# Relativistic Elasticity II

Robert Beig
Faculty of Physics, University of Vienna, Austria

###### Abstract

This paper is based on a series of talks given at the ESI program on
’Mathematical Perspectives of Gravitation Beyond the Vacuum Regime’ in
February 2022. It is meant to be an introduction to the field of relativistic
elasticity for readers with a good base in the mathematics of General
Relativity with no necessary previous of knowledge of elasticity either in the
classical or relativistic domain. Despite its introductory purpose, the
present work has new material, in particular related to the formal structure
of the theory.

Keywords: Relativistic Elasticity

Dedicated to the memory of Bernd G. Schmidt

##  1 Introduction

Elasticity theory is the branch of continuum mechanics dealing with deformable
solids, which are ideal in the sense that dissipative processes can be
ignored. Good introductions to the nonrelativistic theory for the
mathematically oriented reader are Ciarlet [1] and Marsden, Hughes [2].
In the context of relativity elasticity was first treated (in special
relativity) by Herglotz [3] as early as 1911. For many years, with papers by
many different authors, the subject remained somewhat diffuse until the
insightful work of Carter and Quintana [4], which became particularly
influential among authors seeking astrophysical applications [5]. The paper
[4] had been remarkably silent about what should be considered the basic
dynamical variables of the theory. Then, in the work of Kijowski and Magli
[6], came the realization that the relativistic version of the classical
theory of hyperelastic materials is fundamentally a theory - derivable from an
action principle - of maps subject to some conditions, which are sometimes
called back-to-label maps and called here configurations, from spacetime into
a fictitious 3-dimensional space called material space111A lucid, but largely
unnoticed, account had previously been given in the textbook of Soper [7]..
The recent work of Brown [8], similarly to Herglotz, describes the theory in
the material (often called ’Lagrangian’) perspective with configurations
replaced by time dependent maps from material space into physical space
relative to some slicing of spacetime.
The present paper should be viewed as an introduction to the geometric
structure of the theory. A guiding principle throughout is the role of
invariance under spacetime diffeomorphisms. This gives rise to a considerable
departure from practically all the standard treatments of nonrelativistic
elasticity. So, for example, by working with a diffeomorphism invariant theory
on a general spacetime, the issue of ’material frame indifference’ (see [1]
and [2]) is completely bypassed. Furthermore the standard literature sees the
theory primarily as a branch of mechanics, which results in a preference of
the material picture.
In contrast we view elasticity as a field theory on spacetime, which makes the
spacetime (’Eulerian’) - as opposed to the material - viewpoint, where
diffeomorphism invariance is manifest, the method of choice. Moreover: for
whatever field is present in the vacuum region surrounding the body, the
material picture is a priori not even defined. On the other hand the natural
(in a sense described later) boundary conditions at the matter-vacuum
interface render this a free boundary, and this, for the actual solution of
problems, makes an intermediate use of the material picture - in which this
boundary is fixed - indispensable.
Our treatment will be formal in the sense that there will be no function
spaces - and also in the sense of staying away from astrophysical applications
(for a recent account see e.g. [9]) or detailed equations of state suitable
for describing compact elastic bodies in astrophysics.
The plan of this paper is as follows. In Sect.2 we give a quick account of the
Lagrangian theory of maps from spacetime to some other manifold. This has
applications ranging from wave maps to continuum mechanics. In Sect.3 we study
the kinematics of relativistic elasticity, namely the geometry of
configurations. In Sect.4 we apply the results of Sect.2 to a theory of
configurations, namely relativistic elasticity. Barotropic perfect fluids turn
out to be a special case. But fluids in the Eulerian picture are usually
described by the 4-velocity, energy density and pressure, all viewed as fields
on $M$, which obey a 1st order system of partial differential equations
(PDE’s). There is no material manifold and no configurations.
In Sect.5 we describe a formulation in this spirit, due to Tahvildar-Zadeh
[10], of elasticity in the so-called isotropic case. The next two sections
concern the subjects of the nonrelativistic limit and linearization at a
stressfree state. In Sect.8 there is a quick account of causality and local
wellposedness, which largely draws on [11]. In Sect.9 we treat conditions at
the matter-vacuum interface appropriate for freely floating bodies. Our
treatment is based on the variational identity (95), which is in turn based on
(5). These, to the best of our knowledge, are new. The resulting boundary
condition takes the form of the vanishing of normal stress on an a priori
unknown timelike hypersurface in spacetime, namely the boundary of the world
tube swept out by the congruence of timelike curves of the particles making up
the material manifold. In Sect.10 we for completeness derive the elastic
action in the material picture, although that is not used explicitly in the
present work. In Sect.11 we review some time independent problems which have
so far been solved. Finally, in Sect.12, we give a short description of what
is known so far about the fully coupled Einstein-elastic system. These last
two sections exclusively treat the case of finite bodies surrounded by vacuum.
Note, finally, that we have tried our presentation to be mostly self-contained
- with the exception of sections 8,11 and 12 which have review character.

##  2 Lagrangian theory of maps

A careful introduction to the Lagrangian theory of maps is given in [11], see
also [12]. Absent from these works is however a discussion of the role of
diffeomorphism invariance, which is a main focus here. The material in the
present section is largely a generalization of [13].
Let $f^{A}(x^{\mu})\,,A=1,..m$ be maps from spacetime $(M,g_{\mu\nu})$ with
$g$ of signature $(-+++)$ to an $m$-dimensional manifold $N$. The Lagrange
function $L$ should be $L=L(f^{A},\partial_{\mu}f^{A};g^{\mu\nu})$. For the
time being think of $N$ as $\mathbb{R}^{m}$ or a manifold in a fixed chart.
Our basic requirement is: $L$ is a scalar under general diffeomorphisms of
$M$. Infinitesimally this means that

$\xi^{\mu}\partial_{\mu}\,L=\frac{\partial L}{\partial
f^{A}}\mathcal{L}_{\xi}f^{A}(x)+\frac{\partial
L}{\partial\partial_{\mu}f^{A}}\mathcal{L}_{\xi}\partial_{\mu}f^{A}(x)+\frac{\partial
L}{\partial g^{\mu\nu}}\mathcal{L}_{\xi}g^{\mu\nu}$ (1)

Expanding the left side in (1) and using the arbitrariness of the vector field
$\xi$, we find that

$(\partial_{\mu}f^{A})\frac{\partial
L}{\partial_{\nu}f^{A}}=2g^{\lambda\nu}\frac{\partial L}{\partial
g^{\lambda\mu}}$ (2)

The relation (2) is in particular satisfied when

$L=L(f^{A},H^{BC})\hskip 14.22636pt\mathrm{where}\hskip
11.38092ptH^{AB}=(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})g^{\mu\nu}$ (3)

Suppose $H^{AB}$ is non-degenerate, which in particular entails that $f$ is a
submersion, i.e. $\partial_{\mu}f^{A}$ has maximal rank (in particular $m\leq
4$). Then (2) actually implies $L=L(f^{A},H^{BC})$. To see this first note
that in that case the tangent space of $(M,g)$ at the point $x$ in the fiber
over $X=f(x)$ splits into an $(n-m)$ dimensional vertical (i.e. spanned by
vectors in the null space of $\partial_{\mu}f^{A}$) distribution and an
orthogonal $m$ dimensional horizontal distribution, and $H^{AB}$ can be seen
as the inverse metric on the horizontal distribution. But (2) implies that

$\frac{\partial L}{\partial g^{\mu\nu}}v^{\nu}=0$ (4)

for $v^{\mu}$ any vertical direction. Thus $L$ can depend on $g^{\mu\nu}$ only
via $H^{AB}$, so $L=L(f,^{A},\partial_{\mu}f^{B},H^{CD})$. Inserting this back
into (2) yields that actually $L=L(f^{A},H^{BC})$, which ends the proof.
Note that (4) also means that $T_{\mu\nu}$ has no mixed vertical-horizontal
components. The identity (2) has further important implications.
Theorem:
(F1) the canonical stress-energy tensor
$\mathcal{T}_{\mu}{}^{\nu}=(\partial_{\mu}f^{A})\frac{\partial
L}{\partial_{\nu}f^{A}}-L\,\delta_{\mu}{}^{\nu}$ satisfies
$g_{\nu\rho}\mathcal{T}_{\mu}{}^{\rho}=T_{\mu\nu}$, where
$T_{\mu\nu}=2\frac{\partial L}{\partial g^{\mu\nu}}-Lg_{\mu\nu}$, that is the
r.h. side of the Einstein equations. This identity is a special case of the
Belinfante-Rosenfeld theorem (see [14] and references therein.)
(F2) The Euler-Lagrange equations for the action
$S=\int_{M}L\sqrt{-g}\,d^{4}x$ imply $\nabla_{\nu}T_{\mu}{}^{\nu}=0$ and the
latter 4 equations are in general not independent. More precisely, there holds
the (Noether-type) ’off-shell’ identity (see also [14])

$\nabla_{\nu}T_{\mu}{}^{\nu}=-(\partial_{\mu}f^{A})\,\mathcal{E}_{A}\,,$ (5)

where

$-\mathcal{E}_{A}=\frac{1}{\sqrt{-g}}\,\partial_{\mu}\left(\sqrt{-g}\,\frac{\partial
L}{\partial_{\mu}f^{A}}\right)-\frac{\partial L}{\partial f^{A}}$ (6)

In the case $f$ is a submersion, i.e. $\partial_{\mu}f^{A}$ has rank
$m=\mathrm{dim}N$, $\nabla_{\nu}T_{\mu}{}^{\nu}=0$ is actually sufficient for
$\mathcal{E}_{A}=0$.
Proof: (F1) is a straightforward verification based on (2). (F2) is also easy,
using $\mathcal{T}_{\mu}{}^{\nu}$ on the left side.
One way to think of the identity (5) is that it is a direct expression of
diffeomorphism invariance of the action in the same way as the contracted
Bianchi identity for the Einstein tensor is a consequence of the
diffeomorphism invariance of the Hilbert action. To see this vary the action
$S[f,\partial f;g]$ under a 1-parameter family of diffeomorphisms generated by
a vector field $\xi^{\mu}$, there results

$0=\frac{1}{2}\int
T_{\mu\nu}(\mathcal{L}_{\xi}g^{\mu\nu})\sqrt{-g}\,d^{4}x+\frac{d}{d\epsilon}|_{\epsilon=0}\,S[f+\epsilon\mathcal{L}_{\xi}f;g]\,,$
(7)

where integration is carried out over some fixed domain in spacetime and
$\xi^{\mu}$ is compactly supported there. Now the 2nd term in (7) equals
$\int\mathcal{E}_{A}(\xi^{\mu}\partial_{\mu}f^{A})dV_{g}$. Using that
$\mathcal{L}_{\xi}g^{\mu\nu}=-2\nabla^{(\mu}\xi^{\nu)}$ and integrating by
parts the 1st term in (7) gives $\int(\nabla^{\nu}T_{\mu\nu})\xi^{\mu}dV_{g}$.
So (5) follows using the arbitrariness of $\xi^{\mu}$.
(F2) implies: Provided that the matter system has a well-posed Cauchy problem
in the background of any given spacetime (’test case’), the combined matter-
Einstein system is also well-posed (see e.g.[15]).

Clearly

$-\mathcal{E}_{A}=\frac{\partial^{2}L}{\partial(\partial_{\mu}f^{A})\partial(\partial_{\nu}f^{B})}\,\partial_{\mu}\partial_{\nu}f^{B}+\mathrm{l.o.}$

More explicitly (setting
$M^{\mu\nu}_{AB}=\frac{\partial^{2}L}{\partial(\partial_{\mu}f^{A})\partial(\partial_{\nu}f^{B})}$)

$M^{\mu\nu}_{AB}=2\,\frac{\partial L}{\partial
H^{AB}}\,g^{\mu\nu}+4\,\frac{\partial^{2}L}{\partial H^{AC}\partial
H^{BD}}(\partial_{\rho}f^{C})(\partial_{\sigma}f^{D})g^{\mu\rho}g^{\nu\sigma}$

Note: only $M^{(\mu\nu)}_{\,AB}$ contributes to the equations of motion (EOM).
So far the $f^{A}$’s were maps to some manifold $N$ in a fixed coordinate
system $X^{A}$: $f^{A}:x^{\mu}\in M\mapsto X^{A}=f^{A}(x^{\mu})\in N$. We
consider 2 important cases:
(i) Homogenous case: $N$ is viewed as an affine vector space and $f^{A}$ a
collection of scalars subject only to affine transformations. Suppose
furthermore that $L=L(H^{AB})$. Then the exact Euler-Lagrange expression can
be written as

$-\mathcal{E}_{A}=M^{\mu\nu}_{AB}\nabla_{\mu}\partial_{\nu}f^{B}$

Proof: note $M^{\mu\nu}_{AB}(f,\partial f;g)$ transforms as a $(2,0)$-tensor
under spacetime coordinate transformations and $\mathcal{E}_{A}(f,\partial
f,\partial^{2}f;g,\partial g)(x)$ as a scalar. Then, taking normal coordinates
centered at $x$ gives the result. The ’fully explicit’ proof is remarkably
tricky.

(ii) Isotropic case: the target space $N$ is a manifold equipped with a
Riemannian metric $G_{AB}(X)$, and $L$ is a function of the principal
invariants (or what is the same: eigenvalues) of $H^{A}{}_{B}$, where
$H^{D}{}_{A}=G_{AC}(f)(\partial_{\mu}f^{C})(\partial_{\nu}f^{D})g^{\mu\nu}$.
For example, when $m=3$, the Lagrangian is then of the form

$L=(H^{A}{}_{A},H^{A}{}_{B}H^{B}{}_{A},\mathrm{det}(H^{A}{}_{B}))$ (8)

Then the theory is also invariant under diffeomorphisms of $N$. Furthermore
the exact Euler-Lagrange expression is

$-\mathcal{E}_{A}=M^{\mu\nu}_{AB}\overline{\nabla}_{\mu}\partial_{\nu}f^{B}$
(9)

where

$\overline{\nabla}_{\mu}\partial_{\nu}f^{A}=\partial_{\mu}\partial_{\nu}f^{A}-\Gamma^{\rho}_{\mu\nu}\partial_{\rho}f^{A}+\Gamma^{A}_{\\!BC}(\partial_{\mu}f^{B})(\partial_{\nu}f^{C})\,,$
(10)

where $\Gamma^{A}_{\\!BC}$ are the Christoffel symbols of $G_{AB}$. The
isotropic and homogenous case is where $G_{AB}=\delta_{AB}$. Suppose, in the
isotropic case, we take $L=H^{A}{}_{A}=G_{AB}(f)H^{AB}$. The EOM is then
linear in $f$ when $G_{AB}$ is flat and semilinear otherwise - these are
called wave maps in the mathematics [16] and sigma models in the physics [17]
literature. One can, as done here, also consider the general isotropic case.
This happens e.g. for the Skyrme model without the mass term [18] or isotropic
relativistic elasticity. In this case the EOM are quasilinear.

##  3 Geometry of configurations

We now specialize to the case $N=\mathcal{B}$ called material manifold or
body, a domain in $\mathbb{R}^{3}$ with smooth boundary or all of
$\mathbb{R}^{3}$, thought of as the collection of elements making up the
material. Note: no further structure is imposed on $\mathcal{B}$ to start
with. For bodies of everyday life that is defined by the ’making’ of the body,
for astrophysical objects it is largely determined by gravity.
Definition: The field $f:(M,g)\rightarrow\mathcal{B}$ is called a
configuration if $\partial_{\mu}f^{A}$ has everywhere maximal rank and $f$ is
surjective with level sets timelike curves in $M$.
In other words: a configuration is a surjective submersion with timelike
fibers. Codimension 1 means we are dealing with media composed of pointlike
objects (’particles’). Timelike fibers means we are dealing with massive
particles.
Given a configuration $f^{A}$, there is then - subject to time orientation - a
unique vector field $u^{\mu}$, s.th.

$u^{\mu}\partial_{\mu}f^{A}=0\,,\hskip
28.45274ptg_{\mu\nu}u^{\mu}u^{\nu}=-1\,.$ (11)

Note that the concept of configuration is intrinsically nonlinear.
Let $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ be the metric on the ’horizontal
distribution’ defined by $u^{\mu}$. There then exists a unique field
$\psi^{\mu}{}_{A}$ with $\psi^{\mu}{}_{A}u_{\mu}=0$ such that

$\psi^{\mu}{}_{A}\,\partial_{\nu}f^{A}=h^{\mu}{}_{\nu}\,,\hskip
28.45274pt\psi^{\mu}{}_{B}\,\partial_{\mu}f^{A}=\delta^{A}{}_{B}\,.$ (12)

Furthermore $H^{AB}=(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})g^{\mu\nu}$ is
now positive definite.
Note: $H^{AB},u^{\mu},h_{\mu\nu},\psi^{\mu}{}_{A}$ are all algebraic functions
of $(\partial f,g)$. Some representative formulae in this regard are:

$\psi^{\mu}{}_{A}\psi^{\nu}{}_{B}\,H^{AB}=h^{\mu\nu}\,,\hskip
14.22636pt\partial_{\nu}f^{A}=\psi^{\mu}{}_{B}\,h_{\mu\nu}\,H^{AB}$ (13)

$\frac{\partial
u^{\mu}}{\partial(\partial_{\nu}f^{A})}=-u^{\nu}\psi^{\mu}{}_{A}\,,\hskip
39.83368pt\frac{\partial\psi^{\mu}{}_{A}}{\partial(\partial_{\nu}f^{B})}=-\psi^{\mu}{}_{B}\psi^{\nu}{}_{A}-H_{AB}u^{\mu}u^{\nu}\,,$
(14)

where $H_{AB}H^{BC}=\delta^{C}{}_{A}$ or

$\frac{\partial u^{\mu}}{\partial
g^{\nu\rho}}=-\frac{1}{2}u^{\mu}u_{\nu}u_{\rho}\,,\hskip
42.67912pt\frac{\partial\psi^{\mu}{}_{A}}{\partial
g^{\rho\sigma}}=-u^{\mu}h_{\nu(\rho}u_{\sigma)}\psi^{\nu}{}_{A}$ (15)

We will later write down some coordinate formulae.

###  3.1 Uncontorted configurations

In field theory time independent fields are often used as reference fields.
For configurations this means that there is a timelike Killing field
$\xi^{\mu}$ on $(M,g)$ such that
$\mathcal{L}_{\xi}f^{A}=\xi^{\mu}\partial_{\mu}f^{A}=0$, which in turn says
that the particle flow defined by $f^{A}$ is tangent to Killing trajectories:
$u^{\mu}\sim\xi^{\mu}$ with $\xi$ a Killing field on $(M,g)$. In the case of
configurations there is available a slightly weaker concept, which, for
reasons explained below, we call uncontorted configurations and which is the
analogue of Riemannian submersions in the case $g_{\mu\nu}$ is Riemannian.
Uncontorted configurations are as close as configurations can get to being
isometries.
Definition: A configuration is called uncontorted if $u^{\mu}$ is Born-rigid
(see [19]), i.e.

$\mathcal{L}_{u}h_{\mu\nu}=0$ (16)

Clearly time independence implies uncontortedness. To explain the relation
with more standard terminology recall the decomposition

$h_{\mu}{}^{\mu^{\prime}}h_{\nu}{}^{\nu^{\prime}}\,\nabla_{(\mu^{\prime}}u_{\nu^{\prime})}=\sigma_{\mu\nu}+\frac{1}{3}h_{\mu\nu}\Theta$
(17)

into shear $\sigma_{\mu\nu}$ and divergence $\Theta=\nabla_{\mu}u^{\mu}$. So a
configuration is uncontorted iff its particle world lines are shear- and
expansionfree. To explain the geometric meaning of non-contortedness calculate

$\mathcal{L}_{u}h_{\mu\nu}=\mathcal{L}_{u}[(\partial_{\mu}f^{A})(\partial_{\nu}f^{B}H_{AB})]=(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})u^{\rho}\partial_{\rho}H_{AB}\,,$
(18)

where we were using (13) in the first equality and $[\mathcal{L},d]=0$ in the
second. So uncontortedness is equivalent to

$u^{\mu}\partial_{\mu}H_{AB}=0\hskip 14.22636pt\mathrm{or}\hskip
14.22636ptu^{\mu}\partial_{\mu}H^{AB}=0\,,$ (19)

which in turn means that there exists a Riemannian metric $G_{AB}(X)$ on
$\mathcal{B}$ so that $H_{AB}(x)=G_{AB}(f(x))$, i.e. $h_{\mu\nu}$ is the pull-
back of G under $f$.
In a similar fashion a configuration whose particle flow is shearfree defines
a conformal metric on $\mathcal{B}$ which pulls back to the conformal metric
defined by $h$, and a configuration which is incompressible (i.e. has non-
expanding flow: $\Theta=0$) defines a volume form $\Omega_{ABC}$ on
$\mathcal{B}$ which under $f$ pulls back to the 3-form
$\epsilon_{\mu\nu\rho\sigma}u^{\sigma}$ on $M$. Shear and expansion are pieces
of information referring to the horizontal distribution defined by $f$. The
remaining pieces of horizontal data are the acceleration
$u^{\nu}\nabla_{\nu}u^{\mu}$ and rotation $\omega_{\mu\nu}$ defined by

$\omega_{\mu\nu}=h_{\mu}{}^{\mu^{\prime}}h_{\nu}{}^{\nu^{\prime}}\,\nabla_{[\mu^{\prime}}u_{\nu^{\prime}]}\,,$
(20)

which is the obstruction to integrability of the horizontal distribution.
Note finally that non-contorted configurations which are not time independent
are rare, since Born-rigid congruences are generically Killing. For example it
is a theorem of Herglotz and Fritz Noether (see [19]) that a congruence on
Minkowski space which is Born-rigid and non-rotating has to be Killing.

###  3.2 Back and forth between $\mathcal{B}$ and $M$ under a configuration

The relation between $h_{\mu\nu}$ and $G_{AB}$ in the previous subsection is a
special case of the following observation: Let $\Omega_{A..}$ be a covariant
tensor field on $\mathcal{B}$. Then its pull-back $(f^{\star}\Omega)_{\mu..}$
under a configuration $f^{A}$ is a covariant tensor field on $M$ of the same
type and is horizontal, i.e. gives zero under contraction with $u^{\mu}$ over
any index. Furthermore $(\mathcal{L}_{u}f^{\star}\Omega)_{\mu..}=0$. The
latter fact can be easily checked explicitly by noting that
$\mathcal{L}_{u}\partial_{\mu}f^{A}=0$.
Conversely, let $o_{\mu..}$ be a covariant, horizontal tensor field on $M$
with $\mathcal{L}_{u}o_{\mu..}$ vanishing for the flow vector field $u^{\mu}$
corresponding to some configuration $f^{A}$. Then $o_{\mu..}$ arises as pull
back of a covariant tensor field $\Omega_{A..}$. To see this pick a
hypersurface $\Sigma$ in $M$ which is everywhere transversal to $u^{\mu}$,
e.g. spacelike. Thus the map $\bar{f}_{\Sigma}:\Sigma\rightarrow\mathcal{B}$
is invertible and onto. Define a tensor field $\Omega_{A..}$ on $\mathcal{B}$
by pull back of $o_{\mu..}|_{\Sigma}$ under $\bar{f}^{-1}$. The pull back of
$\Omega_{A..}$ along all of $M$ is Lie derived under $u^{\mu}$ and coincides
with $o_{\mu..}$ initially, thus everywhere. This ends the proof.

###  3.3 Some coordinate identities

Note first the map $f^{A}$ is a configuration onto its image, iff $H^{AB}$ is
positive definite and, in local coordinates $(t,x^{i})$ with $t=$ constant
being spacelike hypersurfaces, the matrix $\partial_{i}f^{A}$ is
nonsingular222Actually it is not hard to see that the latter condition follows
from the former..
The coordinate velocity
$v^{\mu}\partial_{\mu}=\partial_{t}+V^{i}\partial_{i}$, since
$v^{\mu}\partial_{\mu}f^{A}=0$, is in terms of the configuration given by
$V^{i}=-\phi^{i}{}_{A}\,\partial_{t}f^{A}$, where
$\phi^{i}{}_{A}\,\partial_{i}f^{B}=\delta^{B}{}_{A}$.
Take more specifically ADM coordinates, i.e.$(t,x^{i})$, where
$g(\partial_{t}-Y^{j}\partial_{j},\partial_{i})=0$. Setting
$g^{-1}(dt,dt)=-N^{-2}$ and $g(\partial_{i},\partial_{j})=g_{ij}$:

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-N^{2}dt^{2}+g_{ij}(dx^{i}+Y^{i}dt)(dx^{j}+Y^{j}dt)$
(21)
$g^{\mu\nu}\partial_{\mu}\partial_{\nu}=-\frac{1}{N^{2}}(\partial_{t}-Y^{i}\partial_{i})^{2}+g^{ij}\partial_{i}\partial_{j}$
(22)
$u^{\mu}\partial_{\mu}=\frac{(\partial_{t}-Y^{i}\partial_{i})+W^{i}\partial_{i}}{(N^{2}-W^{2})^{\frac{1}{2}}}\,,\hskip
28.45274ptW^{i}=Y^{i}+V^{i}$ (23)

and $W^{2}=W^{i}W^{j}g_{ij}=W^{i}W_{i}$. We call the quantity $W^{i}$ relative
shift, following a suggestion by L.Andersson (private communication).
Furthermore

$(\partial_{\mu}f^{A})dx^{\mu}=(\partial_{i}f^{A})[-W^{i}dt+(dx^{i}+Y^{i}dt)]$
(24)

$H^{AB}=(\partial_{i}f^{A})(\partial_{j}f^{B})\,\left(g^{ij}-\frac{W^{i}W^{j}}{N^{2}}\right)$
(25)

$u_{\mu}dx^{\mu}=\frac{-N^{2}dt+W_{i}(dx^{i}+Y^{i}dt)}{(N^{2}-W^{2})^{\frac{1}{2}}}$
(26)

$\psi^{\mu}{}_{A}\,\partial_{\mu}=\phi^{i}{}_{A}\left[\frac{W_{i}}{N^{2}-W^{2}}(\partial_{t}-Y^{l}\partial_{l})+\left(\delta^{j}{}_{i}+\frac{W^{j}W_{i}}{N^{2}-W^{2}}\right)\partial_{j}\right]$
(27)

###  3.4 Galilean configurations

A Galilean spacetime is affine $\mathbb{R}^{4}$ with the degenerate
contravariant metric
$e^{\mu\nu}\partial_{\mu}\partial_{\nu}=\delta^{ij}\partial_{i}\partial_{j}$
and covector $\tau_{\mu}dx^{\mu}=dt$ annihilated by $e^{\mu\nu}$. Affine
transformations leaving $M=(\mathbb{R}^{4},e,\tau)$ invariant form the 10
dimensional Galilei group.
Associated with a configuration $f:M\rightarrow\mathcal{B}$ is the 4-velocity
$v^{\mu}\partial_{\mu}$, uniquely given by $v^{\mu}\partial_{\mu}f^{A}=0$ and
$v^{\mu}\tau_{\mu}=1$, so

$v^{\mu}\partial_{\mu}=\partial_{t}+V^{i}\partial_{i}=\partial_{t}-\phi^{i}{}_{A}\,\partial_{i}f^{A}\,,\,\,\mathrm{where}\,\,\,\phi^{i}{}_{A}\,\partial_{j}f^{A}=\delta^{i}{}_{j}$
(28)

and

$(\partial_{\mu}f^{A})dx^{\mu}=(\partial_{i}f^{A})(dx^{i}-V^{i}dt)$ (29)

The field $\psi^{\mu}{}_{A}$ is now defined by

$\psi^{\mu}{}_{A}(\partial_{\nu}f^{A})=\delta^{\mu}{}_{\nu}-v^{\mu}\tau_{\nu}$
(30)

which gives

$\psi^{\mu}{}_{A}\partial_{\mu}=\phi^{i}{}_{A}\partial_{i}$ (31)

Note: both $\psi^{\mu}{}_{A}$ and
$K^{AB}=(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})e^{\mu\nu}$ depend only on
spatial derivatives of $f$.
The analogue of uncontorted configurations are now those for which $v^{\mu}$
leaves $e^{\mu\nu}$ and $\tau_{\mu}$ invariant. These are the infinite
dimensional group of rigid body motions, i.e. a time dependent rotations plus
time dependent spatial translations plus time translations. Like in the
relativistic case, the uncontorted configurations considered here will always
be ones constant along time translations or rigid rotations.

###  3.5 Time independent configurations

For a time independent background the ADM variables $(N,Y^{i},g_{jk})$ are
unsuitable. We rather take $(V,Y_{i},h_{jk})$ defined by

$V^{2}=-\xi^{\mu}\xi^{\nu}g_{\mu\nu}=N^{2}-Y^{2},\,\,\,Y_{i}=g_{ij}Y^{j},\,\,\,h_{ij}=g_{ij}-\frac{1}{N^{2}}Y_{i}Y_{j}$
(32)

whence

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-V^{2}(dt+Y_{i}dx^{i})^{2}+h_{ij}dx^{i}dx^{j}$
(33)

Geometrically $h_{ij}$ is the metric on the space $Q$, which is the quotient
space of $(M,g_{\mu\nu})$ under the action of $\xi^{\mu}$.
The Killing vector $\xi^{\mu}\partial_{\mu}=\partial_{t}$ satisfies
($u^{\mu}=V^{-\frac{1}{2}}\xi^{\mu}$)

$u^{\mu}\partial_{\mu}=\frac{1}{V}\partial_{t}\,,\,\,\,u_{\mu}dx^{\mu}=-V(dt+Y_{i}dx^{i})$
(34)
$g^{\mu\nu}\partial_{\mu}\partial_{\nu}=-\frac{1}{V^{2}}\,\partial_{t}^{2}+h^{ij}(\partial_{i}-Y_{j}dt)(\partial_{j}-Y_{j}dt)$
(35) $(\partial_{\mu}f^{A})dx^{\mu}=(\partial_{i}f^{A})dx^{i}\,,\hskip
19.91684ptH^{AB}=(\partial_{i}f^{A})(\partial_{j}f^{B})h^{ij}$ (36)

We add 2 cautionary remarks on rotation:

  * •

If the Killing vector is rotating, i.e.

$\omega_{\mu\nu}\,dx^{\mu}dx^{\nu}=-V\,D_{[i}Y_{j]}\,dx^{i}dx^{j}$ (37)

is non-zero, time independence does not necessarily mean ’stationarity’ in the
usual sense. Take Minkowski case with $\xi$ the ’helical Killing vector
$\xi^{\mu}\partial_{\mu}=\partial_{t}+\omega\,\eta^{i}\partial_{i}$, where
$\eta$ is a spatial rotation, so the configuration is ’time independent in a
rotating frame’.

  * •

Secondly, the theory as described breaks down if $f^{-1}(\mathcal{B})$ is too
large in the sense of extending across the ’light cylinder’ where $\xi$
becomes null.

##  4 Relativistic elasticity

Elasticity is a diffeomorphism invariant theory of configurations on a
spacetime $(M,g_{\mu\nu})$, thus defined by an action

$S[f;g]=\int\rho(f^{A},\partial_{\mu}f^{B};g^{\mu\nu})\,\sqrt{-g}\,d^{4}x$
(38)

with the $f^{A}$’s configurations and with the Lagrangian (called $L=\rho$ for
reasons which will become clear shortly) diffeomorphism invariant as described
in Sect.2. We next derive an intriguing-looking variational identity which
among diffeomorphism invariant theories of maps is specific to elasticity.
Theorem: Let $\delta\rho$ be the first variation of $\rho$ w.r. to $f$, i.e.

$\delta\rho=\frac{\partial\rho}{\partial f^{A}}\,\delta
f^{A}+\frac{\partial\rho}{\partial\partial_{\mu}f^{A}}\,\partial_{\mu}\delta
f^{A}\,.$ (39)

Then

$\delta\rho-\nabla_{\mu}(\rho\,\psi^{\mu}{}_{A}\delta
f^{A})=T^{\mu}{}_{\nu}\nabla_{\mu}(\psi^{\nu}{}_{A}\delta f^{A})\,.$ (40)

Proof: First note that

$\delta\rho=\mathcal{E}_{A}\delta
f^{A}+\nabla_{\mu}\left(\frac{\partial\rho}{\partial\partial_{\mu}f^{A}}\,\delta
f^{A}\right)=-(\psi^{\nu}{}_{A}\nabla_{\mu}T^{\mu}{}_{\nu})\delta
f^{A}+\nabla_{\mu}\left(\frac{\partial\rho}{\partial\partial_{\mu}f^{A}}\right)\,,$
(41)

where (5) was used in the 1st term on the right. Writing

$-(\psi^{\nu}{}_{A}\nabla_{\mu}T^{\mu}{}_{\nu})\partial
f^{A}=T^{\mu}{}_{\nu}\nabla_{\mu}(\psi^{\nu}{}_{A}\delta
f^{A})-\nabla_{\mu}(T^{\mu}{}_{\nu}\psi^{\nu}{}_{A}\delta f^{A})\,,$ (42)

using
$T^{\mu}{}_{\nu}=(\partial_{\nu}f^{A})\frac{\partial\rho}{\partial\partial_{\mu}f^{A}}-\rho\,\delta^{\mu}{}_{\nu}$
in the 2nd term in (42) and inserting into (41), there is one cancellation,
and (40) results. This ends the proof.
Eq.(40) of course entails the result that the EOM are given by
$\psi^{\mu}{}_{A}\nabla_{\nu}T^{\nu}{}_{\mu}=0$, which in turn is equivalent
to $\nabla_{\nu}T^{\nu}{}_{\mu}=0$. The total divergence on the l.h. side of
(40) will play a role when boundary conditions are treated in Sect.9.
To further develop the theory as a branch of continuum mechanics, the missing
piece of structure is a notion of ’particle number density’. This is done by
picking a volume form $\Omega_{ABC}(X)$ on $\mathcal{B}$ and setting

$(f^{\star}\Omega)_{\mu\nu\lambda}:=J_{\mu\nu\lambda}=(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})(\partial_{\lambda}f^{C})\Omega_{ABC}=n\,\epsilon_{\mu\nu\lambda\sigma}u^{\sigma}\,,\hskip
19.91684ptn>0$ (43)

Note that $n>0$ in (43) entails the assumption that $f^{A}$ preserve
orientation. Eq.(43) defines $n$ algebraically in terms of $(f,\partial f;g)$
and is invariant under diffeomorphisms preserving orientation. Moreover

$6\,n^{2}=H^{AA^{\prime}}H^{BB^{\prime}}H^{CC^{\prime}}\Omega_{ABC\,}\Omega_{A^{\prime}B^{\prime}C^{\prime}}$
(44)

and

$\frac{\partial n}{\partial\partial_{\mu}f^{A}}=n\psi^{\mu}{}_{A}\,.$ (45)

Most importantly, $\nabla_{\mu}(nu^{\mu})=0$ holds as an identity. To see this
recall that the exterior differential $d$ commutes with pull back and note
that $(d\Omega)_{ABCD}$ is zero.

Taking $\Omega_{ABC}=\varepsilon_{ABC}$, and in a local Lorentzian frame
$(t,x^{i})$ at $p$ of $M$ where
$v^{\mu}\partial_{\mu}|_{p}=(\partial_{t}+V^{i}\partial_{i})|_{p}$, there
holds

$n=\mathrm{det}(\partial_{i}f^{A})(1-V^{2})^{\frac{1}{2}}$ (46)

More generally, in the language of Sect.(3.2)

$n^{2}=\mathrm{det}(H^{AB})=[\mathrm{det}(\partial_{k}f^{B})]^{2}\,\mathrm{det}(g^{ij}-\frac{W^{i}W^{j}}{N^{2}})$
(47)

The importance of the concept of particle number density will become apparent
soon. Furthermore, as opposed to the nonrelativistic theory, particle number
carries information about kinetic energy.
We now come back to the elastic Lagrangian. From Sect.2 we know that
diffeomorphism invariance implies

$\frac{\partial\rho}{\partial g^{\mu\nu}}u^{\mu}=0$ (48)

so $\frac{\partial\rho}{\partial g^{\mu\nu}}$ is ’horizontal’. For example

$\frac{\partial n}{\partial
g^{\mu\nu}}=\frac{n}{2}h_{\mu\nu}\,\,\Longrightarrow\,\,\,2\,\frac{\partial
n}{\partial g^{\mu\nu}}-n\,g_{\mu\nu}=nu_{\mu}u_{\nu}\,.$ (49)

In other words, the energy momentum tensor of the Lagrangian $\rho=n$ is
’purely vertical’, i.e. $T_{\mu\nu}=nu_{\mu}u_{\nu}$ \- which is the energy
momentum tensor of dust. Thus, if we factorize $\rho$ as $\rho=n\,\epsilon$,
this gives the linear decomposition

$T_{\mu\nu}=n\epsilon u_{\mu}u_{\nu}+2n\frac{\partial\epsilon}{\partial
g^{\mu\nu}}$ (50)

into a vertical ’energy- momentum part’ and a horizontal ’stress part’. The
latter is of course the relativistic version of (minus) the Cauchy stress
tensor. Furthermore $T^{\mu}{}_{\nu}u^{\nu}=\rho u^{\mu}$ which justifies the
terminology $L=n\epsilon=\rho$, the rest energy density of the elastic
material. The quantity $\epsilon$ is the relativistic version of the stored
energy function of nonrelativistic elasticity. Note that $T^{\mu\nu}$ can
never be zero, provided that $\epsilon>0$.

Consider the special case where $\rho=n\epsilon(n)$. It turns out that

$T^{\mu\nu}=n\epsilon\,u^{\mu}u^{\nu}+n^{2}\epsilon^{\prime}h^{\mu\nu}$ (51)

i.e. a barotropic fluid with energy density $\rho=n\epsilon$, internal energy
$\epsilon$ and pressure $p=n^{2}\epsilon^{\prime}$. The function $\epsilon(n)$
for fluids and $\epsilon(f^{A},H^{BC})$ in general is determined by the
material in question, i.e. plays the role of equation of state - ’stored
energy’ in classical elasticity. Finding the ’right’ EOS for a particular
material is a mixture of phenomenology together with ellipticity conditions
hopefully guaranteeing good causal properties or existence of equilibria in
the time independent case.
It might occasionally be useful to change the Lagrangian by changing the
volume form $\Omega_{ABC}$ for fixed stored energy $\epsilon$. Here is how the
EOM changes under $\overline{L}=\bar{\Omega}(f)L$:

$\overline{\mathcal{E}}_{A}=\overline{\Omega}\,\mathcal{E}_{A}-\left[(\partial_{\nu}f^{B})\frac{\partial\rho}{\partial_{\nu}f^{A}}-\rho\,\delta_{A}{}^{B}\right]\partial_{B}\overline{\Omega}$
(52)

The quantity
$M^{\mu\nu}_{AB}=\frac{\partial^{2}\rho}{\partial(\partial_{\mu}f^{A})\partial(\partial_{\nu}f^{B})}$,
similarly to $T_{\mu\nu}$, will have a vertical part, determining the leading-
order $\partial_{t}^{2}$ term in the EOM plus a horizontal part containing the
spatial derivatives. In more detail (setting
$\tau_{AB}=2\,\frac{\partial\epsilon}{\partial H^{AB}}$)

$T^{\mu\nu}=n\epsilon
u^{\mu}u^{\nu}+n\,\tau_{AB}(\partial_{\sigma}f^{A})(\partial_{\lambda}f^{B})g^{\mu\sigma}g^{\nu\lambda}\,,$
(53)

then $M^{\mu\nu}_{AB}=M^{\nu\mu}_{BA}$ is given by

$M^{\mu\nu}_{AB}=-n(\epsilon\,H_{AB}+\tau_{AB})u^{\mu}u^{\nu}+U_{ACBD}(\partial_{\sigma}f^{C})(\partial_{\lambda}f^{D})g^{\mu\sigma}g^{\nu\lambda}$
(54)

where $U_{ACBD}=U_{BDAC}$ satifies

$U_{ACBD}=n\left(\tau_{AB}H_{CD}+\tau_{AC}H_{BD}+\tau_{BD}H_{AC}+2\,\frac{\partial\tau_{AB}}{\partial
H^{CD}}+2\,\epsilon H_{A[C}H_{D]B}\right)$ (55)

Note $M^{\mu\nu}_{AB}$ depends on $(f,\partial f)$, so

$-\mathcal{E}_{A}=0=M^{\mu\nu}_{AB}\partial_{\mu}\partial_{\nu}f^{B}+\mathrm{l.o.}$
(56)

is a 2nd order quasilinear system of PDE’s for $f^{A}$.

The local initial value problem in the analytic case with initial surface
$\Psi(x)=0$ is solvable iff
$M^{\mu\nu}_{AB}(\partial_{\mu}\Psi)(\partial_{\nu}\Psi)$ is invertible. This
can certainly be achieved when $|\tau_{AB}|$ and
$|\frac{\partial\tau_{AB}}{\partial H^{CD}}|$ are both $\ll\epsilon$ and
$|H^{AB}-\delta^{AB}|$ is small. Note in particular that conditions on
$T_{\mu\nu}$ like the dominant energy condition do not imply wellposedness -
which is of course known already for perfect fluids, since sound speed needs
2nd derivatives of $\epsilon$.
Finally in this section we turn to the issue of simple solutions, which do not
require solving any partial differential equations. In some field theories
such are easy to find, typically ’vacuum’ solutions e.g. spacetimes with
constant curvature as solutions for the Einstein vacuum equations with
cosmological constant. In the case of elasticity, even in the background of
Minkowski spacetimes, this requires assumptions on the stored energy function.
As an example consider $(M,g)$ an ultrastatic spacetime, i.e. with a timelike
Killing vector which is hypersurface orthogonal and geodesic, so that in
suitable coordinates

$g=-dt^{2}+h_{ij}(x)dx^{i}dx^{j}$ (57)

For the body we take $(\mathcal{B},G_{AB})$, where $G_{AB}(X)dX^{A}dX^{B}$ is
isometric to $h_{ij}$, that is there exists a diffeomorphism
$\mathring{f}:x^{i}\mapsto X^{A}=\mathring{f}^{A}(x)$, s.th.

$G^{AB}(\mathring{f}(x))=(\partial_{i}\mathring{f}^{A})(x)(\partial_{j}\mathring{f}^{A})(x)h^{ij}(x)\hskip
11.38092pt\mathrm{or}\hskip
11.38092ptG_{AB}=(\mathring{\Phi}^{\star}h)_{AB}\,,$ (58)

where $\mathring{\Phi}^{\star}$ is pull-back under
$\mathring{\Phi}=(\mathring{f})^{-1}$, and for the volume element on
$\mathcal{B}$ we take $\sqrt{\mathrm{det}G}\,\varepsilon_{ABC}$.
Clearly $f^{A}(t,x)=\mathring{f}^{A}(x)$ is an allowable, in fact: non-
contorted, configuration with

$\mathring{H}^{AB}(t,x)=G^{AB}(\mathring{f}(x))$ (59)

Next require that the function $\epsilon(f^{A},H^{BC})$ satisfies

$\frac{\partial\epsilon(X,H^{CD})}{\partial H^{AB}}|_{H^{CD}=G^{CD}(X)}=0$
(60)

Then $\mathring{f}$ solves the field equations. To prove this assertion, first
write

$\mathring{T}^{\mu\nu}=\mathring{n}\mathring{\epsilon}u^{\mu}u^{\nu}+\mathring{t}^{\mu\nu}\,,$
(61)

where $u^{\mu}\partial_{\mu}=\partial_{t}$ and
$\mathring{t}^{\mu\nu}\partial_{\mu}\partial_{\nu}=2\mathring{n}\frac{\partial\epsilon(\mathring{f},H^{CD})}{\partial
H^{AB}}|_{H^{CD}=\mathring{H}^{CD}}(\partial_{i}\mathring{f}^{A})(\partial_{j}\mathring{f}^{B})h^{ik}h^{jl}\,\partial_{k}\partial_{l}$
is clearly zero. Then note that $u^{\mu}\partial_{\mu}\mathring{H}^{AB}=0$,
whence $\mathring{n}$ and $\mathring{\epsilon}$ are also invariant under
$u^{\mu}$. Furthermore $u^{\mu}$ is geodesic. Thus the first term in
$\mathring{T}^{\mu\nu}$ has zero divergence, which ends the proof.

##  5 Elasticity à la Tahvildar-Zadeh

As we have seen in the last section, perfect fluids are a special case of our
equations. However those are usually, and preferably, described in terms of
the quantities 4-velocity and energy density as function of pressure, all
viewed as fields on $M$, which obey a 1st order system of PDE’s. We repeat
that in our present framework the 4-velocity is a derived quantity, namely an
algebraic function of the partial derivatives of the configuration and of the
spacetime metric. Still one may ask if there is a formulation of elasticity in
the spirit of hydrodynamics, from which configurations whence the material
manifold have disappeared. In this section we describe such a formulation, due
to Tahvildar-Zadeh [10], corresponding to elasticity in the so-called
isotropic case. Here the $3\times 4=12$ partial derivatives of configurations
are replaced by the 3+6=9 independent quantities given by 4-velocity $u^{\mu}$
and a symmetric tensor $\gamma_{\mu\nu}$ describing ’strain’, constrained by
the conditions $u_{\mu}u^{\mu}=-1$ and $\gamma_{\mu\nu}u^{\nu}=0$, all subject
to a 1st order system of PDE’s. Missing from this formulation are the 3
degrees of freedom corresponding to rotations.
Suppose $\rho$ depends only on eigenvalues of $H^{AB}$ w.r. to some a priori
given metric $G_{AB}(X)$ on $\mathcal{B}$ (isotropic case). The pull-back of
$G$ under $f$,

$\gamma_{\mu\nu}(x)=(\partial_{\mu}f^{A})(x)(\partial_{\nu}f^{B})(x)\,G_{AB}(f(x))$
(62)

by fiat satisfies

$\mathcal{L}_{u}\gamma_{\mu\nu}=0\,,\,\hskip
36.98866pt\gamma_{\mu\nu}u^{\mu}=0$ (63)

and the non-zero eigenvalues of $\gamma_{\mu}{}^{\nu}$ are identical with the
eigenvalues of $H_{A}{}^{B}=G_{AC}H^{BC}$. By the Cayley-Hamilton theorem
$T_{\mu}{}^{\nu}$ is a linear combination of $(1,1)$-tensors
$u_{\mu}u^{\nu},h_{\mu}{}^{\nu},\gamma_{\mu}{}^{\nu},\gamma^{2}_{\mu}{}^{\,\nu}$
with coefficients depending only on the eigenvalues of $\gamma_{\mu}{}^{\nu}$.
Field equations are $\nabla_{\nu}T_{\mu}{}^{\nu}=0$ together with (63) and the
condition $u^{\mu}u_{\mu}=-1$. The identity
$u_{\mu}\nabla_{\nu}T_{\mu}{}^{\nu}=0$ will of course still hold, but particle
number conservation needs checking. Namely there holds

$6\,\gamma_{\mu[\mu^{\prime}}\gamma_{|\nu|\nu^{\prime}}\gamma_{|\rho|\rho^{\prime}]}=J_{\mu\nu\rho}\,J_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}$
(64)

Thus

$(\mathcal{L}_{u}J_{\mu\nu\rho})\,J_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}+J_{\mu\nu\rho}\,(\mathcal{L}_{u}J_{\mu^{\prime}\nu^{\prime}\rho^{\prime}})=0$
(65)

with $J_{\mu\nu\rho}$ nowhere zero. This further implies
$\mathcal{L}_{u}J_{\mu\nu\rho}=0$. But

$\mathcal{L}_{u}J=u\lrcorner\,dJ+d(u\lrcorner\,J)=u\lrcorner\,dJ$ (66)

Thus, since $dJ$ is a 4-form, $dJ=0$, which we set out to prove. Thus we have
a total of 6 independent 1st order equations for the 6 quantities
$(u^{\mu},\gamma_{\nu\rho})$, that is a closed system from which the material
manifold has disappeared. In [10] it is shown that, under reasonable
conditions on the stored energy, the system is weakly hyperbolic in the sense
that the characteristic speeds are real, i.e. the characteristic polynomial is
hyperbolic in the sense of Sect.8. Unfortunately this does not always imply
wellposedness. On the other hand, given a solution with data on some spacelike
hypersurface $\Sigma$ so that the pull back to $\Sigma$ of $\gamma_{\mu\nu}$
is the pull back of $G_{AB}$ under some map
$\bar{f}:\Sigma\rightarrow\mathcal{B}$, we can reconstruct the configuration
$f^{A}$ by solving $u^{\mu}\partial_{\mu}f^{A}=0$ with initial data
$f^{A}|_{\Sigma}=\bar{f}^{A}$.

##  6 Nonrelativistic limit

Take $(M,g)$ to be Minkowski spacetime $(\mathbb{R}^{4},\eta_{\mu\nu})$ with
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}=-\frac{1}{c^{2}}(\partial_{t})^{2}+\delta^{ij}\partial_{i}\partial_{j}$
and take

$\epsilon=m_{0}c^{2}+W(f,H^{AB})=m_{0}c^{2}+W(f,K^{AB})+O(\frac{1}{c^{2}})\,,$
(67)

where $m_{0}$ is (rest-mass)/(particle)(volume). We set $m_{0}=1$ from now on.
Furthermore

$H^{AB}=(\partial_{i}f^{A})(\partial_{j}f^{B})\left(\delta^{ij}-\frac{V^{i}V^{j}}{c^{2}}\right)$
(68)

with $K^{AB}=(\partial_{i}f^{A})(\partial_{j}f^{B})\delta^{ij}$ the so-called
inverse Cauchy-Green strain tensor. $W$ is called stored energy 333This $W$
should not be confused with $W$ appearing in $W^{2}=W^{i}W_{i}$ in
Sect.(3.1)and theories of this type are called ’hyperelastic, frame-
indifferent’. Hyperelasticity means that the theory is derivable from a
Lagrangian, frame indifference corresponds in our setting to diffeomorphism
invariance.
Furthermore (recall from (28) that $V^{i}=-\phi^{i}{}_{A}\,\partial_{t}f^{A}$)

$n=(\mathrm{det}H^{AB})^{\frac{1}{2}}=\mathrm{det}(\partial_{i}f^{A})\left(1-\frac{V^{2}}{c^{2}}\right)^{\frac{1}{2}}$
(69)
$S[f]=\int\mathrm{det}(\partial_{i}f^{A})(c^{2}-\frac{V^{2}}{2}+W)\,dt\,d^{3}x+O(\frac{1}{c^{2}})$
(70)

Note: the divergent term is a Null Lagrangian, i.e. the EL-equations for that
term are identically satisfied, i.e.

$\partial_{i}[\mathrm{det}(\partial_{j}f^{B})\phi^{i}{}_{A}]=0\,,$ (71)

which is the famous Piola identity.

The ’renormalized’ Lagrangian, i.e. with the divergent term omitted, breaks
Galilean invariance, but not of course the EOM. In fact the Galilean equations
have a spacetime form. The definition of the particle number density, now
called $k$, is as before is

$(\partial_{\mu}f^{A})(\partial_{\nu}f^{B})(\partial_{\lambda}f^{C})\Omega_{ABC}=k\,\epsilon_{\mu\nu\lambda\sigma}u^{\sigma}\,,\hskip
19.91684ptn>0$ (72)

or $k=\mathrm{det}(\partial_{i}f^{A})$, when $\Omega_{ABC}=\varepsilon_{ABC}$.
Note $k$ contains no time derivatives. The stress-energy tensor $T^{\mu\nu}$
is then given by

$T^{\mu\nu}=kv^{\mu}v^{\nu}+2\,k\,\frac{\partial W}{\partial
H^{AB}}(\partial_{\rho}f^{A})(\partial_{\sigma}f^{B})e^{\mu\rho}e^{\nu\sigma}$
(73)

Then $\partial_{\mu}(kv^{\mu})=0$ and $\tau_{\mu}\partial_{\nu}T^{\mu\nu}=0$
hold as identities, and the 3 EOM are $\partial_{\nu}T^{\mu\nu}=0$.
In the presence of a gravitational potential $U$ we define the ’gravitational
stress’ by

$\Theta^{\mu\nu}=\frac{1}{4\pi
G}(e^{\mu\rho}e^{\nu\sigma}-\frac{1}{2}e^{\mu\nu}e^{\rho\sigma})(\partial_{\rho}U)(\partial_{\sigma}U)$
(74)

Then

$\partial_{\nu}\Theta^{\mu\nu}=\frac{1}{4\pi
G}(e^{\mu\nu}\partial_{\nu}U)\,\Delta
U\,\,\mathrm{where}\,\,\Delta=h^{\mu\nu}\partial_{\mu}\partial_{\nu}\,,$ (75)

and the coupled system of equations can concisely be written as

$\partial_{\nu}(T^{\mu\nu}+\Theta^{\mu\nu})=0\,,\hskip 14.22636pt\Delta U=4\pi
G\,n$ (76)

##  7 Linearization at a stressfree state

We assume for simplicity that the material is homogenous and isotropic, so the
equations of motion are

$-\mathcal{E}_{A}=M^{\mu\nu}_{AB}\nabla_{\mu}\partial_{\nu}f^{B}=0$ (77)

Here: we linearize at a stressfree configuration in Minkowski, i.e.
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}=-\frac{1}{c^{2}}\partial_{t}^{2}+\delta^{ij}\partial_{i}\partial_{j}$
and take

$\mathring{f}^{A}=\delta^{A}{}_{i}x^{i}\,\,\Rightarrow\partial_{\mu}\partial_{\nu}\mathring{f}^{A}=0\,\,\Rightarrow\,\,\mathring{\mathcal{E}}_{A}=0$
(78)

and the linearized equations are of the form

$\mathring{M}^{\mu\nu}_{AB}\,\partial_{\mu}\partial_{\nu}\,\delta f^{B}=0$
(79)

Further constitutive assumptions are:

  * •

$\epsilon|_{H=\delta}=c^{2}$

  * •

$\frac{\partial\epsilon}{\partial H^{AB}}|_{H=\delta}=0$

  * •

$\frac{\partial^{2}\epsilon}{\partial H^{AB}\partial
H^{CD}}|_{H=\delta}=\frac{1}{4}(q\,\delta_{AB}\delta_{CD}+2r\,\delta_{C(A}\delta_{B)D})$
$r,q$ constants

So (note
$\mathring{n}=1,\,\mathring{H}^{AB}=\delta^{AB}\,,\,\,\mathring{u}^{\mu}=\frac{1}{c}\,\delta^{\mu}{}_{0}\,,\,\,\,\mathring{\tau}^{AB}=0$)

$\mathring{M}^{\mu\nu}_{AB}=[-\delta_{AB}\,\delta^{\mu}{}_{0}\delta^{\nu}{}_{0}+(q+r)\,\delta^{\mu}{}_{A}\delta^{\nu}{}_{B}+r\,\delta^{\nu}{}_{A}\delta^{\mu}{}_{B}]\,,$
(80)

so that

$\delta\mathcal{E}_{i}=-\partial_{t}^{2}\,\delta\\!f_{i}+\partial_{j}\,(L_{ijkl}\,\partial_{(k}\delta
f_{l)})\,,$ (81)

where $L_{ijkl}=L_{(ij)(kl)}=L_{klij}$ is given by

$L_{ijkl}=q\,\delta_{ij}\delta_{kl}+2r\,\delta_{i(k}\delta_{l)j}$ (82)

The following ellipticity conditions for the spatial operator play a role:
(i) $L_{ijkl}\lambda_{i}\mu_{j}\lambda_{k}\mu_{l}>0$: this is called rank-one
positivity or the Legendre-Hadamard condition. In the present case this means
$r=c_{2}^{2}>0,\,2r+q=c_{1}^{2}>0$. It then follows that the characteristic
polynomial $P(k)$ is given by (setting $c=1$)

$\mathrm{det}(\mathring{M}^{\mu\nu}_{AB}k_{\mu}k_{\nu})=(g_{1}^{\mu\nu}k_{\mu}k_{\nu})(g_{2}^{\mu\nu}k_{\mu}k_{\nu})^{2}\,,$
(83)
$g_{1}^{\mu\nu}=\eta^{\mu\nu}+\mathring{u}^{\mu}\mathring{u}^{\nu}\left(1-\frac{1}{c_{1}^{2}}\right)\,,\hskip
17.07182ptg_{2}^{\mu\nu}=\eta^{\mu\nu}+\mathring{u}^{\mu}\mathring{u}^{\nu}\left(1-\frac{1}{c_{2}^{2}}\right)$
(84)

The $g_{1}$ \- cone determines the phase speed of longitudinal sound waves,
the $g_{2}$ one the tranversal (’shear’) waves.

(ii) $L_{ijkl}m_{ij}m_{kl}>0$: ’pointwise stability’. This requires
$r>0,\,\frac{2r}{3}+q>0$.

(ii) plays the following role. Consider the Neumann-type boundary value
problem on $\mathbb{R}^{3}$ given by

$\partial_{j}(L_{ijkl}\,\partial_{(k}\,\delta
f_{l)})=b_{i}\,\,\,\,\mathrm{in}\,\,\mathring{f}^{-1}(\mathcal{B})\in\mathbb{R}^{3},\hskip
22.76228ptL_{ijkl}\,\partial_{(k}\,\delta
f_{l)}\,n_{j}|_{\mathring{f}^{-1}(\partial\mathcal{B})}=\tau_{i}$ (85)

Then, in appropriate function spaces, this has the ’right’ kernel and
cokernel, namely kernel spanned by Euclidean Killing vectors $\delta
f_{i}=\xi_{i}$, and the range spanned by

$\int_{\mathring{f}^{-1}(\mathcal{B})}\xi^{i}b_{i}\,d^{3}\\!x+\int_{\mathring{f}^{-1}(\partial\mathcal{B})}\xi^{i}\tau_{i}\,dS(x)=0$
(86)

In the language of nonrelativistic elasticity: ’load $b_{i}$ and surface
traction $\tau_{i}$ have to be equilibrated’, see Sect.11.

##  8 Elastic causality, wellposedness

There is, for equations of the form

$M^{\mu\nu}_{AB}[f,\partial
f;x]\,\partial_{\mu}\partial_{\nu}f^{B}+\,\mathrm{l.o.}=0$ (87)

with $M^{\mu\nu}_{AB}=M^{\nu\mu}_{BA}$ a theory of local wellposedness by
Hughes, Kato, Marsden [20] and in geometrical form by Christodoulou in [11]:
Definition: The partial differential operator
$M^{\mu\nu}_{AB}\,\partial_{\mu}\partial_{\nu}f^{B}+\mathrm{l.o.}$ is called
regularly hyperbolic w.r. to $(\xi_{\mu},X^{\nu})$, if there exists a
’subcharacteristic’ covector $\xi_{\mu}$ and a ’timelike’ vector $X^{\mu}$
with $X^{\mu}\xi_{\mu}>0$, s.th.
(i)$\,\,\,M^{\mu\nu}_{AB}\,\xi_{\mu}\xi_{\mu}$ is negative definite.
(ii) $M^{\mu\nu}_{AB}\lambda^{A}\lambda^{B}\eta_{\mu}\eta_{\nu}>0$ for all
$\eta_{\mu}$ with $X^{\mu}\eta_{\mu}=0$.

Our case as above is regular hyperbolic w.r. to
$(\xi_{\mu}=-u_{\mu},X^{\mu}=u^{\mu})$ provided $\epsilon\,H_{AB}+\tau_{AB}$
is positive definite and $U_{ACBD}$ is rank-1 positive. The set of pairs
$(\xi_{\mu},X^{\nu})$ consists of 2 connected components of opposite pairs
$(\xi^{+},X^{+})\cup(\xi^{-},X^{-})$ all of which are convex.
The connection with the characteristic polynomial
$P(k)=\mathrm{det}(M^{\mu\nu}_{AB}\,k_{\mu}k_{\nu})$ is as follows. A
homogenous polynomial in $k_{\mu}$ is called a hyperbolic polynomial (Garding
[21]) w.r. to a covector $\xi_{\mu}$ if the equation

$P(\eta+\lambda\xi)=\mathrm{det}[M^{\mu\nu}_{AB}(\eta_{\mu}+\lambda\xi_{\mu})(\eta_{\nu}+\lambda\xi_{\nu})]=0\hskip
28.45274pt$ (88)

has only real (in our case $2\times 3=6$) solutions $\lambda$ corresponding to
the 6 sheets of the characteristic set $\mathfrak{C}$. That can in general be
complicated, have singularities and is perhaps ultimately a matter for
algebraic geometry.
For a regularly hyperbolic operator the set of $\xi$’s turns out to be equal
to the union $\\{\xi^{+}\\}\cup\\{\xi^{-}\\}$. For our previous example 2
pairs of these solutions cones coincide.
Note: for the $\xi^{+}$’s the faster ($c_{1}>c_{2}$) cone lies inside the
slower one. This is because the covectors $\xi$ describe ’acoustically
spacelike’ hypersurfaces, i.e. hypersurfaces foliating a domain of dependence,
which is the smaller the faster the phase velocity. The $X$’s are dual objects
describing ’rays’, i.e. domains of influence. The faster the phase speed, the
larger the domain of influence. Formally

$g_{1,2}^{\mu\nu}=\eta^{\mu\nu}+u^{\mu}u^{\nu}(1-\frac{1}{c_{1,2}^{2}})\,,\hskip
11.38092ptg_{\mu\nu}^{1,2}=\eta_{\mu\nu}+u_{\mu}u_{\nu}(1-c_{1,2}^{2})$ (89)

In nonrelativistic elasticity the same relations hold after replacing
$\eta^{\mu\nu}+u^{\mu}u^{\nu}$ with the Galilean $h^{\mu\nu}$ and $u^{\mu}$ by
$v^{\mu}$ with $v^{\mu}\tau_{\mu}=1$ and

$g_{\mu\nu}^{1,2}=h_{\mu\nu}-\tau_{\mu}\tau_{\nu}\,c_{1,2}^{2}\,\,\,\,\mathrm{with}\,\,\,\,h^{\mu\nu}h_{\nu\rho}=\delta^{\mu}{}_{\rho}-v^{\mu}\tau_{\rho}$
(90)

Christodoulou in [11] proved a uniqueness theorem for regularly hyperbolic
equations in domains of dependence foliated by spacelike surfaces. For
existence one should be able to use the results of [20].
As for the connection with symmetric hyperbolicity (Beig [22]): replace

$M^{\mu\nu}_{AB}\,\partial_{\mu}\partial_{\nu}f^{B}=\mathrm{l.o.}$ (91)

with

$W^{\mu\nu}{}_{AB}{}^{(\sigma)}\partial_{\sigma}F^{B}{}_{\nu}=\mathrm{l.o.}\hskip
28.45274ptw^{\mu}\partial_{\mu}f^{A}=F^{A}{}_{\mu}w^{\mu}$ (92)
$W^{\mu\nu}{}_{AB}{}^{(\sigma)}=w^{\mu}M^{\sigma\nu}_{AB}+2w^{[\nu}M^{\sigma]\mu}_{BA}$
(93)

for a suitable vector $w^{\mu}$. Clearly
$W^{\mu\nu}{}_{AB}{}^{(\sigma)}=W^{\nu\mu}{}_{BA}{}^{(\sigma)}$, so the new
system is symmetric. Sometimes, when $w^{\mu}=X^{\mu}$,
$W^{\mu\nu}{}_{AB}{}^{(\rho)}\xi_{\rho}$ is positive definite, but that is not
guaranteed, e.g. for elasticity this requires pointwise stability. We remark
that none of the above works for fluids since the ’spacelike’ term is
degenerate there.
A local wellposedness result for elasticity coupled to gravity should follow
the standard path (see e.g. Ringström [15]), but a detailed presentation seems
to be lacking. Whether the gravitational cone is the fastest one or not should
play no role, when only local wellposedness is concerned.
Finally let us mention that for nonrelativistic elasticity there are even
global results on $\\{t\\}\times\mathbb{R}^{3}$ for certain equations of state
which lead to a null condition (see Sideris [23, 24]).

##  9 Bodies surrounded by vacuum

When $\mathcal{B}$ is compact, the Lagrangian density can only be integrated
over $f^{-1}(\mathcal{B})$, which is the infinite region of spacetime ’inside’
the timelike hypersurface $f^{-1}(\partial\mathcal{B})$. We write

$S[f]=\int_{f^{-1}(\mathcal{B})}\\!\\!\\!\\!\rho\,\sqrt{-g}\,d^{4}x=\int_{f^{-1}(\mathcal{B})}\\!\\!\\!\\!\rho\,dV_{g}\,,$
(94)

where we assume that $f^{-1}(\mathcal{B})$ is cut off in the future and past
by two non-intersecting spacelike hypersurfaces $\Sigma_{1}$ and $\Sigma_{2}$.
Suppose we vary (94) w.r. to $f$ with the understanding that the $f^{A}$’s are
all equal to a given map outside a compact proper sub-region which however may
include $f^{-1}(\partial\mathcal{B})$. Then we have the
Theorem: There holds the identity

$\delta
S=\int_{f^{-1}(\mathcal{B})}\\!\\!\\!T^{\mu}{}_{\nu}\nabla_{\mu}(\psi^{\nu}{}_{A}\delta
f^{A})\,dV_{g}\,,$ (95)

where $\delta f^{A}$ has ’compact support in time’, i.e. vanish on
$\Sigma_{1}$ and $\Sigma_{2}$, but need not satisfy any further condition on
$f^{-1}(\partial\mathcal{B})$.
Proof: Using the identity (40) together with the formula

$\delta\int_{f^{-1}(\mathcal{B})}\\!\\!\\!dV_{g}=-\int_{f^{-1}(\partial\mathcal{B})}\\!\\!\psi^{\mu}{}_{A}\delta
f^{A}d\Sigma_{\mu}\,,$ (96)

the contribution from the total divergence in (40) drops out, and we get (95).
Alternatively one can prove (95) directly from diffeomorphism invariance of
(94).

Suppose now we ask for an extremum of the action $S[f]$ subject only to the
condition that variations have compact support in time. This is similar, in
the calculus of variations (see e.g. [25]), to asking for ’natural boundary
conditions’. Then, using the identity (94), it is easy to see that this
implies both the Euler-Lagrange equation

$\psi^{\mu}{}_{A}\nabla_{\nu}T_{\mu}{}^{\nu}=0\hskip
11.38092pt\mathrm{in}\hskip 2.84544ptf^{-1}(\mathcal{B})$ (97)

and the boundary condition

$\psi^{\mu}{}_{A}T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}=0\hskip
5.69046pt\Leftrightarrow\hskip
5.69046pt\frac{\partial\epsilon}{\partial\partial_{\mu}f^{A}}\,m_{\mu}|_{f^{-1}(\partial\mathcal{B})}=0$
(98)

to be satisfied, where $m_{\mu}$ is the (spacelike) conormal of
$f^{-1}(\partial\mathcal{B})$. Recall that the vertical component of
$\nabla_{\nu}T_{\mu}{}^{\nu}$ namely $u^{\mu}\nabla_{\nu}T_{\mu}{}^{\nu}$ is
identically zero. Similarly the vertical component of
$T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}$, namely
$u^{\mu}T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}$ also vanishes
due to (4) since $u^{\mu}m_{\mu}=0$.
Summing up, we take as requirement on the matter-vacuum interface the
condition that $T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}$ be zero
with the idea that this corresponds to bodies ’floating freely in spacetime’.
This is supported by the following fact: let the spacetime $(M,g_{\mu\nu})$
have a Killing vector $\xi^{\mu}$. Then the integral

$E_{\xi}=\int_{\Sigma\cap
f^{-1}(\mathcal{B})}\xi^{\mu}T_{\mu}{}^{\nu}d\Sigma_{\nu}$ (99)

does not depend on the choice of spacelike slice $\Sigma$.
Let us point out that
$T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}=0$ is also natural when
elasticity is coupled to gravity, i.e.

$G_{\mu\nu}=\kappa\,T_{\mu\nu}\,\chi_{f^{-1}(\mathcal{B})}\,,$ (100)

where $\chi_{f^{-1}(\mathcal{B})}$ is the characteristic function of
$f^{-1}(\mathcal{B})$. Now for elastic solids $T_{\mu\nu}$ necessarily drops
to zero sharply at the matter-vacuum interface. For regularity it is sensible
to require that the first and second fundamental for the metric change
continuously across $f^{-1}(\partial\mathcal{B})$. But since
$G_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}$, as is well known,
contains only up to 1st normal derivatives of the metric and $G_{\mu\nu}$ is
zero in the vacuum region, it follows that
$G_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}$ has to vanish, in
accordance with (98).

##  10 Material picture

Trying to solve the elastic equations in the presence of the ’free’ boundary
$f^{-1}(\partial\mathcal{B})$ makes it tempting (almost obligatory?) to
rewrite these equations in terms of the inverse of $f^{A}$ relative to some
slicing of $M$ by spacelike hypersurfaces. We first write the action in terms
of the ADM variables introduced previously.

$\sqrt{-g}\,d^{4}x=N[\mathrm{det}(g^{ij}-\frac{Y^{i}Y^{j}}{N^{2}})]^{-\frac{1}{2}}dtd^{3}x$
(101)

We had

$H^{AB}=(\partial_{i}f^{A})(\partial_{j}f^{B})\,(g^{ij}-\frac{W^{i}W^{j}}{N^{2}})$
(102)

so that

$n^{2}=\mathrm{det}(H^{AB})\Omega^{2}(f)=[\mathrm{det}(\partial_{k}f^{A})]^{2}\,\mathrm{det}(g^{ij}-\frac{W^{i}W^{j}}{N^{2}})\Omega^{2}(f)$
(103)

Thus

$S=\int_{f^{-1}(\mathcal{B})}\\!\\!\\!\\!\epsilon(f^{A},H^{BC})\,\sqrt{\frac{N^{2}-W^{2}}{N^{2}-Y^{2}}}\,N\,\Omega(f)\,\mathrm{det}(\partial_{i}f^{A})\,dtd^{3}x=\\\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\int_{\\{T\\}\times\mathcal{B}}\\!\\!\\!\epsilon(X,\overline{H}^{AB})\,\sqrt{\frac{\overline{N}^{2}-\overline{W}^{2}}{\overline{N}^{2}-\overline{Y}^{2}}}\,\,\overline{N}\,\Omega(X)\,dTd^{3}X\,,$
(104)

where an overbar denotes composition with $\Phi^{i}$ defined by

$f^{A}(T,\Phi(T,X))=X^{A}$ (105)

$\overline{H}^{AB}=\rho^{A}{}_{i}\rho^{B}{}_{j}\left(\overline{g}^{ij}-\frac{\overline{W}^{i}\overline{W}^{j}}{\overline{N}^{2}}\right)\,,\hskip
14.22636pt\rho^{A}{}_{j}\,\partial_{A}\Phi^{i}=\delta^{i}{}_{j}$ (106)

Note that $\overline{V}^{i}=\partial_{T}\Phi^{i}$. Thus the ’prefactor’ in the
material action has no space derivatives. Observe metric components are pulled
back under $\Phi$ as scalars. So

$L_{mat}=\epsilon(X,\overline{H}^{AB})\,\sqrt{\frac{\overline{N}^{2}-\overline{W}^{2}}{\overline{N}^{2}-\overline{Y}^{2}}}\,\,\overline{N}\,\Omega(X)$
(107)

is the material Lagrangian and the material EOM are

$-\partial_{A}\,\left(\frac{\partial
L_{mat}}{\partial(\partial_{A}\Phi^{i})}\right)-\partial_{T}\,\left(\frac{\partial
L_{mat}}{\partial(\partial_{T}\Phi^{i})}\right)+\frac{\partial
L_{mat}}{\partial\Phi^{i}}=0$ (108)

with boundary conditions

$\left(\frac{\partial\epsilon}{\partial(\partial_{A}\Phi^{i})}\right)n_{A}\big{|}_{\partial\mathcal{B}}=0\,,$
(109)

which are of course the equivalent of (97,98) in the material picture.
Local wellposedness follows from a theorem by Koch [26], provided the initial
configuration satifies the material version of the previous hyperbolicity
condition, involving e.g.
$\frac{\partial^{2}\epsilon}{\partial(\partial_{A}\Phi^{i})\partial(\partial_{B}\Phi^{j})}$
(see also [27] and [28]).

##  11 Time independent problems

Suppose $(M;g)$ has a timelike Killing vector $\xi^{\mu}$. Recall the quotient
space $Q$ is endowed with the metric $h_{ij}$. From (33) it follows that

$\sqrt{-g}\,d^{4}x=Vdt\,\sqrt{h}\,d^{3}x\,,$ (110)

where $-V^{2}=g_{\mu\nu}\xi^{\mu}\xi^{\nu}$. So the reduced action for time
independent configurations $f^{A}(t,x^{i})=f^{A}(x^{i})$ is

$S[f;h^{ij},V]=\int_{f^{-1}(\mathcal{B})}\rho\,V\,\sqrt{h}\,d^{3}x\,,$ (111)

where $\rho$ is function of
$(f^{A},H^{BC}=(\partial_{i}f^{B})(\partial_{j}f^{C})h^{ij})$. Note that the
presence of the potential $V$ means that the diffeomorphism invariance of the
action is now broken. In fact, let $\Phi$ be a diffeomorphism from $Q$ into
itself. Then

$S[f\circ\Phi;\Phi^{\star}h,V]=S[f;h,V\circ\Phi^{-1}]$ (112)

So

$-\int_{f^{-1}(\mathcal{B})}\\!\\!\\!\rho\,\mathcal{L}_{\eta}V\,\sqrt{h}\,d^{3}x=\frac{1}{2}\int_{f^{-1}(\mathcal{B})}\\!\\!\\!t_{ij}\mathcal{L}_{\eta}h^{ij}\,\sqrt{h}\,d^{3}x+\frac{d}{d\epsilon}|_{\epsilon=0}\,S[f+\epsilon\mathcal{L}_{\xi}f;g]\,,$
(113)

where $t_{ij}=2\frac{\partial\rho}{\partial h^{ij}}-\rho h_{ij}$. Setting
again $\eta^{i}=\psi^{i}{}_{A}\delta f^{A}$, where
$\psi^{i}{}_{A}\,\partial_{i}f^{B}=\delta^{B}{}_{A}$, we find that

$\delta S=\int_{f^{-1}(\mathcal{B})}V\,t_{ij}D^{i}(\psi^{j}{}_{A}\delta
f^{A})\sqrt{h}\,d^{3}x-\int_{f^{-1}(\mathcal{B})}\\!\\!\rho\,D_{i}V\psi^{i}{}_{A}\delta
f^{A}\,\sqrt{h}\,d^{3}x\,,$ (114)

where $D_{i}$ is the covariant derivative associated with $h_{ij}$. So the
field equations plus boundary conditions are

$D_{j}(Vt_{i}{}^{j})+\rho\,D_{i}V=0\hskip 8.5359pt\mathrm{in}\hskip
2.84544ptf^{-1}(\mathcal{B})\,,\hskip
17.07182ptt_{i}{}^{j}n_{j}|_{f^{-1}(\mathcal{\partial B})}=0\,.$ (115)

Note the ’force term’ $\rho D_{i}V$ in (115) is zero if and only if the
Killing vector $\xi^{\mu}$ is geodesic.
As an example consider Minkowski spacetime $(M,\eta_{\mu\nu})$ and
$\xi^{\mu}\partial_{\mu}=\partial_{t}+\omega\,\partial_{\phi}$ with
$\omega=\mathrm{const}$, the ’helical’ Killing vector corresponding to rigid
rotations at angular frequency $\omega$. Then $V^{2}=1-\omega^{2}\rho^{2}$
where $\rho^{2}=(x^{1})^{2}+(x^{2})^{2}$ and we have set $c=1$. Furthermore

$h_{ij}dx^{i}dx^{j}=\delta_{ij}dx^{i}dx^{j}+\frac{\omega^{2}}{1-\omega^{2}\rho^{2}}(x^{1}dx^{2}-x^{2}dx^{1})^{2}\,\,\,\,\,\mathrm{for}\,\,\rho<\frac{1}{\omega}\,.$
(116)

Here $D_{i}V$ describes the centrifugal force. One can [29] study (115) for
small $\Omega$ (for which $V=1$ and $h_{ij}=\delta_{ij}$) and $f$ close to a
stressfree state $\mathring{f}$ (for which $t_{i}{}^{j}$ vanishes). Now the
linearized operator given by the pair

$\delta
f^{A}\mapsto\left(\partial_{j}\delta_{f}t_{i}{}^{j},\,\delta_{f}t_{i}{}^{j}\,n_{j}|_{\mathring{f}^{-1}(\mathcal{\partial
B})}\right)$ (117)

has a kernel consisting (at least) of the 6 elements of the form $\delta
f^{A}=\xi^{i}\partial_{i}\mathring{f}^{A}$, where $\xi^{i}$ is a Euclidean
Killing vector. This operator also does not have full range, but the set of
pairs $(b_{i},\tau_{i})$ in its image, due to the symmetry of $t_{ij}$ plus
the Killing equation, has to satisfy Eq. (86) namely

$\int_{\mathring{f}^{-1}(\mathcal{B})}\xi^{i}b_{i}\,d^{3}\\!x+\int_{\mathring{f}^{-1}(\partial\mathcal{B})}\xi^{i}\tau_{i}\,dS(x)=0\,,$
(118)

so the implicit function theorem does not directly apply. Still one can show
existence of solutions for small $\omega^{2}$ (see [29]), provided that there
hold the constitutive conditions in Sect. 6 with pointwise stability on the
elastic constants. Furthermore: the undeformed body in space, i.e. the domain
$\mathring{f}^{-1}(\mathcal{B})$ should be such that the rotation axis (a)
goes through the centroid and (b) coincides with an axis of inertia. Finally
(this can be relaxed) the 3 axes of inertia should be all different444The
additional condition (97) in [29] turns out to be superfluous.. Similar
results [30] can be proved for a sufficiently small body moving along a
circular geodesic in Schwarzschild spacetime (note that such circular
geodesics are orbits of a helical Killing vector).

##  12 Coupling to gravity

###  12.1 Time independent case

The problems in the previous section with constructing time independent
solutions near stressfree configurations was ultimately due to the fact that
these systems were not closed: an external force acted on the system, so
precautions had to be made in order for this force to lie in the non-trivial
range of the linearized operator. When gravity is coupled the system becomes
closed, so one might expect a difference - and there is. For simplicity we
explain this difference in the Newtonian case, see the end of Sect.6. Here the
equations are (when $\mu(f)=1$)

$\partial_{j}t_{i}{}^{j}=-n\,\partial_{i}U\,\,\,\mathrm{in}\,\,\,f^{-1}(\mathcal{B})\subset\mathbb{R}^{3}\,,\hskip
28.45274ptt_{i}{}^{j}n_{j}|_{f^{-1}(\partial\mathcal{B})}=0$ (119)

with $t_{ij}=2n\,\frac{\partial W}{\partial
H^{AB}}\,(\partial_{i}f^{A})(\partial_{j}f^{B})$, $n=\mathrm{det}(\partial f)$
and

$\Delta U=4\pi G\,n\,\chi_{f^{-1}(\mathcal{B})}\,,\hskip
28.45274ptU\rightarrow 0\,\,\mathrm{at}\,\infty$ (120)

The range problem just mentioned results from the fact that (119) implies

$\int_{f^{-1}(\mathcal{B})}\\!\\!n\,\xi^{i}\partial_{i}U\,d^{3}x=0$ (121)

for the 6 Euclidean Killing vectors. Put differently, the total self-force and
the total self-torque on the body due to gravity should be zero. But that is
in fact true due to (120), as can either be checked explicitly from

$U(x)=-G\int_{f^{-1}(\mathcal{B})}\frac{n(x^{\prime})}{|x-x^{\prime}|}d^{3}x^{\prime}$
(122)

or by first recalling

$\partial_{j}\Theta^{ij}=n\,\partial^{i}U\,\chi_{f^{-1}(\mathcal{B})}$ (123)

Now integrate the l.h. side against $\xi^{i}$ over $\mathbb{R}^{3}$: that is
finite since $\xi=O(r),\Theta^{ij}=O(r^{-4})$ and gives zero after partial
integration and using the Killing equations. Using this fact one obtains [31]
an existence theorem for small $G$ and $f^{A}$ close to a stressfree
configuration under appropriate conditions on the stored energy $W$. In
spherical symmetry and for bodies $\mathcal{B}$ with the topology of a shell
an analogous theorem was proved in [32]. Calogero and Leonori in [33] obtained
solutions without these smallness assumptions, using the calculus of
variations.
The Einstein-elastic equations for an elastic body in an asymptotically flat
gravitational field are a more complicated matter. But an argument in a
similar spirit still works, where the Bianchi identity plays a crucial role.
In this manner Andersson et al [34] obtained existence theorems for static
elastic bodies with small $G$ and with the configuration close to stressfree,
similarly for rigidly rotating ones for small $G,\omega$ in [35]. In the
former case that provided the first construction of static self-gravitating
bodies in GR without symmetries.
In spherical symmetry the subject started with [36] (see also [37], [5] and
[38]) and has meanwhile advanced much further, see the recent [39, 40] and
references therein. In these works there is no assumption of closeness to a
stressfree configurations.

###  12.2 Dynamics

We finally make some remarks on the evolution problem for (possibly several)
elastic bodies interacting through Einstein gravity. For more material consult
the review paper [41]. The problem here lies mainly in the following facts.
The first problem is that of ’corner conditions’ for the matter variable, i.e.
$f^{A}$. Namely the matter boundary condition

$T_{\mu}{}^{\nu}m_{\nu}|_{f^{-1}(\partial\mathcal{B})}=0$ (124)

in order for a sufficiently regular solution to exist, requires that the
initial data for $f^{A}$ satisfy the restriction to the initial surface and
(124) and that of a sufficient number of its time (i.e. tangential-
to-$f^{-1}(\partial\mathcal{B})$ derivatives): this is already present for
bodies moving in a given background and is addressed in [26, 28]555There is an
incorrect argument in the Appendix of [28], which can easily be remedied..
The second problem comes from what Andersson, Oliynyk call the transmission
conditions [42]: these, loosely speaking result from the regularity condition
mentioned earlier that first and second fundamental form of the metric behave
continuously across the timelike surface $f^{-1}(\partial\mathcal{B})$. In a
similar vein as with the matter variable this condition implies a sequence of
- in this case - regularity conditions on the gravitational initial data. The
corresponding evolution problem has been solved in the works [42, 43], but
compatibility with the GR initial value constraints is still an open issue.

### Acknowledgement

I am indebted to my long-term collaborator Bernd G. Schmidt for initiating my
interest in this field 22 years ago.

## References

  * [1] Philippe G Ciarlet.  Mathematical elasticity: Three-dimensional elasticity.  SIAM, 2021.
  * [2] Jerrold E Marsden and Thomas JR Hughes.  Mathematical foundations of elasticity.  Courier Corporation, 1994.
  * [3] Gustav Herglotz.  Über die mechanik des deformierbaren körpers vom standpunkte der relativitätstheorie.  Annalen der Physik, 341(13):493–533, 1911.
  * [4] Brandon Carter and H Quintana.  Foundations of general relativistic high-pressure elasticity theory.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 331(1584):57–83, 1972.
  * [5] Max Karlovini and Lars Samuelsson.  Elastic stars in general relativity: I. foundations and equilibrium models.  Classical and Quantum Gravity, 20(16):3613, 2003.
  * [6] Jerzy Kijowski and Giulio Magli.  Relativistic elastomechanics as a lagrangian field theory.  Journal of Geometry and Physics, 9(3):207–223, 1992.
  * [7] DE Soper.  Classical field theory.  1976\.
  * [8] J David Brown.  Elasticity theory in general relativity.  Classical and Quantum Gravity, 38(8):085017, 2021.
  * [9] AJ Penner, N Andersson, Lars Samuelsson, I Hawke, and DI Jones.  Tidal deformations of neutron stars: The role of stratification and elasticity.  Physical Review D, 84(10):103006, 2011.
  * [10] A Shadi Tahvildar-Zadeh.  Relativistic and nonrelativistic elastodynamics with small shear strains.  In Annales de l’IHP Physique théorique, volume 69, pages 275–307, 1998.
  * [11] D Christodoulou.  The action principle and partial differential equations, no. 146 in.  Ann Math. Studies. Princeton, NJ, 2000.
  * [12] Willie Wai-Yeung Wong.  Regular hyperbolicity, dominant energy condition and causality for lagrangian theories of maps.  Classical and Quantum Gravity, 28(21):215008, 2011.
  * [13] Robert Beig and Bernd G Schmidt.  Relativistic elasticity.  Classical and Quantum Gravity, 20(5):889, 2003.
  * [14] Jerzy Kijowski and Giulio Magli.  Unconstrained variational principle and canonical structure for relativistic elasticity.  Reports on mathematical physics, 39(1):99–112, 1997.
  * [15] Hans Ringström.  The Cauchy problem in general relativity, volume 6.  European Mathematical Society, 2009.
  * [16] Daniel Tataru.  The wave maps equation.  Bulletin of the American Mathematical Society, 41(2):185–204, 2004\.
  * [17] Sergei V Ketov.  Nonlinear sigma model.  Scholarpedia, 4(1):8508, 2009.
  * [18] THR Skyrme.  Particle states of a quantized meson field.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 262(1309):237–245, 1961.
  * [19] FAE Pirani and Gareth Williams.  Rigid motion in a gravitational field.  Séminaire Janet. Mécanique analytique et mécanique céleste, 5:1–16, 1962.
  * [20] Thomas JR Hughes, Tosio Kato, and Jerrold E Marsden.  Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity.  Archive for Rational Mechanics and Analysis, 63(3):273–294, 1977\.
  * [21] Lars Gårding.  An inequality for hyperbolic polynomials.  Journal of Mathematics and Mechanics, pages 957–965, 1959.
  * [22] Robert Beig.  Concepts of hyperbolicity and relativistic continuum mechanics.  In Analytical and Numerical Approaches to Mathematical Relativity, pages 101–116. Springer, 2006.
  * [23] Thomas C Sideris.  The null condition and global existence of nonlinear elastic waves.  Inventiones mathematicae, 123(2):323–342, 1996.
  * [24] Thomas C Sideris.  Nonresonance and global existence of prestressed nonlinear elastic waves.  Annals of Mathematics, 151(2):849–874, 2000.
  * [25] Mariano Giaquinta and Stefan Hildebrandt.  Calculus of variations, vol. 1 and 2.  Grundlehren der mathematischen Wissenschaften, (311), 1995.
  * [26] Herbert Koch.  Mixed problems for fully nonlinear hyperbolic equations.  Mathematische Zeitschrift, 214(1):9–42, 1993.
  * [27] Michael Wernig-Pichler.  Relativistic elastodynamics.  arXiv preprint gr-qc/0605025, 2006.
  * [28] Robert Beig and Michael Wernig-Pichler.  On the motion of a compact elastic body.  Communications in mathematical physics, 271(2):455–465, 2007.
  * [29] Robert Beig and Bernd G Schmidt.  Relativistic elastostatics: I. bodies in rigid rotation.  Classical and Quantum Gravity, 22(11):2249, 2005.
  * [30] Robert Beig and Bernd G Schmidt.  Celestial mechanics of elastic bodies ii.  Classical and Quantum Gravity, 34(8):085001, 2017.
  * [31] Robert Beig and Bernd G Schmidt.  Static, self–gravitating elastic bodies.  Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2029):109–115, 2003.
  * [32] CM Kroon.  Static elastic shells in einsteinian and newtonian gravity.  arXiv preprint gr-qc/0603103, 2006.
  * [33] Simone Calogero and Tommaso Leonori.  Ground states of self-gravitating elastic bodies.  Calculus of Variations and Partial Differential Equations, 54(1):881–899, 2015.
  * [34] Lars Andersson, Robert Beig, and Bernd G Schmidt.  Static self-gravitating elastic bodies in einstein gravity.  Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 61(7):988–1023, 2008.
  * [35] Lars Andersson, Bernd G Schmidt, and Robert Beig.  Rotating elastic bodies in einstein gravity.  Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 63(5):559–589, 2010.
  * [36] Giulio Magli and Jerzy Kijowski.  A generalization of the relativistic equilibrium equations for a non-rotating star.  General relativity and gravitation, 24(2):139–158, 1992.
  * [37] Jiseong Park.  Spherically symmetric static solutions of the einstein equations with elastic matter source.  General Relativity and Gravitation, 32(2):235–252, 2000.
  * [38] Jörg Frauendiener and Alexander Kabobel.  The static spherically symmetric body in relativistic elasticity.  Classical and Quantum Gravity, 24(18):4817, 2007.
  * [39] Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo.  Compact elastic objects in general relativity.  Physical Review D, 105(4):044025, 2022.
  * [40] Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo.  Erratum: Compact elastic objects in general relativity [phys. rev. d 105, 044025 (2022)].  Physical Review D, 105(12):129903, 2022.
  * [41] Lars Andersson.  Self-gravitating elastic bodies.  In Equations of Motion in Relativistic Gravity, pages 543–559. Springer, 2015.
  * [42] Lars Andersson and Todd A Oliynyk.  A transmission problem for quasi-linear wave equations.  Journal of Differential Equations, 256(6):2023–2078, 2014.
  * [43] Lars Andersson, Todd A Oliynyk, and Bernd G Schmidt.  Dynamical compact elastic bodies in general relativity.  arXiv preprint arXiv:1410.4894, 2014.