# Time-like hypersurfaces of prescribed mean
extrinsic curvature.

Helmut Friedrich
Max-Planck-Institut für Gravitationsphysik
Am Mühlenberg 1
14476 Golm, Germany

###### Abstract

The results on the initial boundary value problem for Einstein’s vacuum field
equation obtained in [7] rely on an unusual gauge. One of the defining gauge
source functions represents the mean extrinsic curvature of the time-like
leaves of a foliation that includes the boundary and covers a neighbourhoood
of it. The others steer the development of a frame field and coordinates on
the leaves. In general their combined action is needed to control in the
context of the reduced field equations the evolution of the leaves. In this
article are derived the hyperbolic equations implicit in that gauge. It is
shown that the latter are independent of the Einstein equations and well
defined on arbitrary space-times. The analysis simplifies if boundary
conditions with constant mean extrinsic curvature are stipulated. It
simplifies further if the boundary is required to be totally geodesic.

##  1 Introduction

In this article we consider a question that arises in the context of the
initial boundary value problem for Einstein’s vacuum field equation formulated
in [7]. The setting is that of a smooth Lorentz metric $g$ on a manifold $M$
with boundary $S\cup T$ where $S$ and $T$ are smooth hypersurfaces of $M$
which are space- and time-like respectively and intersect in the space-like
surface $\Sigma=S\cap T$ that represents the common boundary of $S$ and of
$T$. The manifold $T$ is assumed to be diffeomorphic to
$\mathbb{R}^{+}_{0}\times\Sigma$ with $\\{0\\}\times\Sigma$ identified with
$\Sigma$. The assumptions on $S$ are usually chosen according to the desired
application. For definiteness we assume it to be compact with boundary
$\Sigma$, though this is not needed for the following arguments. The set
$M\setminus S$ is supposed to be in the future of $S$ and on one side of $T$
so that all past directed non-extendible time-like curves in $M$ acquire an
endpoint on $S\cup T$. As in [6] a smooth space-time $(M,g)$ with these
properties is referred to as an $ST$-space-time. The initial boundary value
problem asks for the existence and uniqueness of solutions to Einstein’s
vacuum field equations $R_{\mu\nu}=0$ that induce suitably prescribed Cauchy
data on $S$ and boundary data on $T$. Such a solution will be called an
$ST$-vacuum solution.

The analysis of the standard Cauchy problem for Einstein’s equations is
typically based on coordinates $x^{\nu}$ that obey a wave gauge characterized
by gauge source functions $F^{\mu}=F^{\mu}(x^{\nu})$ [4]. In this gauge, often
referred to as ‘harmonic’ if $F^{\mu}=0$ and (somewhat absurdly) as
‘generalized harmonic’ if $F^{\mu}\neq 0$, Einstein’s equations take the form
of a system of wave equations of second order, the reduced equations. If $g$
is a solution to this system for Cauchy data that satisfy the constraints and
the gauge conditions on a space-like initial hypersurface, the reduced
equations imply with the Bianchi identity a subsidiary system which allows one
to conclude that the coordinates satisfy the semi-linear system of wave
equations

$\Box_{g}x^{\mu}=F^{\mu}(x^{\nu}).$ (1.1)

This ensures that $g$ is in fact a solution to the Einstein equations.

As pointed out in [4], the concept of a gauge source function gives access to
a huge class of useful gauge conditions. In principle any coordinate system
(ignoring situations of very weak differentiability) can be used to get
hyperbolic reduced equations. If a solution admits coordinates that exist
globally, these can be characterized in terms of gauge source functions. Since
they combine by (1.1) information on the coordinates as well as the metric,
the corresponding functions $F^{\mu}$ are usually not known a priori. A
successful use of the concept thus requires a clever choice of the functions
$F^{\mu}(x^{\nu})$ or suitable generalizations thereof. Interesting examples
of such applications can be found in [14] in a purely analytical context and
in [10], [12], [13] in numerical contexts.

The analysis of the initial boundary value problem in [7] rests on an unusual
gauge and is characterized by unusual gauge source functions $f$ and $F^{A}$,
$A=1,2$. Here $f$ corresponds to a function of the connection coefficients
that controls in a neighbourhood of the boundary $T=T_{0}$ the evolution of a
family of time-like hypersurfaces $T_{c}$, $0\leq c<\epsilon$, which define a
smooth space-time foliation that extends into the interior of the solution
space-time. The two functions $F^{A}$, $A=1,2$, correspond to connection
coefficients that control the evolution of a time-like vector field tangential
to the hypersurfaces $T_{c}$, that serves to define coordinates on these
hypersurfaces.

With these particular fields singled out as the gauge source functions there
can be extracted from the Einstein equations, in the representation used in
[7], a hyperbolic system of reduced equations that allows one to formulate
with suitably prescribed data a well-posed initial boundary value problem. The
choice of this system is motivated by the fact that Einstein’s equations do
not supply evolution equations for $f$ and $F^{A}$. Its final justification
follows from the existence of a hyperbolic subsidiary system which implies
that the reduced system preserves the constraints and gauge conditions and
that the latter do what they have been chosen for.

This leads to a well-posedness result, local in time, for the initial boundary
value problem for Einstein’s field equations and one could leave it at that.
In this article we would like to show, however, that there are explicit
equations, analogous to (1.1), that reveal the relation between the gauge
source functions $f$ and $F^{A}$, the gauged structures, and the metric. Such
equations will show that the desired gauge can be established under general
assumptions and without any particular equation imposed on $g$. They give, in
particular, additional confirmation that the gauge conditions do not impose
restrictions on the solutions. Moreover, the knowledge of these equations
could help to find gauge source functions that extend the life time of the
gauge and give control on the long term behaviour or other desired features of
the solutions.

On the solution space-time the restriction of $f$ to $T_{c}$ represents the
mean extrinsic curvature induced on this hypersurface and thus encodes an
implicitly evolution law for $T_{c}$. This is in general not independent of
the functions $F^{A}$. The mean extrinsic curvature $\chi$ of the boundary $T$
constitutes in the setting of [7] a boundary datum which can be prescribed
freely. Thus $f$ should be smooth and coincide with $\chi$ on $T$ but can be
chosen rather arbitrary elsewhere. In the special case where the boundary data
are given so that $\chi=\chi_{*}=const.$ on $T$, one can choose $f=\chi_{*}$
on the foliation. As pointed out in [7], this leads to a considerable
simplification. If $\chi$ is point-dependent, however, $f$ must be point-
dependent and the part of the gauge controlled by $F^{A}$ comes into play.
This problem is much more involved and has not been analyzed so far.

The situation is reminiscent of the problem of constructing standard Cauchy
data for Einstein’s field equations with point-dependent mean extrinsic
curvature. For a long time it was customary to construct Cauchy data with
constant mean extrinsic curvature (CMC). The main reason was that this led to
technical simplifications but it had even been assumed occasionally that
asymptotically flat solutions always admit such slices [2]. The analysis of
the Einstein evolution equations, however, does not require such an assumption
on the initial data. In 1982 Dieter Brill showed that there exist
asymptotically flat solutions to Einstein’s field equations which do not admit
a Cauchy hypersurface with vanishing mean extrinsic curvature [2] and in 1988
Robert Bartnik showed that there exist cosmological space-times which do not
admit CMC Cauchy slices [1]. Neglecting data with point-dependent mean
extrinsic curvature may thus exclude large classes of important space-times.
Only recently have been obtained results on such data of some generality [9],
[11].

Similarly, we shall miss out on large classes of space-times developing from
initial and boundary data if the mean extrinsic curvature on the boundary is
required to be constant, a point dependence of $\chi$ could induce the
boundary $T$ to shrink or bulge or oscillate. The nature and origin of the
difficulties arising in the case of a point-dependent mean extrinsic curvature
on the time-like boundary are, however, quite different from those arising in
the construction of standard Cauchy data on space-like slices.

To explain the way the well posed initial boundary value problem is set up in
[7], some considerations of [7] will be recalled in section 2. In section 3
are discussed various aspects of the special case of constant mean extrinsic
curvature $\chi=\chi_{*}$. This will in particular shed additional light on
the result of Grigorios Fournodavlos and Jacques Smulevici [3], who studied
the case of boundaries that are totally geodesic. In the section 4 we finally
consider the case of a point-dependent datum $\chi$. It is shown in which
sense the functions $f$ and $F^{A}$ are related to an implicit quasi-linear,
symmetric hyperbolic system that fixes the gauge.

In the context of the initial boundary value problem long term evolution
projects are beset with all the difficulties known from the standard Cauchy
problem. There are issues, however, that are specific to this problem. If a
solution $(M,g)$ to Einstein’s equation is given, a manifold of the form
$T=\mathbb{R}^{+}_{0}\times\Sigma$ can be smoothly embedded as a time-like
hypersurface but it can also just be immersed so that the pull-back of $g$
still defines a smooth Lorentzian metric on $T$ but the image shows self-
intersections. The ambient space-time then still induces on $T$ smooth vacuum
data of the type considered below. We can thus imagine situations where $T$ is
smoothly embedded close to its initial boundary $\Sigma$ but then, as the
space-time evolves, opposite sides of $T$ start to move towards each other and
threaten to touch and intersect. The point is that the data on $T$ alone may
not allow one to decide whether they are induced by an embedding or an
immersion and even if they are induced by an embedding they may represent
situations where discrete points of $T$ are mapped to points in $M$ lying
quite close to each other. If one is ambitious enough to analyze such
situations the gauge problem may need reconsiderations. In the following we
shall not be ambitious and only consider the initial boundary value problem
local in time.

##  2 The setting.

In the following we consider $4$-dimensional $ST$-space-times $(M,g)$ with
boundary $S\cup T$ and edge $\Sigma=S\cap T$ as described in the introduction.
Our goal is to construct solutions to Einstein’s vacuum field equation
$R_{\mu\nu}[g]=0$ that arise from suitably prescribed initial data on $S$ and
boundary data on $T$. On the domain of dependence of the initial hypersurface
$S$ they are uniquely determined by the standard Cauchy problem for Einstein’s
vacuum field equation with Cauchy data on $S$. Seeking to construct local in
time solutions to initial boundary value problems, a first step is to control
the solutions in a neighbourhood of the edge $\Sigma$ that simultaneously
represents the boundary of $S$ and of $T$. The choice of data on the boundary
$T$ depends very much on the chosen representation of the field equations. We
follow here the discussion of [7], which employs a frame formalism and uses
the vacuum Bianchi equations for the conformal Weyl tensor.

###  2.1 Formalism and gauge conditions.

On any smooth ST-space-time $(M,g)$ can be chosen as follows an $ST$-adapted
gauge, consisting of a smooth coordinate system $x^{\mu}$, $\mu=0,\,.\,.\,,3$,
and smooth local orthonormal frames $e_{k}$, $k=0,\,.\,.\,,3$ near $T$.

The function $x^{3}$ satisfies $x^{3}=0$ on $T$, $x^{3}>0$ elsewhere, so that
the sets $T_{c}=\\{x^{3}=c=const.\\}$ with $0\leq c<c_{*}$ and $T_{0}=T$ are
time-like hypersurfaces diffeomorphic to $T$ that smoothly foliate some
neighbourhood $W$ of $T$ in $M$ with $dx^{3}\neq 0$ on $W$.

The unit vector field $e_{0}$ on $W$ is time-like, future directed, tangential
to the hypersurfaces $T_{c}$, and orthogonal to the 2-surfaces $S_{c}=S\cap
T_{c}$ with $S_{0}=\Sigma$. The space-like vector field $e_{3}$ on $W$
represents the inward directed unit normals to the $T_{c}$.

The local vector fields $e_{A}$, $A=1,2$, define together with $e_{0}$ and
$e_{3}$ a local orthonormal frame on some open subset of $S\cap W$, so that

$g(e_{i},e_{j})=g_{ij}=\eta_{ij}=\mbox{diag}(1,-1,-1,-1).$

The $e_{A}$ are then tangential to the surfaces $S_{c}$. Denoting by $D$ the
Levi-Civita connection of the Lorentz-3-metric induced by $g$ on $T_{c}$ we
require the fields $e_{A}$ to be $D$-Fermi transported in the direction of
$e_{0}$ so that

$g(e_{0},e_{0})\,D_{e_{0}}\,e_{A}+g(e_{A},D_{e_{0}}\,e_{0})\,e_{0}-g(e_{A},e_{0})\,D_{e_{0}}\,e_{0}=0.$

The fields $e_{A}$ are then everywhere tangential to $T_{c}$.

The function $x^{0}$ defines a natural parameter on the integral curves of
$e_{0}$ so that

$<e_{0},dx^{0}>\,=1\quad\mbox{on}\quad W,\quad x^{0}=0\quad\mbox{on}\quad
S\cap W.$ (2.1)

Consider the vector field
$X=(q^{\\#}(dx^{3},dx^{3}))^{-1}\,\mbox{grad}_{q}x^{3}$ tangential to $S$ and
orthogonal to the $S_{c}$, where $q$ denotes the metric induced on $S$ by $g$.
An integral curve $\gamma(\sigma)$ of $X$ with $\gamma(0)\in\Sigma$ satisfies

$\frac{d}{d\sigma}(x^{3}(\gamma(\sigma)))=\,<dx^{3},\frac{d}{d\sigma}\gamma(\sigma)>\,=\,<dx^{3},(q^{\\#}(dx^{3},dx^{3}))^{-1}\,\mbox{grad}_{q}x^{3}>\,=1,$

so that $\sigma=x^{3}(\gamma(\sigma))$, whence $\gamma(c)\in S_{c}$. The flow
of $X$ thus maps $\Sigma$ diffeomorphically onto the $S_{c}$ and we have a
parametrization $W=\\{(x^{0},p,x^{3})|\,\,x^{0}\geq 0,\,\,p\in\Sigma,\,\,0\leq
x^{3}<c_{*}\\}$. Choose local coordinates $x^{\alpha}$, $\alpha=1,2$, on
$\Sigma$, assume them to be dragged into the interior of $S$ with the flow of
$X$ so that

$q^{\\#}(dx^{\alpha},dx^{3})=0,\quad\alpha=1,2,$ (2.2)

and then dragged along with the flow of $e_{0}$ so that

$<e_{0},dx^{\alpha}>\,=0\quad\mbox{on}\quad W,\quad\alpha=1,2.$ (2.3)

The $x^{\alpha}$, $\alpha=1,2$, define local coordinates on the $S_{c}$ and
the $x^{\alpha}$, $\alpha=0,1,2$, define local coordinates on $T_{c}$ for
$0\leq c<c_{*}$. For the frame coefficients satisfying
$e_{k}=e^{\mu}\,_{k}\,\partial_{\mu}$ holds

$e^{\mu}\,_{0}=\delta^{\mu}\,_{0},\quad\,\,\,e^{3}\,_{A}=0,\quad\,\,\,e^{3}\,_{3}>0\quad\mbox{on
\,\,$T_{c}$ \,\,and}\quad e^{0}\,_{A}=0,\quad A=1,2,\quad\mbox{on
\,\,$S_{c}$}.$ (2.4)

We will have to consider three types of projections. Since our frame is well
adapted to the geometrical situation, corresponding projection formalisms can
be avoided by distinguishing three groups of indices. They are given, with the
values they take, by

$a,c,d,e,f=0,1,2;\quad\quad i,j,k,l,m,n=0,1,2,3;\quad\quad A,B,C,D=1,2.$

For each group the summation rule is assumed. If $\nabla$ denotes the
connection defined by $g$, the connection coefficients
$\Gamma_{j}\,^{i}\,{}_{k}$ in the frame $e_{k}$ satisfy
$\nabla_{j}e_{k}\equiv\nabla_{e_{j}}e_{k}=\Gamma_{j}\,^{i}\,{}_{k}\,e_{i}$ and
$\Gamma_{jlk}=-\Gamma_{jkl}$ with
$\Gamma_{jlk}=g_{li}\,\Gamma_{j}\,^{i}\,{}_{k}$. The second fundamental form
induced on $T_{c}$ in the frame $e_{a}$ and the mean extrinsic curvature of
the hypersurfaces $T_{c}$ are given by

$\chi_{ab}\equiv
g(\nabla_{e_{a}}\,e_{3},e_{b})=-g(e_{3},\nabla_{e_{a}}\,e_{b})=\Gamma_{a}\,^{3}\,{}_{b}=\Gamma_{(a}\,^{3}\,{}_{b)},$
$\chi\equiv
g^{ab}\,\chi_{ab}=g^{jk}\,\Gamma_{j}\,^{3}\,{}_{k}=\nabla_{\mu}\,e^{\mu}\,_{3},$
(2.5)

respectively. Because

$D_{a}\,e_{c}\equiv D_{e_{a}}\,e_{c}=\Gamma_{a}\,^{b}\,{}_{c}\,e_{b},$ (2.6)

the $\Gamma_{a}\,^{b}\,{}_{c}$ define the inner connection $D$ on $T_{c}$. The
Fermi condition implies

$\Gamma_{0}\,^{A}\,{}_{B}=0,$ (2.7) $D_{e_{0}}\,e_{0}=F^{A}\,e_{A},\quad
D_{e_{0}}\,e_{A}=-F_{A}\,e_{0}\quad\mbox{with}\quad
F^{A}=\Gamma_{0}\,^{A}\,{}_{0},\quad F_{A}=\eta_{AB}\,F^{A}.$ (2.8)

The freedom of choosing the function $x^{3}$ and the time-like vector field
$e_{0}$ we started with finds new expression in this formalism. With $e_{a}$
as given above the functions $F^{A}=F^{A}(x^{\mu})$ follow from these
formulas. We can, however, also think of the functions $F^{A}$ as being at our
free disposal. If we solve equation (2.8) on $T_{c}$ with arbitrarily
prescribed functions $F^{A}=F^{A}(x^{\alpha},c)$ and initial data satisfying
$g(e_{a},e_{b})=\eta_{ab}$ on $S_{c}$ and $e_{0}$ orthogonal to $S_{c}$, the
solution will satisfy the relation $g(e_{a},e_{b})=\eta_{ab}$ on $T_{c}$.
Since the field equations (2.11), (2.12), (2.13) do not provide propagation
equations for the $F^{A}$ this suggests to consider these functions in the
reduced field equations as smooth gauge source function that can be freely
prescribed. With the special choice

$F^{A}=0,$ (2.9)

the field $e_{0}$ will be $D$-geodesic and the fields $e_{A}$ parallel
propagated. More general choices of $F^{A}$ may allow one to avoid the
development of caustics.

The freedom of choosing the leaves $T_{c}$ of the foliation defined by the
function $x^{3}$ is encoded in the mean extrinsic curvature
$\chi=\chi(x^{\alpha},c)$ induced on $T_{c}$. It will be discussed below in
detail how $T_{c}$ is determined by $\chi=\chi(x^{\alpha},c)$ and the initial
data on $S\cap W$. Since equations (2.11), (2.12), (2.13) do not provide a
propagation equation for $\chi$ this function will be considered as a gauge
source function that will be represented by a smooth function $f=f(x^{\mu})$
in the reduced field equations. There is, however, a slight difference with
the $F^{A}$. The function $\chi(x^{\alpha},0)$ will be used as a free boundary
datum on $T$ that determines the form of $T$. While the function $f$ can be
freely extended into the interior of the solution space-time, it must thus be
given so that $f(x^{\alpha},0)=\chi(x^{\alpha},0)$ on $T$.

Due to the compactness of $\Sigma$, whence of $S_{c}=S\cap T_{c}$, the
coordinates $x^{\alpha}$, $\alpha=1,2$ and, in general, also the frame vector
fields $e_{A}$ are only defined locally on $S_{c}$. If they have to be
redefined on some overlap region $U$ in $W\cap S$ it is important to note that
the corresponding transformation is explicitly controlled along the integral
curves of $e_{0}$ by the propagation laws imposed above and the transformation
on $U$.

If the frame is subject to a transformation $e_{A}\rightarrow
e_{A^{\prime}}=s^{A}\,_{A^{\prime}}\,e_{A}$ in some overlap patch $U\subset
S\cap W$ with some point dependent transformation $s^{A}\,_{A^{\prime}}\in
SO(2)$ on $S_{c}$, we must require

$F^{A}\rightarrow
F^{A^{\prime}}=s^{A^{\prime}}\,_{B}\,F^{B}\,\,\,\mbox{whence}\,\,\,F_{A}\rightarrow
F_{A^{\prime}}=s^{A}\,_{A^{\prime}}\,F_{A}\,\,\,\mbox{with}\,\,\,s^{A}\,_{A^{\prime}}\,s^{A^{\prime}}\,_{B}=\delta^{A}\,_{B},$

to preserve the first of equations (2.8) and the second equation then requires

$D_{e_{0}}s^{A}\,_{A^{\prime}}=0,$

which allows us to control the corresponding transformations

$\eta_{AB}\rightarrow\eta_{A^{\prime}B^{\prime}}=\eta_{AB}\,s^{A}\,_{A^{\prime}}\,s^{B}\,_{B^{\prime}},\quad\beta_{AB}\rightarrow\beta_{A^{\prime}B^{\prime}}=\beta_{AB}\,s^{A}\,_{A^{\prime}}\,s^{B}\,_{B^{\prime}},$

along the integral curves of $e_{0}$.

In the coordinates $x^{\alpha}$, $\alpha=1,2$ described above $f$ is is
assumed to be smooth with local representation $f=f(x^{0},x^{\alpha},x^{3})$
on $W$. If the coordinates are subject on a subset of $\Sigma$ to a
transformation $x^{\alpha}\rightarrow
x^{\beta^{\prime}}=x^{\beta^{\prime}}(x^{\alpha})$ with inverse
$x^{\alpha}=x^{\alpha}(x^{\beta^{\prime}})$, this transformation transfers by
the rules (2.2) and (2.3) into $W$, which allows us to control explicitly the
new coordinate representation
$f^{\prime}=f^{\prime}(x^{0},x^{\alpha^{\prime}},x^{3})=f(x^{0},x^{\beta}(x^{\alpha^{\prime}}),x^{3})$.
In a similar way transform the coordinate representations of the functions
$F^{A}$ on $W$ and of the functions $\chi$, $\eta_{AB}$, $\beta_{AB}$,
$\alpha$, and $\beta$ on $T$ discussed below.

###  2.2 The field equations.

The basic unknowns in the representation of the field equations used in [7]
are

$e^{\mu}\,_{k},\,\,\,\,\,\Gamma_{k}\,^{i}\,{}_{j},\,\,\,\,\,\,C^{i}\,_{jkl},$
(2.10)

where $C^{i}\,_{jkl}$ is a tensor field with the algebraic properties of a
conformal Weyl tensor which for a solution of the equations will in fact
assume that meaning. The field equations are given by the torsion free
condition

$[e_{i},e_{j}]=(\Gamma_{i}\,^{k}\,{}_{j}-\Gamma_{j}\,^{k}\,{}_{i})\,e_{k},$
(2.11)

where the square bracket denotes the commutator of the vector fields, the
curvature relation

$e_{k}(\Gamma_{l}\,^{i}\,{}_{j})-e_{l}(\Gamma_{k}\,^{i}\,{}_{j})+2\,\Gamma_{[k}\,^{i}\,{}_{|m|}\Gamma_{l]}\,^{m}\,{}_{j}-2\,\Gamma_{m}\,^{i}\,{}_{j}\,\Gamma_{[k}\,^{m}\,{}_{l]}=C^{i}\,_{jkl},$
(2.12)

where the left hand side gives the curvature of the connection $\nabla$ in
terms of the connection coefficients and the frame, and the vacuum Bianchi
identity

$\nabla_{i}\,C^{i}\,_{jkl}=0.$ (2.13)

####  2.2.1 Splittings of the conformal Weyl tensor.

In the following we shall need two different decompositions of the conformal
Weyl tensor. The decomposition used in [7] is defined by the time-like frame
vector field $n=e_{0}$. The $n$-electric and the $n$-magnetic part of the
conformal Weyl tensor are given by

$E^{n}_{ik}=p_{i}\,^{m}\,p_{k}\,^{n}\,C_{mjnl}\,n^{j}\,n^{l},\quad\quad
B^{n}_{ik}=p_{i}\,^{m}\,p_{k}\,^{n}\,\frac{1}{2}\,C_{mjpq}\,\epsilon^{pq}\,_{nl}\,n^{j}\,n^{l},$

respectively, where $\epsilon_{ijkl}=\epsilon_{[ijkl]}$ with
$\epsilon_{0123}=1$ and $p_{ij}=g_{ij}-n_{i}\,n_{j}$. These tensors are
symmetric, trace free, and spatial in the sense that $n^{i}\,E^{n}_{ik}=0$ and
$n^{i}\,B^{n}_{ik}=0$. It holds

$C_{ijkl}=2\left(q_{j[k}\,E^{n}_{l]i}-q_{i[k}\,E^{n}_{l]j}-n_{[k}\,B^{n}_{l]m}\,\epsilon^{m}\,_{ij}-n_{[i}\,B^{n}_{j]m}\,\epsilon^{m}\,_{kl}\right),$

with $q_{ij}=g_{ij}-2\,n_{i}\,n_{j}$ and
$\epsilon_{jkl}=n^{i}\,\epsilon_{ijkl}=\epsilon_{0jkl}$, and also

$C_{mnpq}\,n^{m}\,p^{n}\,_{j}\,p^{p}\,_{k}\,p^{q}\,_{l}=C_{0npq}\,p^{n}\,_{j}\,p^{p}\,_{k}\,p^{q}\,_{l}=-B^{n}_{jm}\,\epsilon^{m}\,_{kl},$
$C_{mnpq}\,p^{m}\,_{i}\,p^{n}\,_{j}\,p^{p}\,_{k}\,p^{q}\,_{l}=2\left(p_{j[k}\,E^{n}_{l]i}-p_{i[k}\,E^{n}_{l]j}\right).$

The second decomposition is defined by the space-like frame vector field
$N=e_{3}$. The $N$-electric and the $N$-magnetic part of the conformal Weyl
tensor are given by

$E^{N}_{ik}=k_{i}\,^{m}\,k_{k}\,^{n}\,C_{mjnl}\,N^{j}\,N^{l},\quad\quad
B^{N}_{ik}=k_{i}\,^{m}\,k_{k}\,^{n}\,\frac{1}{2}\,C_{mjpq}\,\epsilon^{pq}\,_{nl}\,N^{j}\,N^{l}.$

These tensors are symmetric, trace free, satisfy $N^{i}\,E^{N}_{ik}=0$,
$N^{i}\,B^{N}_{ik}=0$, and

$C_{ijkl}=2\left(-l_{j[k}\,E^{N}_{l]i}+l_{i[k}\,E^{N}_{l]j}+N_{[k}\,B^{N}_{l]m}\,\bar{\epsilon}^{m}\,_{ij}+N_{[i}\,B^{N}_{j]m}\,\bar{\epsilon}^{m}\,_{kl}\right),$

where $k_{ij}=g_{ij}+N_{i}\,N_{j}$, $l_{ij}=g_{ij}+2\,N_{i}\,N_{j}$,
$\bar{\epsilon}_{ijk}=\epsilon_{ijkl}\,N^{l}=\epsilon_{ijk3}$. It holds

$C_{mnpq}\,N^{m}\,k^{n}\,_{j}\,k^{p}\,_{k}\,k^{q}\,_{l}=C_{3npq}\,k^{n}\,_{j}\,k^{p}\,_{k}\,k^{q}\,_{l}=-B^{N}_{jm}\,\epsilon^{m}\,_{kl},$

i.e.

$C^{3}\,_{abc}=B^{N}\,_{ad}\,\epsilon^{d}\,_{bc3}\quad\mbox{or}\quad
B^{N}\,_{ad}=\frac{1}{2}\,C_{3abc}\,\epsilon^{bc}\,_{3d},$

and

$C_{mnpq}\,k^{m}\,_{i}\,k^{n}\,_{j}\,k^{p}\,_{k}\,k^{q}\,_{l}=2\left(-k_{j[k}\,E^{N}_{l]i}+k_{i[k}\,E^{N}_{l]j}\right).$

The different parts are related by

$B^{N}_{00}=C_{0312}=B^{n}_{33},\quad\quad
B^{N}_{01}=C_{0302}=E^{n}_{23},\quad\quad
B^{N}_{02}=C_{0310}=-E^{n}_{13},\quad\quad$
$B^{N}_{10}=C_{1312}=E^{n}_{32},\quad\quad
B^{N}_{11}=C_{1302}=-B^{n}_{22},\quad\quad
B^{N}_{12}=C_{1310}=B^{n}_{12},\quad\quad$
$B^{N}_{20}=C_{2312}=-E^{n}_{13},\quad\quad
B^{N}_{21}=C_{2302}=B^{n}_{12},\quad\quad
B^{N}_{22}=C_{3201}=-B^{n}_{11},\quad\quad$

$E^{N}_{00}=C_{0303}=E^{n}_{33},\quad\quad
E^{N}_{01}=C_{0313}=-B^{n}_{32},\quad\quad
E^{N}_{02}=C_{0323}=B^{n}_{31},\quad\quad$
$E^{N}_{10}=C_{1303}=-B^{n}_{32},\quad\quad
E^{N}_{11}=C_{1313}=-E^{n}_{22},\quad\quad
E^{N}_{12}=C_{1323}=E^{n}_{12},\quad\quad$
$E^{N}_{20}=C_{2303}=B^{n}_{31}\,\quad\quad
E^{N}_{21}=C_{2313}=E^{n}_{21},\quad\quad
E^{N}_{22}=C_{2323}=-E^{n}_{11}.\quad\quad$

####  2.2.2 The Gauss - Codazzi equations

The tensor

$k_{ij}=g_{ij}+N_{i}\,N_{j}=\eta_{ab}\,\delta^{a}\,_{i}\,\delta^{b}\,_{j},$

represents the metric induced on the hypersurfaces $T_{c}$. We shall need the
well known equations which relate the curvature tensor $R^{i}\,_{jkl}[g]$ of
the metric $g$ to fields living on the hypersurfaces $T_{c}$. Gauss’ equation,
which reads in our formalism

$\bar{R}^{a}\,_{bcd}[k]=R^{a}\,_{bcd}[g]-\chi_{c}\,^{a}\,\chi_{db}+\chi_{d}\,^{a}\,\chi_{cb},$
(2.14)

relates it to the curvature tensor $\bar{R}^{a}\,_{bcd}[k]$ of $k$ and the
second fundamental form. Codazzi’s equation

$D_{c}\chi_{db}-D_{d}\chi_{cb}=R^{3}\,_{bcd}[g],$ (2.15)

relates it to $T_{c}-$intrinsic derivatives of the second fundamental form.
With the well known decomposition of of the curvature tensor on
$3$-dimensional spaces and the relations on the Weyl tensor above, Gauss’
equation can be written on vacuum solutions

$k_{b[d}\,R_{c]a}[k]+k_{a[c}\,R_{d]b}[k]+\frac{1}{3}\,R[k]\,k_{a[d}\,k_{c]b}$
$=2\left(k_{b[d}\,E^{N}_{c]a}+k_{a[c}\,E^{N}_{d]b}\right)-\chi_{ca}\,\chi_{db}+\chi_{da}\,\chi_{cb}.$

where $R_{ab}[k]$ and $R[k]$ denote the Ricci tensor and the Ricci scalar of
$k$. A contraction gives

$\frac{1}{2}\,\left(R_{db}[k]+\frac{R[k]}{3}\,k_{db}\right)=E^{N}_{db}-\chi\,\chi_{bd}+\chi_{d}\,^{c}\,\chi_{bc}.$
(2.16)

Codazzi’s equation takes on a vacuum solution the form

$D_{c}\chi_{db}-D_{d}\chi_{cb}=C^{3}\,_{bcd}=B^{N}\,_{be}\,\epsilon^{e}\,_{cd3}.$
(2.17)

####  2.2.3 Reduced equations and boundary data.

In the following a few remarks will be made on the reduced equations, the
initial and boundary data, the corner conditions, and the subsidiary
equations. For details (which are sometimes slightly rewritten here) the
reader is referred to [7].

With the gauge conditions and the gauge source functions
$F^{A}=F^{A}(x^{\mu})$ and $f=f(x^{\mu})$ equations (2.11), (2.12), (2.13)
imply a symmetric hyperbolic system of reduced equation for the unknowns

$e^{\beta}\,_{a},\quad
e^{\mu}\,_{3},\quad\Gamma_{A}\,^{B}\,{}_{0},\quad\Gamma_{A}\,^{B}\,{}_{C},\quad\Gamma_{3}\,^{A}\,{}_{B},\quad\Gamma_{3}\,^{A}\,{}_{0},\quad\Gamma_{3}\,^{3}\,{}_{A},\quad\Gamma_{3}\,^{3}\,{}_{0},$
$\chi_{01},\quad\chi_{02},\quad\chi_{11},\quad\chi_{12},\quad\chi_{22},\quad
B^{n}_{ab},\quad E^{n}_{ab}.$

The field $\chi_{00}$, which only occurs in non-differentiated form in these
equations, is taken care of by writing $\chi_{00}=\chi_{11}+\chi_{22}+f$.

The initial data on $S$ are given by standard Cauchy data, i.e. a solution to
the vacuum constraints on space-like hypersurfaces, which extend smoothly to
the boundary $\Sigma$ of $S$.

To describe the boundary conditions we consider the trace free parts of the
orthogonal projections of the $n$-electric and $n$-magnetic parts of the
conformal Weyl tensor on $T$ into the planes orthogonal to $e_{3}$ and
$e_{0}$. The corresponding symmetric trace free tensors $\eta_{AB}$ and
$\beta_{AB}$, which by the relations given above can also be expressed in
terms of the $N$-electric and $N$-magnetic parts of the conformal Weyl tensor,
are given by

$\beta_{11}=-\beta_{22}=-\frac{1}{2}\,(C_{3201}+C_{3102})=\frac{1}{2}\,(B^{n}_{11}-B^{n}_{22})=\frac{1}{2}\,(B^{N}_{11}-B^{N}_{22}),$
(2.18) $\beta_{12}=\beta_{21}=C_{3101}=B^{n}_{12}=B^{N}_{12},$

and

$\eta_{11}=-\eta_{22}=\frac{1}{2}\,(C_{1010}-C_{2020})=\frac{1}{2}\,(E^{n}_{11}-E^{n}_{22})=\frac{1}{2}\,(E^{N}_{11}-E^{N}_{22}),$
(2.19) $\eta_{12}=\eta_{21}=E_{12}=C_{1020}=E^{n}_{12}=E^{N}_{12}.$

In terms of the pseudo-orthonormal frame $l$, $k$, $m$ satisfying (with
$e_{3}$ inward pointing)

$\sqrt{2}\,l=e_{0}+e_{3},\,\,\,\,\,\,\sqrt{2}\,k=e_{0}-e_{3},\,\,\,\,\,\,\sqrt{2}\,m=e_{1}-i\,e_{2},$

and the Newman-Penrose notation for the curvature tensor, the relevant
components of the conformal Weyl tensor are given by

$\Psi_{0}=C_{\mu\nu\sigma\pi}\,l^{\mu}\,m^{\nu}\,l^{\sigma}\,m^{\pi}=\eta_{11}+\beta_{12}+i\,(\beta_{11}-\eta_{12}),$
$\Psi_{4}=C_{\mu\nu\sigma\pi}\,\bar{m}^{\mu}\,k^{\nu}\,\bar{m}^{\sigma}\,k^{\pi}=\eta_{11}-\beta_{12}+i\,(\beta_{11}+\eta_{12}).$

The boundary conditions of [7] then take the form

$f=\chi,\quad-\Psi_{4}+\alpha\,\Psi_{0}+\beta\,\bar{\Psi}_{0}=q\quad\mbox{on}\quad
T,$ (2.20)

where $\alpha$ and $\beta$ are complex-valued functions on $T$ that satisfy

$|\alpha|+|\beta|\leq 1,$ (2.21)

and $q$, the main datum besides $\chi$ on $T$, is a smooth complex-valued
function on $T$ that can be prescribed, consistent with the conditions
discussed below, freely. Condition (2.21), which looks simpler than the
corresponding condition given in [7], is obtained from the latter by
diagonalizing the matrix $B$ used there to express the restrictions on
$\alpha$ and $\beta$.

As special examples, which will be of interest below, we note that the
admissible choice $\alpha=0$ and $\beta=1$ results in the boundary condition

$q=2\,(\beta_{12}-i\,\beta_{11})=2\,B^{n}_{12}-i\,(B^{n}_{11}-B^{n}_{22})=2\,B^{N}_{12}-i\,(B^{N}_{11}-B^{N}_{22})\quad\mbox{on}\quad
T.$ (2.22)

It only involves magnetic parts. The choice $\alpha=0$, $\beta=-1$ gives the
boundary condition

$q=-2\,(\eta_{11}+i\,\eta_{12})=E^{n}_{22}-E^{n}_{11}-2\,i\,E^{n}_{12}=E^{N}_{22}-E^{N}_{11}-2\,i\,E^{N}_{12}\quad\mbox{on}\quad
T,$ (2.23)

which only involves electric parts of the conformal Weyl tensor.

To determine a smooth solution, the initial and the boundary data must satisfy
a consistency condition, the so-called corner condition at the edge $\Sigma$.
Because the reduced equations with given $f$ and $F^{A}$ are symmetric
hyperbolic, the Cauchy data on $S$ determine in our gauge a unique formal
expansion type solution on $S=\\{x^{0}=0\\}$. For given functions $\alpha$ and
$\beta$ this expansion determines, in particular, a unique formal expansion of
the fields on the left hand sides of (2.20) at $\Sigma$ . The corner
conditions require the data $\chi$ and $q$ on the right hand sides to be
prescribed consistent with these expansions at $\Sigma$.

For given Cauchy data it is always possible to find boundary data that satisfy
this condition, which leaves the right hand sides of (2.20) essentially
arbitrary away from $\Sigma$. If the boundary data are supposed to satisfy
certain conditions, possibly suggested by some intended application, it
requires an extra effort (and may not be possible) to construct Cauchy data,
i.e. solutions to the constraint equations on $S$, that meet these
requirements.

The main result of [7] says (in the notation introduced above):

Let be given smooth Cauchy data for Einstein’s vacuum field equations on the
compact $3$-manifold $S$ with boundary $\Sigma$ and smooth boundary data

$\chi,\,\,\,q\quad\mbox{and functions}\quad\alpha,\,\,\beta\quad\mbox{on}\quad
T=\mathbb{R}^{+}\times\Sigma,$ (2.24)

satisfying (2.21). Choose on $M=\mathbb{R}^{+}\times S$ smooth gauge source
functions $f$ with $f=\chi$ on $T$ and $F^{A}$ that satisfy together with the
initial and boundary data the corner conditions at $\Sigma$ defined by the
reduced field equations. Then there exists for some $x^{0}_{*}>0$ a unique
smooth solution $g$ to Einstein’s vacuum equations on the manifold
$M^{\prime}=[0,x^{0}_{*}[\,\times\,S\subset\mathbb{R}^{+}\times S$ so that
$S\equiv\\{0\\}\times S$ is space-like,
$T^{\prime}=[0,x^{0}_{*}[\,\times\,\Sigma\subset T$ is time-like, $g$ induces
the given Cauchy data on $S$, $\chi$ acquires on $T^{\prime}$ the meaning of
the mean intrinsic curvature induced on $T^{\prime}$ by $g$ and $q$ coincides
with the function of the conformal Weyl tensor of $g$ on the left hand side of
the second equation of (2.20).

The first step to arrive at this result consists in setting up local initial
boundary value problems for the reduced field equations and showing the
existence of local solutions near given points of $\Sigma$. In a next step the
local solutions are patched together to obtain a solution covering a
neighbourhood of $\Sigma$. Because, as pointed out above, the transformations
between local solutions can be explicitly controlled, there arises no problem.
The solution near $\Sigma$ is then patched together with the solution to the
Cauchy problem for the reduced equations that is determined by the data on
$S$.

This establishes the existence of a unique solution to the reduced equations
on a manifold of the form $M^{\prime}=[0,x^{0}_{*}[\,\times\,S$. In a final
step a hyperbolic subsidary system is derived that supplies an argument that
the solution to the reduced equations is in fact a solution to the Einstein
equations.

Suppose $(M,g)$ is a $ST$-vacuum solution. After choosing a gauge as above one
can read off the data induced on $S$ and $T$ and the gauge source functions
$f$ and $F^{A}$ near $T$ (the corner conditions will, of course, be
satisfied). Our result then shows that local in time the given $ST$-vacuum
solution will be reconstructed uniquely by our method. Thus local in time all
$ST$-vacuum solutions are covered by the existence result above.

##  3 Time-like hypersurfaces and mean extrinsic curvature.

In this section we discuss the basic equation associated with the mean
extrinsic curvature on time-like hypersurfaces. We consider then the case of
constant mean extrinsic curvature and derive the equations which are in this
case implicit in the formulation of the initial boundary value problem of [7].
It follows a discussion of totally geodesic boundaries.

###  3.1 The basic equation.

Let $({\cal M},g)$ be a $4$-dimensional space-time, ${\cal S}$ a space-like
hypersurface which we assume for convenience to be a Cauchy hypersurface of
${\cal M}$, and ${\cal T}$ a time-like hypersurface which intersects ${\cal
S}$ so that it cuts out from it a compact $3$-dimensional manifold $S$ with
compact space-like boundary $\Sigma={\cal S}\cap{\cal T}$. The sets $S$ and
$\Sigma$ may be thought of as identical with the ones labeled by the same
symbols in the previous section. The set ${\cal T}$ does represent a space-
time boundary but serves as a subsidiary hypersurface to establish a certain
equation. It is fairly arbitrary, only when we arrive at condition (4.11) we
will need to restrict it, together with the coordinates $z^{\mu}$ introduced
below, further. The metric $g$ is not required to satisfy any field equation.
In the following the coordinate indices $\alpha,\beta,\gamma,\delta$ take
values $0,1,2$ while the coordinate indices $\kappa,\lambda,\mu,\nu,\pi,\rho$
take values $0,1,2,3$.

Let $z^{0}$, $z^{3}$ with $dz^{0}\neq 0$, $dz^{3}\neq 0$ denote smooth
functions defined on a neighbourhood of $\Sigma$ so that ${\cal
S}=\\{z^{0}=0\\}$, ${\cal T}=\\{z^{3}=0\\}$, and $z^{3}>0$ on $S$ on that
neighbourhood. Set $S_{c}=\\{z^{0}=0,\,\,z^{3}=c=const.\\}$ for $0\leq
c<c_{*}$ so that $S_{0}=\Sigma$. By our assumptions we have
$g^{00}=g^{\\#}(dx^{0},\,dx^{0})>0$ on ${\cal S}$,
$g^{33}=g^{\\#}(dx^{3},\,dx^{3})<0$ on ${\cal T}$, and thus $g^{00}>0$ and
$g^{33}<0$ on the $S_{c}$ for suitably chosen $c_{*}>0$. In the following this
will always be assumed.

The functions $z^{0}$, $z^{3}$ are complemented by functions $z^{1}$, $z^{2}$
so that the $z^{\mu}$ define local coordinates, $z^{1}$, $z^{2}$, $z^{3}$
define local coordinates on $S$, $z^{1}$, $z^{2}$ define local coordinates on
the $S_{c}$, and $z^{0}$, $z^{1}$, $z^{2}$ define local coordinates on ${\cal
T}$. The vector field $\partial_{z^{0}}$ is future directed.

To construct a time-like hypersurface $T_{c}$ with $T_{c}\cap{\cal S}=S_{c}$
whose mean extrinsic curvature coincides with a given function $\chi$ we
assume it to be essentially given as the graph of a function
$\phi=\phi(z^{\alpha})$ over ${\cal T}$ so that

$T_{c}=\\{\Phi(z^{\mu})=c\\}\quad\mbox{where}\quad\Phi(z^{\mu})=z^{3}-\phi(z^{\alpha})\quad\mbox{with}\quad$
$\phi|_{z^{0}=0}=0\quad\mbox{whence}\quad\phi_{,\alpha}|_{z^{0}=0}=\delta^{0}\,_{\alpha}\,\phi_{,0}\quad\mbox{on}\quad
S_{c}.$ (3.1)

For $T_{c}$ to be time-like we need

$\nabla_{\nu}\,\Phi\,\nabla^{\nu}\,\Phi=g^{33}-2\,g^{30}\,\phi_{,0}+g^{00}\,\phi_{,0}\,\phi_{,0}<0\quad\mbox{on}\quad
S_{c}.$ (3.2)

This inequality is satisfied if

$\frac{g^{03}}{g^{00}}-\sqrt{-\frac{g^{33}}{g^{00}}+\left(\frac{g^{03}}{g^{00}}\right)^{2}}<\phi_{,0}<\frac{g^{03}}{g^{00}}+\sqrt{-\frac{g^{33}}{g^{00}}+\left(\frac{g^{03}}{g^{00}}\right)^{2}}.$
(3.3)

Since $g^{33}<0$ and $g^{00}>0$ on $S_{c}$ the roots are real and the unit
normal to $T_{c}$ will be well defined near $S_{c}$ and given by the
restriction to $T_{c}$ of the vector field

$N^{\mu}=\nu\,\nabla^{\mu}\,\Phi=\nu\,(g^{\mu
3}-g^{\mu\alpha}\phi_{,\alpha}),$ (3.4)

with

$\nu=-(-\nabla_{\nu}\,\Phi\,\nabla^{\nu}\,\Phi)^{-\frac{1}{2}}=-(-g^{33}+2\,g^{3\alpha}\,\phi_{,\alpha}-g^{\alpha\beta}\,\phi_{,\alpha}\,\phi_{,\beta})^{-\frac{1}{2}}.$

The gradient fields in brackets on the right hand side of (3.4) are orthogonal
to $\\{x^{3}=c\\}$ and ${\cal S}$ respectively and thus to the tangent spaces
$T_{p}S_{c}$ of $S_{c}$. With $\phi_{,0}$ varying in the range (3.3) the
vector $N^{\mu}$ exhausts all space-like directions in the 2-plane orthogonal
to the tangent space $T_{p}S_{c}$ of $S_{c}$. That plane contains the time-
like vector

$T^{\mu}=(g^{03}-\phi_{,0}\,g^{00})\,g^{\mu
3}-(g^{33}-\phi_{,0}\,g^{03})g^{\mu 0},$ (3.5)

which is orthogonal to $N$ and $T_{p}S_{c}$ and unique up to a factor. It
follows with (3.2)

$g^{33}-\phi_{,0}\,g^{03}<\frac{1}{2}\,(g^{33}-g^{00}\,\phi^{2}_{,0})<0\quad\mbox{on
$\,S_{c}\,$ near $\,\Sigma$}.$ (3.6)

The metric induced on $T_{c}$ is given by

$k_{\mu\nu}=g_{\mu\nu}+N_{\mu}\,N_{\nu},$ (3.7)

the induced second fundamental form is given by

$\chi_{\mu\nu}=k_{\mu}\,^{\kappa}\,k_{\nu}\,^{\lambda}\,\nabla_{\kappa}\,N_{\lambda}=\nu\,k_{\mu}\,^{\kappa}\,k_{\nu}\,^{\lambda}\,\nabla_{\kappa}\,\nabla_{\lambda}\,\Phi,$
(3.8)
$=-\nu\,k_{\mu}\,^{\kappa}\,k_{\nu}\,^{\lambda}\,(\phi_{,\alpha\beta}\,\delta^{\beta}\,_{\kappa}\,\delta^{\alpha}\,_{\lambda}+\Gamma_{\kappa}\,^{\rho}\,{}_{\lambda}(\delta^{3}\,_{\rho}-\phi_{,\alpha}\,\delta^{\alpha}\,_{\rho}))$

and the mean extrinsic curvature by

$\chi=k^{\mu\nu}\,\chi_{\mu\nu}=k^{\mu\nu}\,\nabla_{\mu}\,N_{\nu}=\nabla_{\mu}\,N^{\mu}=\nu\,k^{\mu\nu}\,\nabla_{\mu}\,\nabla_{\nu}\,\Phi.$

The equation which controls the evolution of $\phi$ and thus of $T_{c}$ thus
reads

$-\nu\,k^{\alpha\beta}\,\partial_{\alpha}\,\partial_{\beta}\,\phi-\nu\,k^{\mu\nu}\,(\Gamma_{\mu}\,^{3}\,{}_{\nu}-\Gamma_{\mu}\,^{\alpha}\,{}_{\nu}\,\phi_{,\alpha})=\chi,$
(3.9)

where here and in the following the background fields $g_{\mu\nu}$
$\Gamma_{\mu}\,^{\rho}\,{}_{\nu}$ as well as $\nu$, $k^{\mu\nu}$ etc. have to
be taken at the points $(z^{\alpha},z^{3})=(z^{\alpha},\phi(z^{\alpha})+c)$.

In the following we will consider the $z^{\alpha}$ also as coordinates on
$T_{c}$. A vector field on $T_{c}$ will then be of the form
$X=X^{\alpha}\,\partial_{z^{\alpha}}$. Considered as a vector field in $M$ is
must be written
$X^{\prime}=X^{\alpha}\,\partial_{z^{\alpha}}+\phi_{,\alpha}\,X^{\alpha}\,\partial_{z^{3}}$.
It holds then $g(X^{\prime},Y^{\prime})=\bar{k}(X,Y)$ with $\bar{k}$ the pull
back $\bar{k}$ of $k_{\mu\nu}\,dz^{\mu}\,dz^{\nu}$ to the hypersurface
$T_{c}=\\{(z^{\mu})=(z^{\alpha},\phi(z^{\alpha})+c)\\}$. It coincides with the
pull back of $g_{\mu\nu}\,dz^{\mu}\,dz^{\nu}$ to $T_{c}$ and is given by

$\bar{k}=\bar{k}_{\alpha\beta}\,dz^{\alpha}\,dz^{\beta}=(g_{\alpha\beta}+2\,g_{3(\alpha}\,\phi_{,\beta)}+g_{33}\,\phi_{,\alpha}\,\phi_{,\beta})\,dz^{\alpha}\,dz^{\beta}.$
(3.10)

As long as $T_{c}$ is time-like this metric is Lorentzian.

The principal part of equations (3.9) is governed by the symmetric tensor

$k^{\alpha\beta}=\bar{k}^{\alpha\beta}\equiv
g^{\alpha\beta}+\nu^{2}\,(g^{\alpha
3}-g^{\alpha\gamma}\,\phi_{,\gamma})\,(g^{\beta
3}-g^{\beta\delta}\,\phi_{,\delta})\quad\mbox{on}\quad T_{c},$ (3.11)

with

$\nu^{2}=(-g^{33}+2\,g^{3\alpha}\,\phi_{,\alpha}-g^{\alpha\beta}\phi_{,\alpha}\,\phi_{,\beta})^{-1}.$

It satisfies

$\bar{k}_{\alpha\beta}\,\bar{k}^{\beta\gamma}=\delta_{\alpha}\,^{\beta}.$

Suppose $\chi$ is is a given function of four variables. We write then
$\chi=\chi(z^{\alpha},\phi(z^{\alpha})+c)$ on the right hand side of (3.9).
Equation (3.9) defines then a quasi-linear wave equation for $\phi$. With a
given, sufficiently small constant $c$, a given right hand side $\chi$ as
above, and initial data $\phi$ and $\phi_{,0}$ on $S_{c}$ that satisfy (3.1)
and (3.2) it determines a unique solution $\phi$ near $S_{c}$ for which the
hypersurface $T_{c}=\\{z^{3}=\phi(z^{\alpha})+c\\}$ is time-like. Because of
the smooth dependence of the solutions on $c$, the initial data and $\chi$,
the hypersurfaces $T_{c}$ represent the leaves of a smooth local foliation of
${\cal M}$ near $\Sigma$ that defines a smooth function $x^{3}$ with $x^{3}=c$
on $T_{c}$.

####  3.1.1 Boundaries of constant mean extrinsic curvature.

The coordinate dependence of the function $\chi$ has been specified above in a
somewhat cursory way. In the context of the initial boundary value problem it
requires in general further considerations. There is, however, a case, pointed
out already in [7], where things simplify considerably. If the data (2.20) are
given with

$\chi=\chi_{*}=\mbox{constant on $T$},$ (3.12)

we can set $f=\chi_{*}$ near $T$. Equation (3.9) simplifies and the
construction of the hypersurfaces $T_{c}$ and the coordinate $x^{3}$
completely decouples from setting up the frame vectors $e_{a}$ and coordinates
$x^{\alpha}$ and thus from the choice of the $F^{A}$. The reduced equations
will ensure that their solution satisfies $\chi=\chi_{*}$ near $T$.

Simple though not uninteresting examples of such situations are given by the
hypersurfaces $T=\\{r=const.>2\,m\\}$ of the Schwarzschild solution in
standard Schwarzschild coordinates. Their mean extrinsic curvature is given by

$\chi=-\sqrt{1-\frac{2\,m}{r}}\,\left(\frac{2\,r-3\,m}{r\,(r-2\,m)}\right).$
(3.13)

This still leaves the freedom to prescribe as a datum the function $q$ in
(2.20) on $T$. Its deviation from the Schwarzschild values may be thought of
as representing ingoing or outgoing gravitational radiation. The situation
changes if we consider the Kerr solution in Boyer-Lindquist coordinates. The
mean extrinsic curvature of the hypersurface $T=\\{r=const.>2\,m\\}$ is then
given by

$\chi=-\frac{1}{2}\,\sqrt{\frac{r^{2}-2\,m\,r+a^{2}}{r^{2}+a^{2}\,\cos^{2}\theta}}\left\\{\frac{2\,(r-m)}{r^{2}-2\,m\,r+a^{2}}+\frac{2\,r}{r^{2}+a^{2}\,\cos^{2}\theta}\right\\}.$

####  3.1.2 Totally geodesic boundaries

An even more restricted case is considered by Grigorios Fournodavlos and
Jacques Smulevici [3], who study the existence of solutions with boundaries
that are totally geodesic, emphasizing that the problem of geometric
uniqueness discussed in [6] is absent here. We consider this situation here
again because the present formalism sheds additional light onto the special
nature of these issues in this case.

In general we are free to prescribe three functions on the boundary. On the
face of it, the condition $\chi_{ab}=0$, needed to make the boundary totally
geodesics, thus looks much too strong. Precisely because all components of
$\chi_{ab}$ are required to vanish, it turns out, however, that the problem
can be reduced to the prescription of three functions. Suppose $(M,g)$ is a
$ST$ vacuum solution with totally geodesic time-like boundary $T$ so that
$\chi_{ab}=0$ on $T$. To set up a gauge as described in section 2, we can use
equation (3.9) with $\chi=0$ on the right hand side to construct the function
$x^{3}$, which has been handpicked before, so that $f=0$ near $T$. For
convenience we can then require (2.9) near $T$.

The Codazzi equation (2.17) implies with our assumption that

$B^{N}\,_{ab}=0\quad\mbox{on}\quad T,$ (3.14)

and thus the, by (2.22) admissible, boundary conditions

$\chi=0,\quad\beta_{12}=B^{N}_{12}=0,\quad\beta_{11}=\frac{1}{2}\,(B^{N}_{11}-B^{N}_{22})=0\quad\mbox{on}\quad
T.$ (3.15)

Together with the Cauchy data induced by $g$ on $S$ these boundary conditions
determine the solution uniquely near $S$. Because the solution is smooth, the
corner conditions are satisfied, which guarantees that

$\chi_{ab}=0\quad\mbox{on}\quad\Sigma.$ (3.16)

But then the data (3.15) and (3.16) lead us back to where we started from. The
reduced equations derived in [7] comprise under general assumptions the
subsystem

$D_{0}\,\chi_{01}-D_{1}\,\chi_{11}-D_{2}\,\chi_{12}=D_{1}(f),$
$D_{0}\,\chi_{02}-D_{1}\,\chi_{12}-D_{2}\,\chi_{22}=D_{2}(f),$
$\quad\quad\quad\,\,\,\,D_{0}\,\chi_{11}-D_{1}\,\chi_{01}=-\beta_{12},$
$2\,D_{0}\,\chi_{12}-D_{1}\,\chi_{02}-D_{2}\,\chi_{01}=2\,\beta_{11},\quad$
$\quad\quad\quad\,\,D_{0}\,\chi_{22}-D_{2}\,\chi_{02}=\beta_{12},$

where $f=\chi_{00}-\chi_{11}-\chi_{22}$,
$\beta_{11}=\frac{1}{2}\,(B^{n}_{11}-B^{n}_{22})=\frac{1}{2}\,(B^{N}_{11}-B^{N}_{22})$
and $\,\,\beta_{12}=B^{n}_{12}=B^{N}_{12}$. Because the vector fields $e_{a}$
are tangential to $T$ this subsystem defines a system intrinsic to the
boundary $T$. It is symmetric hyperbolic. With the boundary condition (3.15)
the right hand sides of the equations vanish on $T$. Observing the consistency
condition (3.16), we can conclude, without explicit knowledge of the frame and
the connection coefficients on $T$, that $\chi_{ab}=0$ on $T$.

The argument shows that every $ST$-vacuum solution with totally geodesic
boundary $T$ can be obtained locally in time by solving the initial boundary
value problem with boundary conditions that satisfy (3.15) and Cauchy data
that imply (3.16). The main problem of characterizing all such solutions then
reduces to the construction of Cauchy data for Einstein’s vacuum field
equations on $3$-manifolds $S$ with boundary $\Sigma$ for which the reduced
equations determine a formal expansion with $\chi_{ab}=0$, $B^{N}\,_{ab}=0$ on
$\Sigma$.

The particular choice of the gauge source functions made here is convenient
but nowhere enters the argument. In fact, because of (3.14) conditions (3.15)
will be satisfied in any frame. The obstructions to geometric uniqueness
pointed out in [6] simply do not occur in this particular case. Anyway, the
discussion of [6] does not ask so much for specific cases in which geometric
uniqueness may hold but refers to the general problem with the complete
freedom of prescribing three functions as boundary data.

In the case of anti-de Sitter type solutions that admit a smooth conformal
boundary ${\cal J}$ at space-like and null infinity, the boundary it totally
geodesic in a suitable conformal gauge as consequence of the field equations.
Nevertheless, there is the freedom, in a sense similar to the second of
conditions (2.20), to freely prescribe on ${\cal J}$ two functions derived
from the conformal Weyl tensor. Moreover, with a condition similar to (2.22)
the boundary conditions can be stated in a completely geometric way [5].

Since the Codazzi equation so much simplifies the above argument one may
wonder whether something similar could be done by imposing conditions on the
metric induced on $T$ and using Gauss’ equation (2.16). Because of the
occurrence of the second fundamental form in that relation there appears to be
no obvious way.

##  4 Prescribed mean extrinsic curvature.

The case of point dependent mean intrinsic curvature is complicated because
the function $\chi$ (or the gauge source function $f$) is not given in terms
of some arbitrary coordinates like the $z^{\alpha}$ considered above. A
relation like: $\chi=\chi(x^{\alpha^{\prime}},c)$ on $T_{c}$ or
$f=f(x^{\alpha^{\prime}},c)$ on $T_{c}$ in section 2 is saying that we must
think of $\chi$ or $f$ as being given in the specific coordinates
$x^{\alpha^{\prime}}$ on $T_{c}$ that are obtained by solving equations (2.1),
(2.3), (2.8) on $T_{c}$. We write $x^{\alpha^{\prime}}$ here to distinguish
the index coming with $x$ from the completely unrelated index of $z^{\alpha}$.
While the function $f$ can be chosen freely away from the boundary $T=T_{0}$,
the free data must be specified on $T$ in some distinguished coordinate
system. This coupling between the gauge on $T$ and the way boundary data are
prescribed, which is a specific feature of the initial boundary value problem,
cannot be avoided unless the boundary conditions are completely stated in
terms geometric structures (see the discussion in [6]). The functions
$F^{A}=F^{A}(x^{\alpha^{\prime}})$ will have to play a role when we specify
$\chi$ or $f$.

Equations (2.1), (2.3), (2.8), i.e.

$D_{e_{0}}\,x^{\alpha^{\prime}}=\delta^{\alpha^{\prime}}\,_{0},\quad\quad
D_{e_{0}}\,e_{0}=F^{A}\,e_{A},\quad\quad D_{e_{0}}\,e_{A}=-F_{A}\,e_{0},$
(4.1)

are defined in terms of structures supposed to be induced by the ambient
space-time on the hypersurface $T_{c}$ which we want to determine by solving
equation (3.9). The right hand side of the latter must thus be arranged so as
to correspond in the coordinates $x^{\alpha^{\prime}}$ to the datum
$\chi=\chi(x^{\alpha^{\prime}},c)$ (or $f=f(x^{\alpha^{\prime}},c)$) assumed
as known.

This suggests to consider $F^{A}(x^{\alpha^{\prime}},c)$,
$f(x^{\alpha^{\prime}},c)$, $\chi(x^{\alpha^{\prime}},c)$ as given functions
and use (4.1) with $F^{A}=F^{A}(x^{\alpha^{\prime}}(z^{\beta},c),c)$ as a
system of equations for $x^{\alpha^{\prime}}=x^{\alpha^{\prime}}(z^{\beta},c)$
and the frame vectors $e_{a}$. The latter are thought to be given in the form
$e^{\alpha}\,_{a}\,\partial_{z^{\alpha}}$ with
$e^{\alpha}\,_{a}=e^{\alpha}\,_{a}(z^{\beta})$ and can be expressed in terms
of the coordinates $x^{\alpha^{\prime}}$ once the coordinate transformation is
available on $T_{c}$. The system (4.1) must be coupled to (3.9) with
$\chi=\chi(x^{\alpha^{\prime}}(z^{\beta},c),c)$ or
$f=f(x^{\alpha^{\prime}}(z^{\beta},c),c)$ on the right hand side to determine
$\phi$ and thus $T_{c}$. The dependence of the various functions on $c$ will
often be suppressed in the following because we mostly work with a fixed $c$,.

This recipe does not work immediately. Two of equations (4.1) involve the
Levi-Civita connection $D$ of the metric $\bar{k}_{\alpha\beta}$ on $T_{c}$ by
(3.10), and thus the Christoffel symbols

$\bar{\zeta}\,_{\alpha}\,^{\beta}\,{}_{\gamma}=\frac{1}{2}\,\bar{k}^{\beta\,\delta}\,(\bar{k}_{\delta\gamma,\alpha}+\bar{k}_{\alpha\delta,\gamma}-\bar{k}_{\alpha\gamma,\delta})\quad\mbox{on}\quad
T_{c}.$ (4.2)

They depend on the functions $g_{\mu\nu}(z^{\alpha},\phi(z^{\alpha})+c)$,
their derivatives
$g_{\mu\nu,\beta}(z^{\alpha},\phi(z^{\alpha})+c)+g_{\mu\nu,3}(z^{\alpha},\phi(z^{\alpha})+c)\,\phi_{,\beta}$,
on the $\phi_{,\alpha}$, and in particular on the second derivatives
$\phi_{,\alpha\beta}$. To control these functions we write equation (3.9) in
the form

$-\nu\,k^{\alpha\beta}\,\partial_{\alpha}\,\partial_{\beta}\,\phi-F=\chi(x^{\alpha^{\prime}}(z^{\beta})),$
(4.3)

with

$F=\nu\,k^{\mu\nu}\,(\Gamma_{\mu}\,^{3}\,{}_{\nu}-\Gamma_{\mu}\,^{\alpha}\,{}_{\nu}\,\phi_{,\alpha})=F(\phi,\phi_{,\alpha},\mbox{background}),$

where the background function depend on $(z^{\alpha},\phi(z^{\alpha})+c)$. To
obtain an equation that supplies the $\phi_{,\alpha\beta}$, we apply
$\partial_{z^{\gamma}}$ to the equation above and obtain

$-\nu\,k^{\alpha\beta}\,\partial_{\alpha}\,\partial_{\beta}\,(\partial_{\gamma}\,\phi)-G=\partial_{x^{\alpha^{\prime}}}\chi\,\,\partial_{z^{\gamma}}x^{\alpha^{\prime}}$
(4.4)

with some

$G=G(\phi,\phi_{,\alpha},\phi_{,\alpha\beta},\mbox{background}).$

where the background functions now involve
$(z^{\alpha},\phi(z^{\alpha}),\phi_{,\beta}(z^{\alpha}))$. For given smooth
functions $x^{\alpha^{\prime}}(z^{\beta})$ the two equations above define a
system of wave equations for $\phi$ and $\phi_{,\alpha}$. In addition to
$x^{\alpha^{\prime}}(z^{\beta})$ the right hand side of (4.4) requires,
however, also control of

$\partial_{z^{\gamma}}\,x^{\alpha^{\prime}}=D_{\gamma}x^{\alpha^{\prime}}.$
(4.5)

The solvability of the coupled system (4.1), (4.3), (4.4) thus depends on the
possibility to implement equations for this field. The first of equations
(4.1) implies

$0=D_{\gamma}\,(e^{\alpha}\,_{0}\,D_{\alpha}\,x^{\beta^{\prime}})=e^{\alpha}\,_{0}\,D_{\gamma}\,D_{\alpha}\,x^{\beta^{\prime}}+D_{\gamma}\,e^{\alpha}\,_{0}\,D_{\alpha}\,x^{\beta^{\prime}}$
$=D_{e_{0}}\,(D_{\gamma}\,x^{\beta^{\prime}})+D_{\gamma}\,e^{\alpha}\,_{0}\,(D_{\alpha}\,x^{\beta^{\prime}}).$

To obtain equations for $D_{\gamma}\,e^{\alpha}\,_{a}$ we use

$D_{e_{0}}\,(D_{\gamma}\,e^{\beta}\,_{a})=e^{\alpha}\,_{0}\,D_{\gamma}\,D_{\alpha}\,e^{\beta}\,_{a}+R^{\beta}\,_{\delta\alpha\gamma}[\bar{k}]\,e^{\alpha}\,_{0}\,e^{\delta}\,_{a}$
$=D_{\gamma}\,(D_{e_{0}}\,e^{\beta}\,_{a})-(D_{\gamma}\,e^{\alpha}\,_{0})\,(D_{\alpha}\,e^{\beta}\,_{a})+R^{\beta}\,_{\delta\alpha\gamma}[\bar{k}]\,e^{\alpha}\,_{0}\,e^{\delta}\,_{a}.$

For the first term on the right hand side we get from equations (4.1)

$D_{\gamma}\,(D_{e_{0}}\,e^{\beta}\,_{0})=F^{A}\,_{,x^{\alpha^{\prime}}}\,(D_{\gamma}\,x^{\alpha^{\prime}})\,e^{\beta}\,_{A}+F^{A}\,(D_{\gamma}\,e^{\beta}\,_{A}),$
$D_{\gamma}\,(D_{e_{0}}\,e^{\beta}\,_{A})=-F_{A}\,{}_{,x^{\alpha^{\prime}}}\,(D_{\gamma}\,x^{\alpha^{\prime}})\,e^{\beta}\,_{0}-F_{A}\,(D_{\gamma}\,e^{\beta}\,_{0}).$

Thus together

$D_{e_{0}}\,(D_{\gamma}\,x^{\beta^{\prime}})=-D_{\gamma}\,e^{\alpha}\,_{0}\,(D_{\alpha}\,x^{\beta^{\prime}}),\quad\quad\quad\quad$
(4.6)
$D_{e_{0}}\,(D_{\gamma}\,e^{\beta}\,_{0})=F^{A}\,_{,x^{\alpha^{\prime}}}\,(D_{\gamma}\,x^{\alpha^{\prime}})\,e^{\beta}\,_{A}+F^{A}\,(D_{\gamma}\,e^{\beta}\,_{A})\quad\quad\quad\quad\,$
(4.7)
$\quad\quad\quad-(D_{\gamma}\,e^{\alpha}\,_{0})\,(D_{\alpha}\,e^{\beta}\,_{0})+R^{\beta}\,_{\delta\alpha\gamma}[\bar{k}]\,e^{\alpha}\,_{0}\,e^{\delta}\,_{0},$
$D_{e_{0}}\,(D_{\gamma}\,e^{\beta}\,_{A})=-F_{A}\,{}_{,x^{\alpha^{\prime}}}\,(D_{\gamma}\,x^{\alpha^{\prime}})\,e^{\beta}\,_{0}-F_{A}\,(D_{\gamma}\,e^{\beta}\,_{0})\quad\quad\,\,\,$
(4.8)
$\quad\quad\quad\quad-(D_{\gamma}\,e^{\alpha}\,_{0})\,(D_{\alpha}\,e^{\beta}\,_{A})+R^{\beta}\,_{\delta\alpha\gamma}[\bar{k}]\,e^{\alpha}\,_{0}\,e^{\delta}\,_{A}.$

These equations involve the curvature tensor of $\bar{k}$. Because the
Christoffel symbols depend on $\phi_{,\alpha\beta}$ we can expect third
derivatives of $\phi$ to enter the expression for the curvature tensor of
$\bar{k}$. Because the background is known, however, we can use Gauss’
equation

$R^{\beta}\,_{\delta\alpha\gamma}[\bar{k}]\,e^{\alpha}\,_{0}\,e^{\delta}\,_{a}=$
(4.9)
$\left(\sum_{\rho,\pi,\mu,\nu=0}^{3}R^{\rho}\,_{\pi\mu\nu}[g]\,k^{\beta}\,_{\rho}k^{\pi}\,_{\delta}\,k^{\mu}\,_{\alpha}\,k^{\nu}\,_{\gamma}-\chi^{\beta}\,_{\alpha}\,\chi_{\delta\gamma}+\chi^{\beta}\,_{\gamma}\,\chi_{\delta\alpha}\right)e^{\alpha}\,_{0}\,e^{\delta}\,_{a},$

where the background field $R^{\rho}\,_{\pi\psi\nu}[g]$ is taken at
$(z^{\alpha},\phi(z^{\alpha})+c)$ and (3.7), (3.8) are used. The terms in
large brackets only contain $\phi_{,\alpha}$ and $\phi_{,\alpha\beta}$.

With (4.9) taken into account, the system (4.1), (4.3), (4.4), (4.6), (4.7),
(4.8) provides the desired closed system for the unknowns

$x^{\alpha^{\prime}},\quad x^{\beta^{\prime}}\,_{,\alpha},\quad
e^{\beta}\,_{a},\quad
D_{\alpha}\,e^{\beta}\,_{a},\quad\phi,\quad\phi_{,\alpha}.$ (4.10)

The initial data on $S_{c}$ are determined, respectively chosen, as follows.

As in (3.1) we assume

$\phi|_{S_{c}}=0\quad\mbox{so
that}\quad\phi_{,\alpha}=\delta^{0}\,_{\alpha}\,\phi_{,0}\quad\mbox{on}\quad
S_{c}.$

The function $\phi_{,0}$ must be chosen on $S_{c}$ so as to satisfy (3.3).
Later on we will be led to consider the further condition (4.11).

The unique future directed unit vector field orthogonal to $N$ and $S_{c}$ is
given by

$e^{\mu}\,_{0}=\frac{T^{\mu}}{\sqrt{(g_{\nu\rho}\,T^{\nu}\,T^{\rho})}}\quad\mbox{on}\quad
S_{c},$

where $T^{\mu}$ is the vector field (3.5) on $S_{c}$ which will be tangential
to $T_{c}$. Because
$e^{3}\,_{0}=\phi_{,0}\,e^{0}\,_{0}=\phi_{,\alpha}\,e^{\alpha}\,_{0}$,
$e^{\mu}\,_{0}$ is uniquely determined by

$e^{\alpha}\,_{0}=\frac{T^{\alpha}}{\sqrt{(g_{\nu\rho}\,T^{\nu}\,T^{\rho})}}=\frac{T^{\alpha}}{\sqrt{(\bar{k}_{\alpha\beta}\,T^{\alpha}\,T^{\beta})}}$
$=\nu\,\frac{(g^{03}-\phi_{,0}\,g^{00})\,g^{\alpha
3}-(g^{33}-\phi_{,0}\,g^{03})\,g^{\alpha
0}}{\sqrt{(g^{03})^{2}-g^{33}\,g^{00}}}\quad\mbox{on}\quad S_{c}.$

It satisfies

$e^{0}\,_{0}=\nu\,\sqrt{(g^{03})^{2}-g^{33}\,g^{00}}>0.$

The fields $e^{\alpha}\,_{A}$ are chosen tangential to $S_{c}$ so that
$e^{0}\,_{A}=0$, whence
$\bar{k}_{\alpha\beta}\,e^{\alpha}\,_{0}\,e^{\beta}\,_{A}=0$. They are
required to satisfy

$\bar{k}_{\alpha\beta}\,e^{\alpha}\,_{A}\,e^{\beta}\,_{B}=\bar{k}_{CD}\,e^{C}\,_{A}\,e^{D}\,_{B}=g_{CD}\,e^{C}\,_{A}\,e^{D}\,_{B}=-\delta_{AB}.$

The forms dual to $e_{a}$ are denoted by
$\sigma^{a}=\sigma^{a}\,_{\alpha}\,dz^{\alpha}$ so that
$\sigma^{a}\,_{\alpha}\,e^{\alpha}\,_{b}=\delta^{a}\,_{b}$. We assume

$x^{0^{\prime}}=0,\quad\mbox{whence}\quad
x^{0^{\prime}}\,_{,A}=0\quad\mbox{on}\quad S_{c}.$

With the $z^{A}$ defining local coordinates on $S_{c}$, we choose choose local
coordinates $x^{A^{\prime}}=x^{A^{\prime}}(z^{A})$ on $S_{c}$, which give
$x^{A^{\prime}}\,_{,A}$ on $S_{c}$ with $\det(x^{A^{\prime}}\,_{,A})\neq 0$.
Following (2.4) we need to require
$\delta^{\alpha^{\prime}}\,_{0}=e^{\alpha^{\prime}}\,_{0}=e^{\alpha}\,_{0}\,x^{\alpha^{\prime}}\,_{,\alpha}$
which implies

$x^{0^{\prime}}\,_{,0}=\frac{1}{e^{0}\,_{0}},\quad\quad
x^{A^{\prime}}\,_{,0}=-\frac{1}{e^{0}\,_{0}}\,e^{A}\,_{0}\,x^{A^{\prime}}\,_{,A},\quad\quad
e^{0^{\prime}}\,_{A}=0.$

It remains to determine

$D_{\alpha}\,e^{\beta}\,_{b}=\sigma^{a}\,_{\alpha}\,D_{e_{a}}\,e^{\beta}\,_{b}.$

By (2.8) we must set, with the right hand sides given by the fields obtained
so far,

$D_{e_{0}}\,e^{\beta}\,_{0}=F^{A}\,e^{\beta}\,_{A},\quad\quad
D_{e_{0}}\,e^{\beta}\,_{A}=-F_{A}\,e^{\beta}\,_{0}\quad\mbox{on}\quad S_{c}.$

With the given information we can finally calculate

$D_{e_{A}}\,e^{\beta}\,_{0},\quad\quad
D_{e_{A}}\,e^{\beta}\,_{B}\quad\mbox{on}\quad S_{c},$

by using the Christoffel symbols (4.2). These involve the functions
$\phi_{,\alpha\beta}$ on $S_{c}$. From the data given above we get

$\phi_{,AB}=0\quad\phi_{,0B}=(\phi_{,0})_{,B}\quad\mbox{on}\quad S_{c}.$

To obtain $\phi_{,00}$ we solve (3.9) on $S_{c}$ for
$k^{\alpha\beta}\,\phi_{,\alpha\beta}$ and observe that by (3.11)

$k^{00}=\bar{k}^{00}\equiv
g^{00}+\nu^{2}\,(g^{03}-g^{00}\,\phi_{,0})^{2}>0\quad\mbox{on}\quad S_{c}.$

To discuss the solvability of our system we consider its principal part.

The first four of the unknowns (4.10) can be combined to an
$\mathbb{R}^{j}$-valued unknown $w$ that satisfies an equation of the form

$e^{\alpha}\,_{0}\,w_{,\alpha}=\dots\,,$

while the remaining unknowns combine to an $\mathbb{R}^{l}$-valued unknown $u$
that satisfies an equation of the form

$\bar{k}^{\alpha\beta}\,u_{,\alpha\beta}=\ldots\,.$

That $e_{0}$ is time-like with respect to $\bar{k}_{\alpha\beta}$ suggests
that the coupled system can be written as a quasi-linear, symmetric hyperbolic
system of first order [8].

In fact, in terms of the auxiliary unknowns $v_{\alpha}\equiv u_{,\alpha}$ the
equation for $u$ implies the system of first order

$u_{,0}=v_{0},$
$\bar{k}^{00}\,v_{0,0}+2\,\bar{k}^{0A}\,v_{0,A}+\bar{k}^{AB}\,v_{A,B}=\ldots$
$-\bar{k}^{AB}\,v_{B,0}+\bar{k}^{AB}\,v_{0,B}=0,$

and the equations for $w$ writes

$e^{0}\,_{0}\,w_{,0}+e^{A}\,_{0}\,w_{,A}=\dots\,$

We have seen above that $e^{0}\,_{0}>0$ and $\bar{k}^{00}>0$ on (whence near)
$S_{c}$. If $\bar{k}^{AB}$ defines a negative definite symmetric bilinear form
on (whence near) $S_{c}$, it follows that the combined system is quasi-linear,
symmetric hyperbolic. It also implies then for its solutions the integrability
condition $v_{B,0}=v_{0,B}$ and thus
$(v_{A,B}-v_{B,A})_{,0}=v_{0,AB}-v_{0,BA}=0$. With suitably given initial data
the remaining integrability conditions follow. It remains to see under which
conditions $\bar{k}^{AB}$ is negative definite.

Assume $a,b\in\mathbb{R}$, $y,z\in\mathbb{R}^{k}$, $k\geq 2$, and $A,B$ are
$k\times k$ matrices so that the following matrix equation holds with $k\times
k$ unit matrix $1_{k}$

$\left(\begin{array}[]{ccc}a&{}^{t}y\\\
y&A\end{array}\right)\left(\begin{array}[]{ccc}b&{}^{t}z\\\
z&B\end{array}\right)=\left(\begin{array}[]{ccc}1&{}^{t}0_{k}\\\
0_{k}&1_{k}\end{array}\right).$

If $b>0$ and $A$ is negative definite and symmetric , i.e. ${}^{t}u\,A\,u<0$
for $u\neq 0$ and ${}^{t}A=A$, then $B$ is negative definite and symmetric if
and only if $a>0$.

In fact, the matrix equation is equivalent to the relations

$b\,y+A\,z=0,\quad a\,b+\,^{t}y\,z=1,\quad y\,^{t}z+A\,B=1_{k},\quad
a\,^{t}z+\,^{t}y\,B=0.$

The second equation implies $a>0$ if $y=0$. Assume that $y\neq 0$. Being
symmetric and negative definite, $A$ has an inverse $A^{-1}$, which is also
symmetric and negative definite. The first equation implies $z=-b\,A^{-1}\,y$,
which gives with the second relation $1=b\,(a-\,^{t}x\,A^{-1}\,y)$ and thus
with our assumptions

$a-\,^{t}y\,A^{-1}\,y>0,\quad\quad b=\frac{1}{a-\,^{t}y\,A^{-1}\,y},\quad\quad
z=-\frac{1}{a-\,^{t}y\,A^{-1}\,y}\,A^{-1}\,y.$

The third and fourth equations are then satisfied with the symmetric matrix

$B=A^{-1}+\frac{1}{a-\,^{t}y\,A^{-1}\,y}\,(A^{-1}\,y)\,^{t}(A^{-1}\,y).$

In terms of the positive definite matrix $W=-A^{-1}$ the condition
${}^{t}u\,B\,u<0$ for $u\neq 0$ translates into

$(^{t}u\,W\,y)^{2}<(^{t}y\,W\,y)\,(^{t}u\,W\,u)+a\,(^{t}u\,W\,u).$

The Cauchy-Schwarz inequality with $u=y$ implies that $a>0$. $\Box$

Compare the matrix equation above with the relation
$\bar{k}_{\alpha\beta}\,\bar{k}^{\beta\gamma}=\delta_{\alpha}\,^{\gamma}$.
Since $S$ is space-like and $\bar{k}_{AB}=g_{AB}$ on $S_{c}$, $\bar{k}_{AB}$
is negative definite. We saw above $\bar{k}^{00}>0$ on $S_{c}$. Thus
$\bar{k}^{AB}$ is negative definite if and only if
$\bar{k}_{00}=g_{00}+2\,g_{3(0}\,\phi_{,0)}+g_{33}\,\phi_{,0}\,\phi_{,0}>0$ on
$S_{c}$, or

$\bar{k}_{00}=g_{\mu\nu}\,P^{\mu}\,P^{\nu}>0\quad\mbox{on $S_{c}$ with}\quad
P^{\mu}=\delta^{\mu}\,_{0}+\phi_{,0}\,\delta^{\mu}\,_{3}.$ (4.11)

It holds $N_{\mu}\,P^{\mu}=0$. Thus $P^{\mu}$ is tangential to $T_{c}$ on
$S_{c}$, but without further assumptions it need not be time-like. By suitable
choices of the hypersurface ${\cal T}$ and the coordinates $z^{\mu}$, which
were rather arbitrary so far, it can be arranged that $P^{\mu}$ is in fact
time-like on, whence near $S_{c}$ with $\phi_{,0}$ satisfying (3.3).

The Cauchy problem for the system (4.1), (4.3), (4.4), (4.6), (4.7), (4.8)
with initial data on open subsets of $S_{c}$ as discussed above is then well
posed. The corresponding solutions define pieces of the prospective
hypersurface $T_{c}$. These local pieces can be patched together to obtain a
part of the hypersurface $T_{c}$ diffeomorphic to
$[0,x^{0^{\prime}}_{*}[\times S_{c}$, where $x^{0^{\prime}}_{*}>0$ and the
unknown $x^{0^{\prime}}$ takes values in $[0,x^{0^{\prime}}_{*}[$.

Because these solutions depend smoothly on the initial data and, for
sufficiently small $c_{*}>0$, on $c\in[0,c_{*}[$, the hypersurfaces $T_{c}$
define a foliation that is smooth in the sense that the function
$x^{3^{\prime}}$ obtained by setting $x^{3^{\prime}}=c$ on $T_{c}$ is smooth.
Moreover, the coordinates $x^{\alpha^{\prime}}$ and the vector fields $e_{0}$,
$e_{A}$ obtained on the patches can be glued together to give smooth
coordinates and vector fields (expressed in terms of $x^{\alpha^{\prime}}$)
that satisfy equations (4.1) with the given functions
$F^{A}=F^{A}(x^{\alpha^{\prime}},x^{3^{\prime}})$ on the domain covered by the
foliation. The mean extrinsic curvature of the $T_{c}$ is given there by
$\chi=\chi(x^{\alpha^{\prime}},x^{3^{\prime}})$ (or
$f=f(x^{\alpha^{\prime}},x^{3^{\prime}})$).

##  5 Concluding remarks

It has been shown that the gauge based in section 2 on the choice of the
function $x^{3}$ and the vector field $e_{0}$, both suitably adapted to the
given time-like boundary $T$, can be completely reconstructed, together with
the hypersurface $T$, on the basis of the given gauge source functions $f$ and
$F^{A}$. The construction imposes no conditions on the underlying space-time
and, in particular, does not require the metric to satisfy any field equation.
The system of differential equations required for this turns out to be fairly
complicated and only quasi-linear. Obtaining information about the life time
of a gauge is a notoriously difficult problem. The characterization in terms
of $f$ and $F^{A}$ appears particularly difficult. A closer comparison with
the way the gauge is discussed in section 2, which covers without
complications a whole neighbourhood of the given hypersurface $T$, may give
some insight into this.

Because the system considered in section 4 looks so difficult, it may be
mentioned that fixing the gauge in terms of $f$ and $F^{A}$ introduces no
additional complication into the reduced system extracted in [7] from
equations (2.11), (2.12), (2.13). Part of the reason is that the curvature,
which somewhat unexpectedly enters the system derived in section 4, is already
an unknown in the reduced system. In a similar way as equation (1.1) reduces
to the relation $-g^{\nu\lambda}\,\Gamma_{\nu}\,^{\mu}\,{}_{\lambda}=F^{\mu}$
if it is expressed in terms of the coordinates $x^{\mu}$ that solve the
equation, the equations of section 4 reduce to simpler expressions if they are
expressed in terms of their solution so that ${\cal T}$ coincides with the
hypersurface $T_{0}$ and $x^{3^{\prime}}=z^{3}$ whence $\phi=0$. The
possibility to declare a particular set of functions as gauge source functions
and the usefulness of this choice obviously depends on the chosen
representation of the field equations.

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