantly incompressible
systems. The first equation gives the spherical solution of

$\bm{V}\sim\bm{R}/R^{3}$ (62)

where $\bm{R}$ is the position vector. Taking (62) into account, one can
extract an analytical solution to (60) having the form of

$\displaystyle\bm{R}=A(\bm{R}_{0}+\omega_{\alpha}R_{\alpha}\bm{S}^{\alpha}t)+BR_{0\alpha}e^{\omega_{\alpha}\bm{S}^{\alpha}t}+\psi_{\alpha}\bm{S}^{\alpha}$
(63)

where $R_{0\alpha}$ is arbitrary coordinates of the initial position vector
$\bm{R}_{0}$, $\omega_{\alpha}$ is the coupling constant indicating the
frequency of oscillation in $\alpha$ directions, and $\bm{S}^{\alpha}$ is the
vector component of $\bm{S}$ unit vector, $A,B$ are some constants that can be
defined by initial conditions,
${e^{\omega_{\alpha}\bm{S}^{\alpha}t}=(e^{\omega_{x}\bm{S}^{x}t},e^{\omega_{y}\bm{S}^{y}t},e^{\omega_{z}\bm{S}^{z}t})}$
by definition and ${{\bm{\psi}}}_{\xi}(\xi,t)$ is the wave function of the
form

$\displaystyle{{\bm{\psi}}}_{\xi}(\xi,t)=\sum_{m=1}^{\infty}\,\frac{2}{l}({\int}_{0}^{l}\,{\bm{\psi}}(\xi,0)\,\sin\,\frac{m\pi\xi}{l}d\xi)$
$\displaystyle\cdot\cos(\frac{{v}_{0}m\pi t}{l})\sin(\frac{{v}_{0}m\pi t}{l})$
(64)

The exact derivation of the solutions (60–64) is given in our works [17, 32],
and we do not repeat it here.

Note that (60) and (61) are the solutions of (39, 58) for two-dimensional
manifolds (surfaces) only. The same results were obtained about soup bubbles
and droplets before [13, 17]. The combination of analytic (63) and numerical
solutions of (59) indicates non-linear oscillations among spherical,
oblate/prolate, and non-oblate/prolate shapes, a tendency towards increasing
the radius and inducing shape instabilities by forming singularities, the
growing amplitude in the oscillations of the surface mass density, meaning
increasing the mass instabilities in the surface. Mass instabilities
ultimately lead to the development of singularities, which may induce
divisions [17]. Increased mass density picks can be considered as model black
holes for two-dimensional surfaces. The solutions (60–63) are explicitly
correct only for two manifolds, but the same analyses can be readily extended
for higher dimensional surfaces of pseudo-Riemannian geometries.

Incompresible spacetime – Another limit that deserves special attention is
$C<<V^{\alpha}$, that is, interface velocity relative to tangent ones is
infinitesimally small, then the space-time can be considered as incompressible
or quasi-incompressible $\partial_{i}V^{i}\approx 0$. Then, the second
equation of (39) further simplifies so that

$\displaystyle V^{i}\partial_{i}[\rho
V^{\alpha}V^{\beta}B_{\alpha\beta}+\sigma B_{\alpha}^{\alpha}]$
$\displaystyle=V^{i}\partial_{i}P+\partial_{i}\partial_{t}f^{i}$
$\displaystyle\rho V^{\alpha}V^{\beta}B_{\alpha\beta}$
$\displaystyle=P+\partial_{i}f^{i}-\sigma B_{\alpha}^{\alpha}$ (65)

We refer to the solution as the generalized Young-Laplace law for
incompressible cases. Assuming $f^{i},\sigma\in P\in\Omega$ the (65) can be
rewritten more compactly as

$P=\rho V^{\alpha}V^{\beta}B_{\alpha\beta}$ (66)

which has a trivial solution $B_{\alpha\beta}=P(\rho
V_{\alpha}V_{\beta})^{-1}$ meaning at appropriate pressure conditions any
manifold can spontaneously form. More intriguingly (66) leads to generalized
Gibbs-Thomson equation

$T_{C}=T_{0}(1-\frac{\mu V^{\alpha}V^{\beta}B_{\alpha\beta}}{H})$ (67)

here $H$ is the fusion enthalpy of the universe, $\mu$ is molar mass, $T_{C}$
curvature related melting temperature and $T_{0}$ planar bath melting
temperature [19, 32]. (67) has a trivial solution of the form

$B_{\alpha\beta}=\frac{\gamma}{\mu}H_{\alpha\beta}$ (68)

where $\gamma=(1-T_{C}/T_{0})$ and
$H_{\alpha\beta}=H(V_{\alpha}V_{\beta})^{-1}$. (66) effectively predicts that
the pressure distribution defines the structure of the universe, while (67-68)
shows the geometry of the universe if temperature distribution is given.
Incidentally, (65–67) appears to be incompressible Navier-Stokes equations
solutions as well [32].

If one assumes that temperature is uniformly distributed throughout space-time
so that $\gamma$ is less than one positive constant and wave propagation
velocities in all directions are scaled to the speed of light $c$, then (68)
simplifies as

$B_{\alpha\beta}=\frac{\gamma}{\mu c^{2}}H\delta_{\alpha\beta}$ (69)

Again, in the case of GR approximation that the speed of light is the same
constant in all directions $V_{\alpha}=c$, then (69) leads to

$B_{\alpha\beta}\simeq\delta_{\alpha\beta}$ (70)

(70) points out that space-time aims to achieve a locally flat geometry, but
this is impossible due to the temperature distribution not being homogeneous
and space-time not incompressible.

Constant Velocities on Incompressible Spheres – Following (66), we can
calculate surface pressure dynamics on the incompressible, two-dimensional
unit sphere for the constant spherical velocities. Even though the
calculations are done for two surfaces, its generalizations for any dimensions
are straightforward. Due to the manifold dynamics, pressure distribution on
the surface is given by

$P=\rho_{0}(\omega^{2}+\varphi^{2}\sin^{2}(\omega t+\theta_{0}))$ (71)

where $\rho_{0}$ is the constant surface mass density,
$\omega=V_{\theta},\varphi=V_{\phi}$ are some constants similar to oscillation
frequency and amplitude. The appendix Two-sphere Calculations gives the
derivation of (71). (71) indicates that the material point on the unit sphere,
with constant spherical $V_{\theta},V_{\phi}=const$ velocities, moves like a
wave due to the surface pressures wave-like dynamics induced by the surface
dynamics.

Conclusion – We have proposed a generalization of the GR by CMS’s extension of
differential geometry. The generalization delivered few striking results.
Namely, it indicates that predominantly compressible space-time geometry is
neither collapsing nor expanding. It might non-harmonically oscillate between
inflation and collapse, frequently developing singularities and mass
instabilities that may lead to black hole formation or division. In some
approximations, the dominantly compressible space-time simplifies to GR
equations by having specific solutions of constant mean curvature shapes. In
contrast to predominantly compressible space-time geometry, dominantly
incompressible geometry gives a plethora of diverse shapes with different
patterns, and dilation strives to form locally flat geometry. As a result,
material points with constant spherical velocities on the dominantly
incompressible two-sphere start moving like a wave.

###### Acknowledgements.

This work is supported by Shota Rustaveli Science Foundation of Georgia by
grant no. FR-21-2844 and by Euro Commission’s ERC Erasmus+ collaborative
linkage grant between the University of Copenhagen and New Vision University.
We also benefit from New Vision University’s internal funding. We want to
acknowledge the warm hospitality of the Biocomplexity Department of Niels Bohr
Institute, University of Copenhagen, where the paper was completed.

## Two-sphere calculations

Here, we calculate how the material point moves with constant spherical
velocities on the dynamic sphere. The calculations are done only on two-
dimensional unit spheres, though extensions in higher dimensions are
straightforward. Since we deal with unit spheres, it is convenient to start
with spherical coordinates given as

$\displaystyle x(\theta,\phi,t)$ $\displaystyle=\sin\theta\cos\phi$
$\displaystyle y(\theta,\phi,t)$ $\displaystyle=\sin\theta\sin\phi$
$\displaystyle z(\theta,\phi,t)$ $\displaystyle=\cos\theta$

The shift, metric, and curvature tensors for the sphere are given as matrices

$\eta_{\alpha}^{i}=\begin{pmatrix}\cos\theta\cos\phi&-\sin\theta\sin\phi\\\
\cos\theta\sin\phi&\sin\theta\cos\phi\\\ -\sin\theta&0\end{pmatrix}$ (72)
$g_{\alpha\beta}=B_{\alpha\beta}=\begin{pmatrix}1&0\\\ 0&\sin^{2}\theta\\\
\end{pmatrix};g^{\alpha\beta}=B^{\alpha\beta}\begin{pmatrix}1&0\\\
0&\sin^{-2}\theta\\\ \end{pmatrix}$ (73)

Note here that the metric tensor and the curvature tensors for spheres are
generally different but the same for unit spheres.

By taking the time derivative of coordinates, the ambient velocity components
are calculated as

$\displaystyle V_{x}$
$\displaystyle=\frac{dx(\theta,\phi,t)}{dt}=\sin\theta(-\sin\phi)\frac{d\phi}{dt}+\cos\theta\cos\phi\frac{d\theta}{dt}$
$\displaystyle V_{y}$
$\displaystyle=\cos\theta\sin\phi\frac{d\theta}{dt}+\sin\theta\cos\phi\frac{d\phi}{dt}$
$\displaystyle V_{z}$ $\displaystyle=-\sin\theta\frac{d\theta}{dt}$

Next, doting velocity vector
$\bm{V}=\begin{pmatrix}V_{x}&V_{y}&V_{z}\end{pmatrix}$ on shift tensor
$\eta_{\alpha}^{i}$ matrix one gets the first and second components of the
tangent velocities, that read

$\displaystyle V_{1}=V_{i}\eta_{1}^{i}$
$\displaystyle=(\cos\theta\cos\phi\frac{d\theta}{dt}-\sin\theta\sin\phi\frac{d\phi}{dt})\cos\theta\cos\phi$
$\displaystyle+(\cos\theta\sin\phi\frac{d\theta}{dt}+\sin\theta\cos\phi\frac{d\phi}{dt})\cos\theta\sin\phi$
$\displaystyle+(-\sin\theta\frac{d\theta}{dt})(-\sin\theta)=(\cos^{2}\theta+\sin^{2}\theta)\frac{d\theta}{dt}$
$\displaystyle=\frac{d\theta}{dt}$ $\displaystyle V_{2}=V_{i}\eta^{i}_{2}$
$\displaystyle=(\cos\theta\cos\phi\frac{d\theta}{dt}-\sin\theta\sin\phi\frac{d\phi}{dt})(-\sin\theta\sin\phi)$
$\displaystyle+(\cos\theta\sin\phi\frac{d\theta}{dt}+\sin\theta\cos\phi\frac{d\phi}{dt})(\sin\theta\cos\phi)$
$\displaystyle=(\sin^{2}\theta\sin^{2}\phi+\sin^{2}\theta\cos^{2}\phi)\frac{d\phi}{dt}$
$\displaystyle=\sin^{2}\theta\frac{d\phi}{dt}$

doting $V_{1},V_{2}$ on curvature tensor for the sphere $B^{\alpha\beta}$ and
taking into account summation convention we get

$\displaystyle V_{\alpha}V_{\beta}B^{\alpha\beta}$
$\displaystyle=V_{1}^{2}B^{11}+V_{2}^{2}B^{22}$
$\displaystyle=V_{\theta}^{2}+V_{\phi}^{2}\sin^{2}\theta$

where $V_{\theta}=d\theta/dt,V_{\phi}=d\phi/dt$ are spherical velocities. Note
here that because the surface moves with constant spherical velocities
$\theta=\theta_{0}+V_{\theta}t$. Therefore, for constant surface mass density
$\rho=\rho_{0}=const$ the surface pressure changes as

$P=\rho_{0}(V_{\theta}^{2}+V_{\phi}^{2}\sin^{2}(\theta_{0}+V_{\theta}t))$ (74)

that is exactly the wave motion on the spherical surface.

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