# Analyticity of stationary spacetimes from maximal hypersurfaces††thanks:
Vienna preprint UWThPh 2022-9

Piotr T. Chruściel
Faculty of Physics, University of Vienna
x
Marc Mars
Faculty of Sciences, University of Salamanca Email
<EMAIL_ADDRESS>URL
http://homepage.univie.ac.at/piotr.chrusciel/Email<EMAIL_ADDRESS>
###### Abstract

The standard method of proving analyticity of stationary vacuum metrics
invokes the quotient-space version of Einstein equations. We verify that the
same conclusion can be obtained using the KID equations on maximal surfaces.

###### Contents

  1. 1 Introduction
  2. 2 ADM notation and Killing vector fields
  3. 3 Einstein equations
  4. 4 Maximal surfaces
  5. 5 Analyticity
  6. A pp-waves
  7. B Einstein-Rosen waves

##  1 Introduction

A classical theorem of Morrey [15] asserts that solutions of a class of
elliptic PDEs are real-analytic. We verify that this theorem applies to vacuum
Lorentzian metrics with a Killing vector field which is transverse to a
maximal surface if and only if the Killing vector is timelike. This provides
another example of usefulness of maximal hypersurfaces in general relativity.

For definiteness we restrict attention to the vacuum Einstein equations with
cosmological constant, but the argument applies as is to all matter models
where the field equations do not involve second derivatives of the metric and
become elliptic when stationarity is assumed.

We note that from the point of view of general relativistic applications, e.g.
to black hole uniqueness theorems, the usefulness of this result is unique
continuation of solutions, which follows from analyticity, even though
analyticity may not be necessary in some cases to obtain unique continuation;
see [7] and references therein.

##  2 ADM notation and Killing vector fields

Any Lorentzian metric $\mathfrak{g}$ can be locally written in the Arnowitt-
Deser-Misner form (cf., e.g., [13, Equation (21.40), p. 507])

$\mathfrak{g}=-N^{2}dt^{2}+g_{ij}(dx^{i}+Y^{i}dt)(dx^{j}+Y^{j}dt)\,,$ (2.1)

where $g_{ij}dx^{i}dx^{j}$ is a family of possibly $t$-dependent Riemannian
metrics on a manifold $M$, $N$ is a function on spacetime and
$Y=Y^{i}\partial_{i}$ is a family of possibly $t$-dependent vector fields on
$M$.

Let $T$ be the field of unit normals to the level sets of $t$, one finds

$T=N^{-1}(\partial_{t}-Y^{i}\partial_{i})\qquad\Longleftrightarrow\qquad\partial_{t}=NT+Y\,.$
(2.2)

Since $Y$ is tangent to $\mycal S$, the vector $\partial_{t}$ is transverse to
$\mycal S$ if and only if $N$ has no zeros.

It follows from (2.1) that

$\mathfrak{g}_{tt}\equiv\mathfrak{g}(\partial_{t},\partial_{t})=-N^{2}+g_{ij}Y^{i}Y^{j}\equiv-N^{2}+g(Y,Y)\,,$
(2.3)

which shows that $\partial_{t}$ is spacelike if and only if the $g$-length of
$Y$ is larger than $N$.

We have

$\det\mathfrak{g}=-N^{2}\det g\,,$ (2.4)

which shows that the metric (2.1) is manifestly Lorentzian in the coordinate
system above when $N$ has no zeros and $g$ is Riemannian. This holds
regardless of the $g$-length of $Y$.

The extrinsic curvature tensor (second fundamental form) of the level sets of
$t$ is given by

$K_{ij}=\frac{1}{2}\mycal
L_{T}g_{ij}=\frac{1}{2}N^{-1}(\partial_{t}g_{ij}-\mycal L_{Y}g_{ij})\,,$ (2.5)

and we note that sometimes an opposite sign convention is used in (2.5).

If all the functions appearing in (2.1) are $t$-independent the vector field
$\partial_{t}$ is a Killing vector field:

$0=\partial_{t}\mathfrak{g}_{\mu\nu}=\mycal L_{\partial_{t}}g_{\mu\nu}\,.$
(2.6)

In this case (2.5) can be rewritten as

$\mycal L_{Y}g_{ij}=-2NK_{ij}\qquad\Longleftrightarrow\qquad
D_{(i}Y_{j)}=-NK_{ij}\,,$ (2.7)

where $D$ is the Levi-Civita covariant derivative associate with the
Riemannian metric $g=g_{ij}dx^{i}dx^{j}$, and where parentheses over indices
denote symmetrisation.

Given a Lorentzian metric $\mathfrak{g}$ with a Killing vector field $X$ one
can construct local coordinates sastisfying (2.1) and the first equation in
(2.6), this proceeds as follows: Let $\mycal S$ be any spacelike hypersurface
which is transverse to $X$. Let $t$ be a function defined near $\mycal S$ by
solving the problem

$\mycal L_{X}t=1\,,\qquad t|S=0\,.$ (2.8)

Let $y^{i}$ be any local coordinates on a coordinate patch $\mycal
O\subset\mycal S$, we can define functions $x^{i}$ near $\mycal O$ by solving
the equations

$\mycal L_{X}x^{i}=0\,,\qquad x^{i}|S=y^{i}\,.$ (2.9)

One checks that in these coordinates the metric can be written in the form
(2.1). We have $X^{t}=X(t)=\mycal L_{X}t=1$ and $X^{i}=X(x^{i})=\mycal
L_{X}x^{i}=0$, which shows that $X=\partial_{t}$. Transversality of $X$ to
$\mycal O$ is equivalent to the condition that $N$ has no zeros on $\mycal O$.

Since the problem addressed is purely local, for our purposes here we can
without loss of generality assume that $\mycal O=\mycal S$, and that the
coordinate system above is global on $I\times\mycal S$, where
$I\subset\mathbb{R}$ is an interval containing $0$.

##  3 Einstein equations

When $\partial_{t}$ is a Killing vector, so that
$\partial_{t}g_{ij}=0=\partial_{t}K_{ij}$, the vacuum Einstein equations for
the metric (2.1) imply the following set of equations [13, 8, 12, 4,
14]:111The explicit form of these equations differs across references by
conventions on signs; we follow [4]. Our equations (3.1)-(3.2) coincide with
those of [12], with our $Y$ denoted by $X$ there, after taking into account
the constraint equations (3.3)-(3.4).

$\displaystyle D_{i}D_{j}N$ $\displaystyle=$
$\displaystyle\big{(}\mathrm{Ric}\,(g)_{ij}-\frac{2\Lambda}{n-2}g_{ij}-2K^{\ell}{}_{i}K_{j\ell}+\mbox{\rm
tr}\,_{g}KK_{ij}\big{)}N$ (3.1) $\displaystyle-
D_{\ell}K_{ij}Y^{\ell}-2K^{\ell}{}_{(i}D_{j)}Y_{\ell}\,,$ $\displaystyle
D_{(i}Y_{j)}$ $\displaystyle=$ $\displaystyle-K_{ij}N\,.$ (3.2)

Here $\mathrm{Ric}\,(g)_{ij}$ the Ricci tensor of $g$ and $R$ its trace,
$\Lambda$ the cosmological constant and $n$ is the spacetime dimension. These
equations should be complemented by the vacuum constraint equations [3, 6],

$\displaystyle R+(\mbox{\rm tr}\,_{g}K)^{2}-K^{ij}K_{ij}$
$\displaystyle=2\Lambda\,,$ (3.3) $\displaystyle D^{p}K_{\ell
p}-D_{\ell}\mbox{\rm tr}\,_{g}K$ $\displaystyle=0\,.$ (3.4)

Summarising: Let the $t$-independent fields $(N,Y,g)$ parameterise the
spacetime metric $\mathfrak{g}$ as

$\mathfrak{g}=-N^{2}dt^{2}+g_{ij}(dx^{i}+Y^{i}dt)(dx^{j}+Y^{j}dt)\,.$ (3.5)

Then the metric $\mathfrak{g}$ has a Killing vector $X=\partial_{t}$ and
satisfies the vacuum Einstein equations if and only if (3.1)-(3.4) hold. The
function $N$ will have no zeros on $\mycal S=\\{t=0\\}$ if the Killing vector
is transverse to $\mycal S$.

In the situation just described we can use (3.2) to write $K_{ij}$ as

$\displaystyle K_{ij}=-\frac{1}{2N}\left(D_{i}Y_{j}+D_{j}Y_{i}\right)$

and insert this into equations (3.1) and (3.4). A direct computation gives

$\displaystyle\mathrm{Ric}\,(g)_{ij}=$
$\displaystyle\frac{1}{N}\left(D_{i}D_{j}N-Y^{\ell}D_{\ell}\left(N^{-1}D_{(i}Y_{j)}\right)+\frac{1}{2N}(D^{\ell}Y_{i}D_{\ell}Y_{j}-D_{i}Y^{\ell}D_{j}Y_{\ell})\right)$
$\displaystyle-\frac{D^{\ell}Y_{\ell}}{N^{2}}D_{(i}Y_{j)}+\frac{2\Lambda}{n-2}g_{ij},$
(3.6) $\displaystyle D^{i}D_{(i}Y_{j)}=$ $\displaystyle
ND_{j}\left(\frac{D_{i}Y^{i}}{N}\right)+D_{(i}Y_{j)}D^{i}\ln|N|.$ (3.7)

The trace of (3.6) gives, after using the constraint (3.3),

$\displaystyle\Delta_{g}N=\frac{1}{N}D_{i}Y_{j}D^{(i}Y^{j)}+Y^{i}D_{i}\left(\frac{D_{j}Y^{j}}{N}\right)-\frac{2\Lambda}{n-2}N.$
(3.8)

##  4 Maximal surfaces

In the setting above, we claim that we can find in $\mathbb{R}\times\mycal S$
a spacelike hypersurface, say $\,\,\mathring{\\!\\!\mycal S}$, on which it
holds

$\mbox{\rm tr}\,_{g}K=0\,,$ (4.1)

and which is transverse to the Killing vector $X$. (While eventually we will
be interested in timelike Killing vectors, we do not make the assumption that
$X$ is timelike here.) While this should be clear to those which are familiar
with the results of [2], in order to dispell any doubts we give a formal
proof: Let $p\in\mycal S$ and consider a small coordinate ball $B(0,r)$ of
radius $r$, centered at $p$, within $\mycal S$. Let $\epsilon>0$ and let
$\phi$ be any smooth function on $S(0,r)$ satisfying

$|\phi|+|d\phi|_{g}<\epsilon\,.$ (4.2)

There exists $r$ small enough and $\epsilon$ small enough so that for all
functions $\phi$ satisfying (4.2) we can find a spacelike hypersurface
$\,\,\,\widehat{\\!\\!\\!{\mycal S_{\,}}}$ such that:

  1. 1.

$\partial\,\,\,\widehat{\\!\\!\\!{\mycal
S_{\,}}}:=\overline{\,\,\,\widehat{\\!\\!\\!{\mycal
S_{\,}}}}\setminus\,\,\,\widehat{\\!\\!\\!{\mycal S_{\,}}}$ is the graph
$\\{t=\phi(p)\,,p\in\partial B(0,r)\\}$;

  2. 2.

the domain of dependence ${\mycal D}(\,\,\,\widehat{\\!\\!\\!{\mycal
S_{\,}}})$ of $\,\,\,\widehat{\\!\\!\\!{\mycal S_{\,}}}$ is globally
hyperbolic;

  3. 3.

and ${\mycal D}(\,\,\,\widehat{\\!\\!\\!{\mycal S_{\,}}})$ has compact
closure.

By [2, Theorem 4.2] there exists in $\mathbb{R}\times\mycal S$ a spacelike
hypersurface, say $\mycal S_{\phi}$ spanned on
$\partial\,\,\,\widehat{\\!\\!\\!{\mycal S_{\,}}}$ and satisfying (4.1).
Clearly $X$ cannot be tangent to all the hypersurfaces $\mycal S_{\phi}$
obtained in this way. Hence there exists a function $\phi$ such that $X$ will
be transverse to $\mycal S_{\phi}$ somewhere. We define
$\,\,\mathring{\\!\\!\mycal S}$ to be any open subset of $\mycal S_{\phi}$ to
which $X$ is transverse.

Let us use again the symbol $g$ for the metric induced by $\mathfrak{g}$ on
$\,\,\mathring{\\!\\!\mycal S}$. We can carry-out the construction, described
around (2.8)-(2.9), of a new coordinate system, still denoted by $(t,x^{i})$,
by requiring that the new function $t$ vanishes on $\,\,\mathring{\\!\\!\mycal
S}$. The metric $\mathfrak{g}$ expressed in these new coordinates takes again
the form (3.5), with $t$ equal to zero on $\,\,\mathring{\\!\\!\mycal S}$,
with new time-independent function $N$ and vector field $Y$ and metric $g$,
and with a Killing vector $\partial_{t}$, Moreover (4.1) holds.

The vacuum KID equations (3.1)-(3.2) and the vacuum constraint equations
(3.3)-(3.4) continue to hold on $\,\,\mathring{\\!\\!\mycal S}$. Inserting
$D_{i}Y^{i}=0$ into (3.6)-(3.8) yields

$\displaystyle\Delta_{g}N-\frac{1}{N}D_{i}Y_{j}D^{(i}Y^{j)}+\frac{2\Lambda}{n-2}N=0$
(4.3) $\displaystyle D^{i}D_{(i}Y_{j)}-D_{(i}Y_{j)}D^{i}\ln|N|=0$ (4.4)
$\displaystyle\mathrm{Ric}\,(g)_{ij}-\frac{1}{N}\bigg{(}D_{i}D_{j}N-Y^{\ell}D_{\ell}(N^{-1}D_{(i}Y_{j)})+\frac{1}{2N}(D^{\ell}Y_{i}D_{\ell}Y_{j}-D_{i}Y^{\ell}D_{j}Y_{\ell})\bigg{)}$
$\displaystyle\hfill=\frac{2\Lambda}{n-2}g_{ij}$ (4.5)

##  5 Analyticity

In harmonic coordinates the system (4.3)-(4.5) forms an elliptic system for
$(N,Y^{i},g_{ij})$ if and only of $N^{2}>|Y|^{2}_{g}$, as we explicitly show
next. In the notation of [15], the principal symbol $L(x,\lambda)$ of the
system (4.3)-(4.5) is obtained by linearising the equations around the
solution $(N,Y,g)$: for $\lambda\in T^{*}\,\,\mathring{\\!\\!\mycal S}$,

$\displaystyle L(x,\lambda)\left(\begin{array}[]{c}\delta N\\\ \delta Y\\\
\delta g\\\ \end{array}\right)=$ (5.4)
$\displaystyle\left(\begin{array}[]{c}g^{k\ell}\lambda_{k}\lambda_{\ell}\delta
N\\\ \frac{1}{2}\lambda^{i}(\lambda_{i}\delta Y_{j}+\lambda_{j}\delta
Y_{i}{{+\lambda^{k}Y_{k}\delta g_{ij}-\lambda_{i}\delta g_{jk}Y^{k}-\delta
g_{ik}Y^{k}\lambda_{j}}})\\\
-\frac{1}{2}g^{k\ell}\lambda_{k}\lambda_{\ell}\delta
g_{ij}-\frac{1}{N}\lambda_{i}\lambda_{j}\delta
N+\frac{Y^{\ell}}{2N^{2}}\Big{(}\lambda_{\ell}(\lambda_{i}\delta
Y_{j}+\lambda_{j}\delta Y_{i})+{{Y^{k}\lambda_{\ell}\left(\lambda_{k}\delta
g_{ij}-\lambda_{i}\delta g_{jk}-\lambda_{j}\delta
g_{ik}\right)}}\Big{)}\end{array}\right)\,.$ (5.7)

Ellipticity for this system is defined in [16, 11] as the condition that the
determinant of the principal symbol is non-zero. This is equivalent to the
requirement that the equation

$L(x,\lambda)\left(\begin{array}[]{c}\delta N\\\ \delta Y\\\ \delta g\\\
\end{array}\right)=0$ (5.8)

has no non-trivial solutions unless $\lambda\equiv 0$. So suppose that
$\lambda\neq 0$, then the equation

$g^{ij}\lambda_{i}\lambda_{j}\delta N=0$

gives $\delta N\equiv 0$, since $g$ is Riemannian.

Next, consider the equation

$\displaystyle\lambda^{i}(\lambda_{i}\delta Y_{j}+\lambda_{j}\delta
Y_{i}{{+\lambda^{k}Y_{k}\delta g_{ij}-\lambda_{i}\delta
g_{jk}Y^{k}-\lambda_{j}\delta g_{ik}Y^{k}}})=0\,.$ (5.9)

Contracting with $\lambda^{j}$ gives

$0=2|\lambda|^{2}_{g}\left(\,\lambda^{j}\delta Y_{j}-{{\delta
g_{jk}\lambda^{j}Y^{k}}}\right){{+\lambda^{k}Y_{k}\delta
g_{ij}\lambda^{i}\lambda^{j}}}\,,$

which for non-zero $\lambda$ (which we assume from now on) implies

$\lambda^{j}\delta Y_{j}={{\delta
g_{ij}\lambda^{i}Y^{j}-\frac{\lambda^{k}Y_{k}}{2|\lambda|^{2}_{g}}\delta
g_{ij}\lambda^{i}\lambda^{j}}}.$ (5.10)

Inserting this into (5.9) gives

$\displaystyle{{\delta Y_{j}=Y^{i}\delta
g_{ij}+\frac{\lambda^{k}Y_{k}}{|\lambda|^{2}_{g}}\left(-\lambda^{i}\delta
g_{ij}+\frac{1}{2|\lambda|^{2}_{g}}\lambda_{j}\lambda^{i}\lambda^{k}\delta
g_{ik}\right)}}\,.$ (5.11)

which is compatible with (5.10). Replacing $\delta N=0$ and (5.11) into the
third line of (5.8) gives, after a simple manipulation,

$\displaystyle\frac{1}{2}\left(-|\lambda|^{2}_{g}+\frac{(\lambda^{k}Y_{k})^{2}}{N^{2}}\right)\delta
g_{ij}+\frac{(\lambda^{k}Y_{k})^{2}}{2N^{2}|\lambda|^{2}_{g}}\left(-\lambda_{i}\lambda^{k}\delta
g_{jk}-\lambda_{j}\lambda^{k}\delta
g_{ik}+\frac{1}{|\lambda|^{2}_{g}}\lambda_{i}\lambda_{j}\delta
g_{kl}\lambda^{k}\lambda^{l}\right)=0\,.$

Contracting with $\lambda^{j}$ one gets

$\lambda^{j}\delta g_{ij}=0\,,$ (5.12)

which inserted back into the equation yields

$\displaystyle\left(-|\lambda|^{2}_{g}+\frac{(\lambda^{k}Y_{k})^{2}}{N^{2}}\right)\delta
g_{ij}=0\,.$ (5.13)

If $|Y|^{2}_{g}<N^{2}$ the term in parenthesis is negative for all
$\lambda_{i}$, so $\delta g_{ij}=0$ which inserted into (5.11) gives $\delta
Y_{j}=0$ and ellipticity is established. When $|Y|^{2}_{g}\geq N^{2}$, let $s$
be any vector orthogonal to $Y$ with norm

$\displaystyle|s|^{2}_{g}=|Y|^{2}_{g}\left(\frac{|Y|^{2}_{g}}{N^{2}}-1\right)>0\,.$

Then $\lambda_{j}:=Y_{j}+s_{j}$ is non-zero and the parenthesis in (5.13) is
identically zero. Thus, for such $\lambda$ the only restriction on $\delta
g_{ij}$ is (5.12), which leads to a kernel of $L(x,\lambda)$ consisting of
fields $(\delta N,\delta Y_{i},\delta g_{ij})$ satisfying $\lambda^{i}\delta
g_{ij}=0$, $\delta N=0$ and $\delta Y_{i}=\delta g_{ij}Y^{j}$. We conclude
that the system is _not_ elliptic whenever $|Y|^{2}_{g}\geq N^{2}$.

Real-analyticity of solutions for $|Y|^{2}_{g}<N^{2}$ follows from [15] by
choosing the indices $s_{i}$ and $r_{j}$ there as $s_{i}=1=r_{j}$.
Alternatively one can appeal to [11].222 Further references of interest in the
context include [5, 17, 9, 18].

The result is sharp: for completeness we present families of elementary
examples of non-analyticity for non-timelike Killing vectors in Appendices A
and B. The reader might find some interest of its own in the reduction of the
problem to a dynamical system in the setting of Appendix A.

While maximality of the level sets of $t$ is not necessary for analyticity, a
restriction on the level sets of $t$ with an elliptic flavour is certainly
necessary: indeed, given an analytic spacetime we can always change the time
slicing by changing $t$ to $t+f$, where $f$ is smooth but _not analytic_ ,
with $df$ sufficiently small so that the level sets of the new time function
are again spacelike. The resulting new data $(N,Y,g)$ will not be analytic.

##  Appendix A pp-waves

The $pp$-waves provide examples of smooth non-analytic metrics with a null
Killing vector. The metric takes the form

$\displaystyle\mathfrak{g}=2dvdu+H(u,x,y)du^{2}+dx^{2}+dy^{2}.$

The vector field $X=\partial_{v}$ is null, covariantly constant and nowhere
zero, hence transversal to any spacelike hypersurface. We chose the
orientation so that $\partial_{v}$ is future. The metric $\mathfrak{g}$ is
Ricci flat if and only if $H$ satisfies [19, Equation (24.43), p. 384]

$\displaystyle(\partial_{x}^{2}+\partial_{y}^{2})H=0,$

i.e. it is harmonic in the $x,y$ coordinates. Moreover, $\mathfrak{g}$ is
locally flat if and only if $H$ is a polynomial of degree one in the
coordinates $x,y$ [19, Equation (24.43), p. 384].

It is trivial to construct non-analytic solutions. An example to be used below
is $H=1+w(u)xy$, with $w(u)\in C^{\infty}(\mathbb{R})$ satisfying $w(u)=0$ in
$u\in(-\infty,0)$, and $w(u)>0$ for $u>0$. The metric is Minkowski in the
domain $u<0$ and has non-zero curvature for $u>0$, so there exists no
coordinate system in which the metric is analytic near the point $u=x=y=0$.

Consider a hypersurface $\mycal S:=\\{v=f(u,x,y)\\}$. To compute the fields
$(N,Y,g)$ on $\mycal S$ we introduce adapted coordinates, as described in the
main text. Let $t=v-f(u,x,y)$ and let us write $\mathfrak{g}$ in the
coordinate system $\\{t,u,x,y\\}$:

$\displaystyle\mathfrak{g}=2dtdu+\left(H+2\partial_{u}f\right)du^{2}+2\partial_{x}fdudx+2\partial_{x}fdudy+dx^{2}+dy^{2}.$
(A.1)

The hypersurface $\mycal S$ is $\\{t=0\\}$ so the induced metric (the first
fundamental form) is

$\displaystyle
g=\left(H+2\partial_{u}f\right)du^{2}+2\partial_{x}fdudx+2\partial_{y}fdudy+dx^{2}+dy^{2}.$
(A.2)

This is Riemannian (positive definite) provided that

$\displaystyle F:=H+2\partial_{u}f-(\partial_{x}f)^{2}-(\partial_{y}f)^{2}>0,$
(A.3)

which we assume from now on. To compute $N$ and $Y$ we compare (A.1) with
(2.1). The absence of a $dt^{2}$ term gives $N^{2}=g_{ij}Y^{i}Y^{j}$, which is
simply a restatement of the fact that $X$ is null everywhere. The cross-terms
$dtdx^{i}$ give $g_{ij}Y^{j}dx^{i}=du$, which can be solved for $Y$:

$\displaystyle
Y=\frac{1}{F}\left(\partial_{u}-f_{x}\partial_{x}-f_{y}\partial_{y}\right).$
(A.4)

The $g$-norm of this vector is $g_{ij}Y^{i}Y^{j}=\frac{1}{F}=N^{2}$. Selecting
the unit normal $T$ to be future directed requires $N>0$ (cf. (2.2)).
Consequently,

$\displaystyle N=\frac{1}{\sqrt{F}}.$ (A.5)

One can now check by an explicit calculation that (3.1)-(3.2) hold.

The metrics (A.1) provide an interesting example where the maximal surface
equation reduces to a polynomial dynamical system with a source. Specifically,
the trace of the extrinsic curvature, as defined in (2.5), is given by

$\displaystyle\mbox{\rm
tr}\,_{g}K=-\partial_{u}N+\partial_{x}(N\partial_{x}f)+\partial_{y}(N\partial_{y}f).$

As already pointed-out we choose $H=1+w(u)xy$, with $w(u)$ as before, and look
for solutions of the maximal surface equation $\mbox{\rm tr}\,_{g}K=0$ of the
form

$\displaystyle f=f_{0}(u)+f_{1}(u)(x^{2}+y^{2})+f_{2}(u)xy.$

The condition that $\mycal S$ is maximal turns out to be equivalent to the
following system of ODEs, where a dot denotes a derivative with respect to
$u$:

$\displaystyle\ddot{f_{0}}$ $\displaystyle=-4f_{1}(1+2\dot{f}_{0}),$
$\displaystyle\ddot{f_{1}}$
$\displaystyle=2f_{2}\dot{f_{2}}+2f_{1}(4f_{1}^{2}-f_{2}^{2})+\frac{1}{2}f_{2}w$
$\displaystyle\ddot{f_{2}}$
$\displaystyle=2f_{2}\left(4\dot{f}_{1}+4f_{1}^{2}-f_{2}^{2}\right)-2f_{1}w-\frac{1}{2}\dot{w}.$

For $u\leq 0$ we take $f_{0}(u)=f_{1}(u)=f_{2}(u)=0$ and for $u\geq 0$ we take
the unique solution of this system with initial data
$f_{0}(0)=f_{1}(0)=f_{2}(0)=\dot{f}_{0}(0)=\dot{f}_{1}(0)=\dot{f}_{2}(0)=0$.
Since $w(u)$ is $C^{\infty}(\mathbb{R})$ but not analytic, the same holds for
$f_{i}(u)$, $i=0,1,2$. It is immediate that the condition (A.3) (i.e., $\mycal
S$ being spacelike) is satisfied in a neighbourhood of the point $u=x=y=0$.
This provides an essentially explicit example of a maximal hypersurface with
transversal null Killing vector field and non-analytic metric.

##  Appendix B Einstein-Rosen waves

Consider the collection of metrics of the form

$\mathfrak{g}=e^{2(\gamma-\psi)}(-dt^{2}+dr^{2})+e^{-2\psi}r^{2}d\varphi^{2}+e^{2\psi}dz^{2}\,,$
(B.1)

where the functions $\psi$ and $\gamma$ depend only upon $t$ and $r$. Taking
$t\in\mathbb{R}$, $z\in\mathbb{R}$, and viewing $(r,\varphi)$ as polar
coordinates on $\mathbb{R}^{2}$, we thus obtain a family of cylindrically
symmetric metrics on $\mathbb{R}\times\mathbb{R}^{3}$. Regularity of the
metric at the axis of rotation $\\{r=0\\}$ requires the vanishing of $\gamma$
there.

The metric (B.2) will satisfy the vacuum Einstein equations (with vanishing
cosmological constant) if and only if [10, 19, 1]

$\displaystyle\big{(}-\partial_{t}^{2}+\partial_{x}^{2}+\partial_{y}^{2}\big{)}\psi$
$\displaystyle=0\,,$ (B.2) $\displaystyle\partial_{t}\gamma$
$\displaystyle=2r\partial_{t}\psi\partial_{r}\psi\,,$ (B.3)
$\displaystyle\partial_{r}\gamma$
$\displaystyle=r\big{(}(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}\big{)}\,.$
(B.4)

The key equation is (B.2): Indeed, given any rotation-invariant solution of
the wave equation $\psi$ on $\mathbb{R}\times\mathbb{R}^{2}$, we can integrate
(B.3)-(B.4), with $\gamma(0,0)=0$, to obtain $\gamma$. We note that (B.2) is
the integrability condition for (B.3)-(B.4).

The vanishing of $\gamma$ at $\\{r=0\,,\ t=0\\}$ is preserved in time by
(B.3). Smooth initial data lead, by evolution, to smooth spacetime metrics by
general considerations. This is clear from (B.2)-(B.3) in any case except
possibly at $r=0$, and can be directly verified at the axis of rotation by
inspection of asymptotic expansions there.

Hence the set of smooth such metrics can be uniquely parameterised by the set
of smooth rotation-invariant initial data for the $(2+1)$-dimensional wave
equation in Minkowski space-time.

Consider a point $p\in\mathbb{R}^{4}$ at which we have

$(-(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}-r^{-1}\partial_{r}\psi)|_{p}\neq
0\,.$ (B.5)

One checks that the metric is _not flat_ at $p$, and hence in a spacetime
neighborhood of $p$. (This is a sufficient, but not necessary, condition).

As an example, let us take smooth rotation invariant initial data for $\psi$
at $t=0$ such that $\psi|_{t=0}\equiv 0$, $\partial_{t}\psi|_{t=0,\,r\leq
1}\equiv 0$, and $\partial_{t}\psi|_{t=0,\,r>1}>0$. Then (B.5) holds at $t=0$
and $r>1$. This results, by evolution, in a smooth spacetime metric with two
spacelike Killing vectors, $\partial_{\varphi}$ and $\partial_{z}$, with a
spacetime metric which is flat in a spacetime neighborhood of the solid
cylinder $\\{t=0,\,r<1\\}$, but is not flat. Unique continuation of the
curvature tensor for real-analytic metrics shows that the spacetime metric
cannot be real-analytic on a maximal hypersurface which is transverse to some
linear combination of the spacelike Killing vectors and which intersects the
cylinder $\\{t=0,\,r=1\\}.$

## References

  * [1] A. Ashtekar, J. Bičák, and B.G. Schmidt, _Behavior of Einstein-Rosen waves at null infinity_ , Phys. Rev. D (3) 55 (1997), no. 2, 687–694. MR 1435251
  * [2] R. Bartnik, _The existence of maximal hypersurfaces in asymptotically flat spacetimes_ , Commun. Math. Phys. 94 (1984), 155–175.
  * [3] R. Bartnik and J. Isenberg, _The constraint equations_ , The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 1–38. MR MR2098912 (2005j:83007)
  * [4] R. Beig and P.T. Chruściel, _Killing Initial Data_ , Class. Quantum Grav. 14 (1997), A83–A92, A special issue in honour of Andrzej Trautman on the occasion of his 64th Birthday, J.Tafel, editor. MR MR1691888 (2000c:83011)
  * [5] S. Blatt, _On the analyticity of solutions to non-linear elliptic partial differential systems_ , arXiv e-prints (2020), arXiv:2009.08762.
  * [6] A. Carlotto, _The general relativistic constraint equations_ , Living Rev. Rel. 24 (2021), no. 1, 2.
  * [7] P.T. Chruściel and E. Delay, _Unique continuation and extensions of Killing vectors at boundaries for stationary vacuum spacetimes_ , Jour. Geom. Phys. 61 (2011), 1249–1257. MR 2802469 (2012h:53097)
  * [8] P.T. Chruściel and E. Delay, _Exotic hyperbolic gluings_ , Jour. Diff. Geom. 108 (2018), 243–293, arXiv:1511.07858 [gr-qc].
  * [9] C. Cosner, _On the definition of ellipticity for systems of partial differential equations_ , Jour. Math. Analysis and Appli. 158 (1991), 80–93.
  * [10] A. Einstein and N. Rosen, _On gravitational waves_ , Jour. Franklin Inst. 223 (1937), no. 1, 43–54. MR 3363463
  * [11] A. Friedman, _On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations_ , Jour. Math. Mech. 7 (1958), 43–59. MR 0118970
  * [12] D. Maerten, _Killing initial data revisited_ , Jour. Math. Phys. 45 (2004), 2594–2599. MR MR2067575 (2005b:83009)
  * [13] C.W. Misner, K. Thorne, and J.A. Wheeler, _Gravitation_ , Freeman, San Fransisco, 1973.
  * [14] V. Moncrief, _Spacetime symmetries and linearization stability of the Einstein equations I_ , Jour. Math. Phys. 16 (1975), 493–498.
  * [15] C.B. Morrey, Jr., _On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior_ , Amer. Jour. Math. 80 (1958), 198–218. MR 0106336
  * [16] , _On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior_ , Amer. Jour. Math. 80 (1958), 198–218. MR 106336
  * [17] , _On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. II. Analyticity at the boundary_ , Amer. Jour. Math. 80 (1958), 219–237. MR 0107081
  * [18] C.B. Morrey, Jr. and L. Nirenberg, _On the analyticity of the solutions of linear elliptic systems of partial differential equations_ , Commun. Pure Appl. Math. 10 (1957), 271–290. MR 89334
  * [19] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, _Exact solutions of Einstein’s field equations_ , second ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003. MR 2003646