Stationary Black Holes
Piotr T. Chru\'{s}ciel\thanks{D\'epartement de Math\'ematiques
Facult\'e des Sciences
Parc de Grandmont
F37200 Tours
France. Partially supported by a Polish Research Committee grant 2 P03B 073 24; %
URL \protect\url{ www.phys.univ-tours.fr/}$\sim$\protect\url{piotr}}
% Golm
Germany
%Erwann Delay\thanks{Visiting Scientist. Permanent address: %D\'epartement de Math\'ematiques
rue %Louis Pasteur
F84000 Avignon
% D\'epartement de Math\'ematiques
%Facult\'e des Sciences
Vienna
Austria
We review the theory of stationary black hole solutions of vacuum Einstein equations.\\ Keywords: {black holes, event horizons, Schwarzschild metric, Kerr metric, no-hair theorems}
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\begin{document}

\title{Stationary Black Holes}

\author{Robert Beig\thanks{Institute for Theoretical Physics
University of Vienna, Boltzmanngasse5 A-1090 Vienna, Austria}\\
 Piotr T. Chru\'{s}ciel\thanks{D\'epartement
de Math\'ematiques, Facult\'e des Sciences, Parc de Grandmont,
F37200 Tours, France. Partially supported by a Polish Research
Committee
grant 2 P03B 073 24; %, by the Erwin Schr\"odinger Institute, and
%by a travel grant from the Vienna City Council;
email \protect\url{ piotr@gargan.math.univ-tours.fr}, URL
\protect\url{
www.phys.univ-tours.fr/}$\sim$\protect\url{piotr}}\\
% Golm, Germany \\\\
%Erwann Delay\thanks{Visiting Scientist. Permanent address:
%D\'epartement de Math\'ematiques, Facult\'e des Sciences, rue
%Louis Pasteur, F84000 Avignon, France. Partially supported by the
%ACI program of the French Ministry of Research and by the Erwin
%Schr\"odinger Institute; email
%  \protect\url{delay@gargan.math.univ-tours.fr}}
%      \\% D\'epartement de Math\'ematiques\\
%Facult\'e des Sciences\\ Parc de Grandmont\\ F37200 Tours, France
%Erwin Schr\"odinger Institute\\ Vienna, Austria }
}
\date{\today}



\maketitle







%%%%%%%%%%%%%%%%%%%%%%%%% resume %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%�






\begin{abstract}
We review the theory of stationary black hole solutions of vacuum Einstein equations.\\
 Keywords: {black holes, event horizons,
Schwarzschild metric, Kerr metric, no-hair theorems}
\end{abstract}




%\tableofcontents

\section{Introduction}\label{Sin}




This article treats a specific class of stationary solutions to
the Einstein field equations which read
\begin{equation}
\label{einstein} R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8 \pi
G}{c^{4}} T_{\mu\nu}.
\end{equation}
Here $R_{\mu\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$ are respectively
the Ricci tensor and the Ricci scalar  of the spacetime metric
$g_{\mu\nu}$, $G$ is the Newton constant and $c$ the speed of
light. The tensor $T_{\mu\nu}$ is the stress-energy tensor of
matter. Spacetimes, or regions thereof, where $T_{\mu\nu}=0$ are
called vacuum.

 Stationary solutions are of interest for a variety of reasons. As models for
compact objects at rest, or in steady rotation, they play a key
role in astrophysics. They are easier to study than non-stationary
systems because stationary solutions are governed by elliptic
rather than hyperbolic equations. Finally, like in any field
theory, one expects that large classes of dynamical solutions
approach (``settle down to") a stationary state in the final
stages of their evolution.

The simplest stationary solutions describing compact isolated
objects are the spherically symmetric ones. In the vacuum region
these are all given by the Schwarzschild family. A theorem of
Birkhoff shows that in the vacuum region any spherically symmetric
metric, even without assuming stationarity, belongs to the family
of Schwarzschild metrics, parameterized by a positive mass
parameter $m$.
%
%of macroscopic objects we always have $m<< r$, so the possible
%singularity of $g$ at $r=2m$ is irrelevant for such purposes.
Thus, regardless of possible motions of the matter, as long as
they remain spherically symmetric, the exterior metric is the
Schwarzschild one for some constant $m$. This has the following
consequence for stellar dynamics: Imagine following the collapse
of a cloud of pressureless fluid (``dust"). Within Newtonian
gravity this dust cloud will, after finite time, contract to a
point at which the density and the gravitational potential
diverge. However, this result cannot be trusted as a sensible
physical prediction because, even if one supposes that Newtonian
gravity is still valid at very high densities, a matter model
based on non-interacting point particles is certainly not.
Consider, next, the same situation in the Einstein theory of
gravity: Here a new question arises, related to the form of the
Schwarzschild metric outside of the spherically symmetric body:
\begin{equation}
\label{Schwarz} g=-V^{2}dt^{2}+V^{-2}dr^{2}+r^{2}d\Omega^{2}\;,
\quad V^{2}=1-\frac{2Gm}{rc^{2}} \;,\quad t\in \R\;,\ r\in
(\frac{2Gm}{c^{2}},\infty)
\end{equation} Here $d\Omega^{2}$ is the line element of the standard
2-sphere. Since the metric \eq{Schwarz} seems to be singular as
$r=2m %(=\frac{2Gm}{c2})
$ is approached (from now on we use units in which $G=c=1$) there
arises the need to understand what happens at
the surface of the star when the radius $r=2m$ %(=\frac{2Gm}{c2})$
is reached.
%This occurs well before the singular collapse (and possibly, one
%might add, for astronomical values of $r$, so that the matter
%model can still be trusted).
One thus faces the need of a careful study of the geometry of the
metric \eq{Schwarz} when $r=2m$ is approached, and crossed.

The first key feature of the metric \eq{Schwarz} is its
stationarity of course with Killing vector field $X$ given by
$X=\partial_t$. A Killing field, by definition, is a vector field
the local flow of which generates isometries. A
space--time\footnote{The term \emph{space--time} denotes a smooth,
paracompact, connected, orientable and time--orientable Lorentzian
manifold.} is called {\em stationary } if there exists a Killing
vector field $X$ which approaches $\partial_t$ in the
asymptotically flat region (where $r$ goes to $\infty$, see below
for precise definitions) {\em and} generates a one parameter
groups of isometries.   A space--time is called {\em static} if it
is stationary and if the stationary Killing vector $X$ is
hypersurface-orthogonal, i.e. $\xflat \wedge d \xflat = 0$, where
$\xflat = X_\mu dx^\mu = g_{\mu\nu} X^\nu dx^\mu$. A space--time
is called {\em axisymmetric} if there exists a Killing vector
field $Y$, which generates a one parameter group of isometries,
and which behaves like a {\em rotation} in the asymptotically flat
region, with  all orbits $2\pi $ periodic. In asymptotically flat
space-times this implies that there exists an axis of symmetry,
that is, a set on which the Killing vector vanishes. Killing
vector fields which are a non-trivial linear combination of a time
translation and of a rotation in the asymptotically flat region
are called \emph{stationary-rotating}, or \emph{helical}.

There exists a technique, due independently to Kruskal and
Szekeres, of attaching together two regions $r>2m$ and two regions
$r<2m$ of the Schwarzschild metric, in a way
shown\footnote{\label{fNi}We are grateful to J.-P.~Nicolas for
allowing us to use his electronic figures, based on those in
\emph{Dissertationes Math.} {\bf 408} (2002), 1--85.} in
Figure~\ref{Sfig0}, to obtain a manifold with a metric which is
smooth at  $r=2m$. In the extended space-time the hypersurface
$\{r=2m\}$ is a null hypersurface $\mcE$, the Schwarzschild event
horizon. The stationary Killing vector $X=\partial_t$ extends to a
Killing vector in the extended spacetime which becomes tangent to
and null on $\mcE$.
 The global properties of the Kruskal--Szekeres
extension of the exterior Schwarzschild\footnote{The exterior
Schwarzschild space-time \eq{Schwarz} admits an infinite number of
non-isometric vacuum extensions, even in the class of maximal,
analytic, simply connected ones. The Kruskal-Szekeres extension is
singled out by the properties that it is maximal, vacuum,
analytic, simply connected, with all maximally extended geodesics
either complete, or with the area $r$ of the orbits of the
isometry groups tending to zero along them.} spacetime, make this
space-time a natural model for a non-rotating black
hole.\begin{figure}[t]
%\FIGURE[h]
\label{Sfig0}%
\begin{center}
\includegraphics[width=.8\textwidth]{KrusExt.eps}
\end{center}
\caption{The Kruskal-Szekeres extension of the Schwarzschild
solution.}
\end{figure}

We can now come back to the problem of the contracting dust cloud
according to the Einstein theory. For simplicity we take the
density of the dust to be uniform --- the so-called
Oppenheimer-Snyder solution. It then turns out that, in the course
of collapse, the surface of the dust will eventually cross the
Schwarzschild radius, leaving behind a Schwarzschild black hole.
If one follows the dust cloud further, a singularity will
eventually form, but will not be visible from the "outside region"
(where $r>2m$). For astronomical masses the Schwarzschild radius
will be ``astronomically large" (\emph{e.g.}, $2m$ equals three
kilometers when $m$ is the mass of the sun), and standard
phenomenological matter models such as that for dust can still be
trusted, so the previous objection to the Newtonian scenario does
not apply.



There is a rotating generalization of the Schwarzschild metric,
namely the two parameter family of \emph{exterior Kerr metrics},
which in Boyer-Lindquist coordinates take the form
\begin{eqnarray}
\label{Kerr} \lefteqn{ g = -\frac{\Delta - a^{2} \sin^{2}\theta
}{\Sigma}dt^{2} - \frac{2a \sin ^{2} \theta
(r^{2}+a^{2}-\Delta)}{\Sigma}dtd\varphi  + {} }
\nonumber\\
& & {}+ \frac{(r^{2}+a^{2})^{2}-\Delta a^{2}\sin ^{2}\theta
}{\Sigma}\sin ^{2}\theta  d\varphi
^{2}+\frac{\Sigma}{\Delta}dr^{2} +\Sigma d\theta ^{2},
\end{eqnarray}
with $0 \leq a<m$. Here $\Sigma=r^{2}+a^{2}\cos^{2}\theta $,
$\Delta=r^{2}+a^{2}-2mr$ and $r_{+}<r<\infty$ where
$r_{+}=m+(m^{2}-a^{2})^{\frac{1}{2}}$. When $a=0$, the Kerr metric
reduces to the Schwarzschild metric. The Kerr metric is again a
vacuum solution, and it is stationary with $X=\partial_t$ the
asymptotic time translation, as well as axisymmetric with
$Y=\partial_\varphi$ the generator of rotations. Similarly to the
Schwarzschild case, it turns out that the metric can be smoothly
extended across $r=r_{+}$, with $\{r=r_{+}\}$ being a smooth null
hypersurface $\mcE$ in the extension. The null generator $K$ of
$\mcE$ is the limit of the stationary-rotating Killing field $X +
\OOmega Y$, where $\OOmega = \frac{a}{2mr_{+}}$. On the other
hand, the Killing vector $X$ is timelike only outside the
hypersurface $\{r=m+(m^{2}-a^{2} \cos^{2}\theta )^\frac{1}{2}\}$,
on which $X$ becomes null. In the region between $r_+$ and
$r=m+(m^{2}-a^{2}\cos^{2}\theta )^\frac{1}{2}$, which is called
the \emph{ergoregion}, $X$ is spacelike. It is also spacelike on
and tangent to  $\mcE$, except where the axis of rotation meets
$\mcE$, where $X$ is null. By the above properties the Kerr family
provides natural models for rotating black holes.

Unfortunately, as opposed to the spherically symmetric case, there
are no known explicit collapsing solutions with rotating matter,
in particular no known solutions having the Kerr metric as final
state.

The aim of the theory outlined below is to understand the general
geometrical features of stationary black holes, and to give a
classification of models satisfying the field
equations.

\section{Model independent concepts}


We now make precise some notions used informally in the
introductory section.  The mathematical notion of black hole is
meant to capture the idea of a region of space-time which cannot
be seen by ``outside observers". Thus, at the outset, one assumes
that there exists a family of physically preferred observers in
the space-time under consideration. When considering isolated
physical systems, it is natural to define the ``exterior
observers" as observers which are ``very far" away from the system
under consideration.
 The standard way of making this mathematically precise is by using conformal completions,  discussed in
more detail in the article about asymptotic structure in this
encyclopedia: A pair $(\tcM ,\cg)$ is called a \emph{conformal
completion at infinity,} or simply {\em conformal completion}, of
$(\mcM,g)$ if $\tcM $ is a manifold with boundary such that:
\begin{enumerate}\item $\mcM$ is the interior of $\tcM $, \item there exists a function $\Omega$, with the
property that the metric $\tg$, defined as $\Omega^{2}g$ on
$\mcM$, extends by continuity to the boundary of $\tcM$, with the
extended metric remaining of Lorentzian signature, \item $\Omega$
is positive on $\mcM$, differentiable on $\tcM $, vanishes on the
boundary $$\Scri:=\tcM\setminus\mcM\;,$$ with $d\Omega$
\emph{nowhere vanishing} on $\Scri$.
\end{enumerate} The boundary $\scri$ of~$\tcM $ is called Scri, a phonic shortcut for ``script I".
 The idea here is the following: forcing $\Omega$ to vanish on
 $\Scri$ ensures that $\Scri$ lies infinitely far away from any
 physical object ---  a mathematical way of capturing the notion ``very far
 away". The condition that $d\Omega$ does not vanish
 is a convenient technical condition which ensures that $\Scri$ is a smooth
 three dimensional hypersurface, instead of some, say, one or two dimensional
 object, or of a set with singularities here and there. Thus,
 $\Scri$ is an idealized description of a family of observers at
 infinity.

 To distinguish between various points of $\scri$ one sets
$$
\Scri^+=\{\mbox{points in $\Scri$ which are to the future of the
physical space-time}\}\ .
$$$$
\Scri^-=\{\mbox{points in $\Scri$ which are to the past of the
physical space-time}\}\ .
$$
(Recall that a point $q$ is to the future, respectively to the
past, of $p$ if there exists a future directed, respectively past
directed, causal curve from $q$ to $p$. Causal curves are curves
$\gamma$ such that their tangent vector $\dot \gamma$ is causal
everywhere, $g(\dot \gamma,\dot \gamma)\le 0$.)
%Compare the article on causal structure in this encyclopedia.)
 One then defines the black hole
region $\mcB$ as \bean \mcB &:=&  \{\mbox{the set of points in
$\mcM$ from which}
\\ && \phantom{\cM\setminus the}
\mbox {no future directed causal curve in $\tcM$ meets
$\scrip$}\}\;.\eeal{sbh1} By definition, points in the black hole
region cannot thus send information to $\scrip$; equivalently,
observers on $\scrip$ cannot see points in $\mcB$.
 The \emph{white hole} region $\mcW$ is
defined by changing the time orientation in \eq{sbh1}.
%In order to obtain a meaningful definition of black hole, one
%needs to assume further that $\scrip$ satisfies a few regularity
%conditions. For example, if we consider the standard conformal
%completion of Minkowski space-time, then of course $\mcB$ will be
%empty. However, one can remove points from that completion,
%obtaining sometimes a new completion with a non-empty black hole
%regions. (Think of a family of observers who stop to exist at time
%$t=0$, they will never be able to see any event with $t>0$,
%leading to a black hole region with respect to this family.) We
%shall not describe those conditions for general space-times, but
%refer the reader to \cite{Chrusciel:2002mi} and references therein
%for a detailed discussion. Here we are mainly interested in
%stationary space-times, with a Killing vector field $X$ which is
%timelike near $\scrip$. For those space-times the topological
%In \eq{def1} we have implicitly used the fact that a Killing
%vector field defined on $\mcM$ smoothly extends to a vector field
%on $\scrip$, and we have used the same symbol $X$ to denote that
%extension. For the standard completion of Minkowski space-time,
%and for the standard timelike vector field $\partial_t$, the space
%of orbits in \eq{def2} is $S2$, indeed a compact manifold. More
%generally, \eq{def1}-\eq{def2} hold for the standard conformal
%
 A key notion related to the concept of a black hole is
that of  \emph{future} ($\mcE^+)$ and \emph{past ($\mcE^-$) event
horizons}, \bel{eh1} \mcE^+:= \partial \mcB\;,\qquad \mcE^-:=
\partial \mcW\;.\ee Under mild assumptions,
 event horizons in stationary space-times
with matter satisfying the  \emph{null energy condition},
\begin{equation}
T_{\mu \nu }\ell^\mu \ell^\nu \ge 0\quad \mbox{  for all null
vectors $\ell^\mu$,} \label{energy}
\end{equation} are  {smooth} null
hypersurfaces, analytic if the metric is analytic.

 In order to
develop a reasonable theory one also needs a regularity condition
for the interior of space-time.   This has to be  a condition
which does not exclude singularities (otherwise the Schwarzschild
and Kerr black holes would be excluded), but which nevertheless
guarantees a well-behaved exterior region. One such condition,
assumed in all the results described below, is the existence in
$\mcM$ of an asymptotically flat space-like hypersurface $\hyp$
with compact interior. Further, either $\hyp$ has no boundary, or
the boundary of $\hyp$ lies on $\overline{\mcE^+\cup\mcE^-}$. To
make things precise, for any spacelike hypersurface let $g_{ij}$
be the induced metric, and let $K_{ij}$ denote its extrinsic
curvature.
 A space--like hypersurface
$\hyp_{\mathrm{ext}}$ diffeomorphic to
      ${\Bbb R}^3$ minus a
ball will be called \emph{asymptotically flat} if the fields
$(g_{ij},K_{ij})$  satisfy the fall--off conditions
\begin{equation}
|g_{ij}-\delta _{ij}|+r|\partial _\ell g_{ij}|+\cdots
+r^k|\partial _{\ell _1\cdots \ell _k}g_{ij}|+r|K_{ij}|+\cdots
+r^k|\partial _{\ell _1\cdots \ell _{k-1}}K_{ij}|\le Cr^{-1 }\ ,
\label{falloff}
\end{equation}
for some constants $C$,  $k\ge 1$. A hypersurface $ \hyp $ (with
or without boundary) will be said to be {\em asymptotically flat
with compact interior} if $\hyp $ is of the form $\hyp_{int}\cup
\hyp_{\mathrm{ext}}$, with $\hyp_{int}$ compact and $\Sext$
asymptotically flat.


There exists a canonical way of constructing a conformal
completion with good global properties for stationary space-times
which are asymptotically flat in the sense of \eq{falloff}, and
which are vacuum sufficiently far out in the asymptotic region.
This conformal completion is referred to as the \emph{standard
completion}
%It satisfies the following regularity
%conditions: The stationary Killing vector field $X$ defined on
%$\mcM$ smoothly extends to a vector field on $\scrip$, also called
%$X$, such that
%\beal{def1}&&\mbox{The orbits of $X$ on $\scrip$ are complete,
%and}
%\\ &&\mbox{the space of orbits of $X$ on $\scrip$ is a compact
%manifold.}\eeal{def2} (Recall that an orbit of a vector field $X$
%is defined as the set of points which lie on an integral curve of
%$X$.)
and will be assumed from now on.

Returning to the event horizon $\mcE=\mcE^+\cup\mcE^-$, it is not
very difficult to show that every Killing vector field $X$ is
necessarily tangent to $\mcE$. Since the latter set is a null
Lipschitz hypersurface, it follows that $X$ is either null or
spacelike on $\mcE$. This leads to a preferred class of event
horizons, called \emph{Killing horizons}. By definition, a Killing
horizon associated with a Killing vector $K$ is a \emph{null
hypersurface} which coincides with a connected component of the
set \bel{defKH} \cH(K):=\{p\in\mcM:\ g(K,K)(p)=0\;,\ K(p)\ne
0\}\;.\ee A simple example is provided by the ``boost Killing
vector field" $K=z\partial_t+ t\partial_z$ in Minkowski
space-time: $\cH(K)$ has four connected components
$$\cH_{\epsilon \delta} :=\{ t=\epsilon z\;, \delta t >0\}\;,
\quad \epsilon, \delta \in \{\pm 1\}\;.$$ The closure $\overline
\cH$ of $ \cH$ is the set $\{|t|=|z|\}$, which is not a manifold,
because of the crossing of the null hyperplanes $\{t=\pm z\}$ at
$t=z=0$. Horizons of this type are referred to as {\em bifurcate
Killing horizons}, with the set $\{K(p)=0\}$ being called the
\emph{bifurcation surface} of $\cH(K)$. The bifurcate horizon
structure in the Kruszkal-Szekeres-Schwarzschild space-time can be
clearly seen on Figures~\ref{Sfig0} and \ref{FCPd}.

 The Vishveshwara-Carter
lemma % ~\cite{Vishveshwara,CarterJMP}
shows that if a Killing vector $K$ is hypersurface-orthogonal,
$\kflat \wedge d \kflat=0$, then the set $\cH(K)$ defined in
\eq{defKH} is a union of smooth null hypersurfaces, with $K$ being
tangent to the null geodesics threading $\cH$ (``$\cH$ is
generated by $K$"), and so is indeed a Killing horizon. It has
been shown by Carter that the same conclusion can be reached if
the hypothesis of hypersurface-orthogonality is replaced by that
of existence of two linearly independent Killing vector fields.

 In stationary-axisymmetric space-times a
Killing vector $K$ \emph{tangent to the generators} of a Killing
horizon $\cal H$ can be normalised so that
 $K=X+\OOmega  Y$,
 where $X$ is the Killing vector field which asymptotes
 to a time translation in the asymptotic region, and $Y$ is the
 Killing vector field which generates rotations in the asymptotic
 region. The constant $\OOmega $ is called the \emph{angular
 velocity of the Killing horizon $\cal H$}.

On a Killing horizon $\cH(K)$ one necessarily has \bel{kappadef}
\nabla^\mu(K^\nu K_\nu)=-2\kappa K^\mu . \ee Assuming the
so-called dominant energy condition on $T_{\mu\nu}$,\footnote{See
the article by Bray on "Positive Energy Theorem and other
inequalities in General Relativity"} it can be shown that $\kappa$
is constant (recall that Killing horizons are always connected in
our terminology), it is called {\em the surface gravity of $\cH$}.
A Killing horizon is called \emph{degenerate} when $\kappa= 0$,
and non--degenerate otherwise; by an abuse of terminology one
similarly talks of degenerate black holes, \emph{etc}. In Kerr
space-times we have $\kappa=0$ if and only if $m=a$. A fundamental
theorem of Boyer shows that degenerate horizons are closed. This
implies that a horizon $\cH(K)$ such that $K$ has zeros in
$\overline \cH$ is non-degenerate, and is of bifurcate type, as
described above. Further, a \emph{non-degenerate} Killing horizon
with \emph{complete} geodesic generators always contains zeros of
$K$ in its closure. However, it is not true that existence of a
non-degenerate horizon implies that of zeros of $K$: take the
Killing vector field $z\partial_t +t\partial _z$ in Minkowski
space-time from which the 2-plane $\{z=t=0\}$ has been removed.
The universal cover of that last space-time provides a space-time
in which one cannot restore the points which have been
artificially removed, without violating the manifold property.




%probably not needed in the stationary context, remove if we run
%out of space} One difficulty of the notion of event horizon is its
%global character: one needs to know the whole space-time in order
%to be able to define $\mcB$. In physics we can at most observe the
%region of the space-time to our past. At best we can perhaps
%arrange the initial data in a small neighborhood around us to have
%some specific properties. However, we are unable to predict the
%future of the whole universe, because for this the initial data in
%a small neighborhood are not sufficient. It is therefore
%convenient to have a locally defined object that signals the
%presence of black holes when we have a spacelike hypersurface
%$\hyp$ with Cauchy data $(g,K)$ at our disposal. One such object
%is an \emph{apparent horizon}: by definition, this is a
%  compact, boundaryless, smooth, two-sided, two-dimensional surface $S\subset \hyp$ with the
%  property that  \bel{trsur} \left.\theta(S):= \lambda- (g^{ij}-
%  n^in^j) K_{ij} = 0\right.\;,\ee {where $\lambda$ is the (outwards)
%  mean extrinsic curvature of $S$ in $\hyp$, $n^i$ is the outwards pointing
%  unit normal to $S$ in $\hyp$, and $K_{ij}$ is the extrinsic
%  curvature of $\hyp$ in $\cM$.} Under appropriate global conditions on $\tcM$,
%  the existence of an apparent horizon implies that of a non-trivial black
%singularities? but an apparent horizon is not trapped, but only
%marginally so, I am not sure which part of the relevant
%singularity theorem survives then}

%We note that if $K_{ij}=0$, then (\ref{trsur}) is the equation for
%a minimal surface within $\hyp$.


  The \emph{domain of outer communications} (d.o.c.) of a black hole space-time is defined as
  \bel{dext} \doc:= \mcM \setminus\{\mcB\cup\mcW\}\;.\ee
  Thus, $\doc$ is the region lying outside of the white hole region
  and outside of the black hole region; it is the region which can
  both
  be seen by the outside observers and  influenced by
  those.

  The subset of $\doc$
 where $X$ is spacelike is called the
 \emph{ergoregion}. In the Schwarzschild space-time $\OOmega =0$ and
 the ergoregion is empty, but neither of these is true in Kerr with
 $a \ne 0$.

 A very convenient method for visualising
  the global structure of space-times is provided by the
  \emph{Carter-Penrose diagrams}. An example of such a diagram is
  given$^{\mbox{\scriptsize \rm \ref{fNi}}}$ in Figure~\ref{FCPd}.
\begin{figure}[t]
\begin{center}
  \includegraphics[width=.8\textwidth]{SchwaPen.eps}
\end{center}
\caption[FCPd]{The Carter-Penrose diagram for the Kruskal-Szekeres
space-time. There are actually two asymptotically flat regions,
with corresponding $\scri^\pm$ and $\mcE^\pm$ defined with respect
to the second region, but not indicated on this diagram. Each
point in this diagram represents a two-dimensional sphere, and
coordinates are chosen so that light-cones have slopes plus minus
one. Regions are numbered as in Figure~\ref{Sfig0}.} \label{FCPd}
\end{figure}


  A corollary of the \emph{topological censorship theorem} of
Friedman, Schleich and Witt is that d.o.c.'s of regular black hole
space-times satisfying the dominant energy condition are simply
connected. This implies that connected components of event
horizons in stationary space-times have ${\Bbb R}\times S^{2}$
topology.

We end our review of the concepts associated with stationary black
hole spacetimes by summarising the properties of the Schwarzschild
and Kerr geometries: The extended Kerr spacetime with $m
> a$ is a black hole spacetime with the hypersurface  $\{r =
r_+\}$ forming a non-degenerate, bifurcate Killing horizon
generated by the vector field $X + \OOmega  Y$ and surface gravity
given by $\kappa =
\frac{(m^{2}-a^{2})^{1/2}}{2m[m+(m^{2}-a^{2})^{1/2}]}$. In the
case $a=0$, where the angular velocity $\OOmega $ vanishes, $X$ is
hypersurface-orthogonal and becomes the generator of $\cH$. The
bifurcation surface in this case is the totally geodesic 2-sphere,
along which the four regions in Figure 1 are joined.

\section{Classification of stationary solutions (``No hair
  theorems")}

We confine attention to the ``outside region" of black holes, the
domain of outer communications (d.o.c.).\footnote{Except for the
so-called degenerate case discussed later, the ``inside"(black
hole) region is not stationary, so that this restriction already
follows from the requirement of stationarity.} For reasons of
space we only consider vacuum solutions; there exists a similar
theory for electro-vacuum black holes. (There is a somewhat less
developed theory for black hole spacetimes in the presence of
nonabelian gauge fields, see the review by Gal'tsov and Volkov.)
In connection with a collapse scenario the vacuum condition begs
the question: collapse of what? The answer is twofold: First there
are large classes of solutions of Einstein equations describing
pure gravitational waves. It is believed that sufficiently strong
such solutions will form black holes. (Whether or not they will do
that is related to the \emph{cosmic censorship conjecture},
discussed in the article on Spacetime Topology, Global Structure
and Singularities  in this encyclopedia.) Consider, next,  a
dynamical situation in which matter is initially present. The
conditions imposed in this section correspond then to a final
state in which matter has either been radiated away to infinity,
or has been swallowed by the black hole (as in the spherically
symmetric Oppenheimer-Snyder collapse described above).








Based on the facts below, it is expected that the d.o.c.'s of
appropriately regular, stationary, vacuum black holes are
isometrically diffeomorphic to those of  Kerr black holes:

\begin{enumerate}
\item  The \emph{rigidity theorem} (Hawking): event horizons in
regular, \emph{non--degenerate}, stationary, \emph{analytic}
vacuum black holes are  either {\em Killing horizons} for $X$,  or
there exists a second Killing vector in $\doc$.

\item The \emph{Killing horizons theorem}  (Sudarsky-Wald):
\emph{non--degenerate} stationary  vacuum black holes such that
the \emph{event horizon is the union of Killing horizons of $X$}
are {\em static}.
% (cf. }Theorem \ref {T:rigidity2} below).

\item  {} The Schwarzschild black holes exhaust the family of {\em
static}  regular vacuum black holes (Israel, Bunting --
Masood-ul-Alam, Chru\'sciel).

\item  {} The non--degenerate Kerr black holes
satisfying
\begin{equation}
\label{acondition} m^{2}>a^{2} %\;,
\end{equation} exhaust the family of \emph{non--degenerate},
{\em stationary--axisymmetric,} vacuum, \emph{connected}
 black holes.  Here $m$ is the total ADM mass (see
the article by Bray in this encyclopedia), while the product $am$
is
 the total ADM angular
momentum. (Of course these quantities generalize the constants
 $a$ and $m$ appearing in the Kerr metric.) The framework for the proof has been set-up by Carter,
 and the statement above
 is due to Robinson.
 %Bunting and independently Mazur, with the justification of the inequality \eq{acondition}
 %due to Weinstein.
\end{enumerate}


The above results are collectively known under the name of
\emph{no hair theorems}, and they have not provided the final
answer to the problem so far.
 There are no \emph{a priori} reasons
known for the analyticity hypothesis in the rigidity theorem.
Further, degenerate horizons have been completely understood in
the static case only.



%One of the steps of the proof of that theorem consists in proving
%that connected components of black--hole event horizons have
%${\Bbb R}\times S2$ topology and simply--connected domain of
%outer communication. The precise statement  is the
%following:\ptc{the more precise statement not needed in the
%stationary context} Consider a space--time which is asymptotically
%flat at null infinity in the sense of the definition given on
%p.~282 of \cite{Waldbook}. We shall say that $\Scri$ satisfies
%{\em the
%  regularity condition\/} if
%\begin{enumerate}
%\item There exists a neighborhood $\widehat{\cal O}$ of $\Scri^+ \cup
%  \Scri^-$ which is simply connected.
%\item{} There exists a neighborhood ${\cal O}\subset\widehat{\cal O}$
%  of $\Scri^-$ such that for every compact set $K\subset {\cal
%O}\setminus \Scri^-$, the
%  set $\partial J^+(K;\bar M)\cap\Scri^+$ is not empty.
%\end{enumerate}
%(Both conditions are always satisfied by
%  the standard conformal completion of asymptotically flat
%  stationary space-times.) We have the following result of Galloway, that does {\em not\/}
%assume stationarity:
%
%\begin{Theorem}[G. Galloway, 96]
%\label{T:galloway} Consider a space--time which is asymptotically
%flat at null infinity and which has a $\Scri$ which satisfies the
%regularity\footnote{The regularity condition of $\Scri$ can be
%  replaced by other conditions, {\em cf.\
%    \emph{e.g.}\/}~\cite{Galloway:fitopology}.  While the regularity
%  condition given here is not spelled out in detail in
%  \cite{FriedmanSchleichWitt} and in \cite{Galloway:fitopology}, some
%  form of causal regularity of $\Scri$ is needed for the arguments
%  given there to go through. I am grateful to G.~Galloway for
%  discussions concerning this point.} condition. Suppose moreover that
%the Ricci tensor satisfies the null convergence condition:
%\begin{equation}
%R_{\mu \nu }X^\mu X^\nu \ge 0\quad \mbox{  for all null vectors
%$X^\mu$.} \label{energy}
%\end{equation}
%Then every globally hyperbolic domain of outer  communications  is
%simply connected.
%\end{Theorem}
%

%
%\subsection{The rigidity theorem}
%
%\begin{Theorem}
%\label{T:rigidity2} Consider an analytic %stably causal
%stationary electro--vacuum space--time $(M,g_{ab})$ with Killing
%vector field $X$, and let  $\langle \langle
%M_{\mathrm{ext}}\rangle \rangle $ be an asymptotically flat
%globally hyperbolic domain of outer communication in $M$. (Here
%asymptotic flatness is understood in the sense of (\ref{falloff})
%with $\alpha >0$ and $k\ge 1$, together with eq. \eq{Efalloff}.)
%Assume that there exists an achronal asymptotically flat slice
%${\cal S}$ of $\langle \langle M_{\mathrm{ext}}\rangle \rangle $,
%{} whose boundary in $M$ intersects the event horizon ${\cal E}$
%of the black hole region in a compact cross--section $K$.
%%Suppose, next, that ${\cal C}\setminus {\cal C}_{{\mathrm{ext}}}$ has
%%compact closure in M (where ${\cal C}$ and $ {\cal
%%  C}_{{\mathrm{ext}}}$ were defined in Section \ref{sec:topology}
%%above).
%Suppose finally that there exists a connected component of ${\cal
%E}\cap J^{+}(M_{\mathrm{ext}})$, say ${\cal E}_1$, which is an
%analytic submanifold of $M$.  If $X$ is {\em not} tangent to the
%generators of ${\cal E}_1$, then:
%\begin{enumerate}
%\item  The isometry group of $\langle \langle M_{\mathrm{ext}}\rangle \rangle $
%contains ${\Bbb R}\times U(1)$.
%\item %If ${\cal E}_1$ is two--sided, than i
%The event horizon is a Killing horizon in the following sense:
%There exists a neighborhood of
%  $\overline{\langle \langle M_{\mathrm{ext}}\rangle \rangle }$ in
%  $J^{+}(M_{\mathrm{ext}})$ on which a Killing vector field $Z$,
%%  which is
%tangent to the generators of ${\cal E}_1$, is defined.
% %\cap J^+(M_{\mathrm ext})$.
%%\item
%%  there exists a Killing vector field $Z$
%%on  $\langle \langle
%%  M_{\mathrm{ext}}\rangle \rangle $ which can be extended
%\end{enumerate}
%\end{Theorem}
%
%Let us point out that the above result generalizes immediately to
%stationary--rotating space--times. The only supplementary
%hypothesis needed is that of asymptotic flatness with $\alpha>1/2$
%and $k\ge 3$ in the sense of eq. (\ref{D1.new}) (in the stationary
%case that property follows from the fall--off hypotheses
%spelled--out above).


Yet another key open question is that of existence of
\emph{non-connected} regular stationary-axisymmetric vacuum black
holes. The following result is due to Weinstein:  Let
$\partial\hyp _a$, $a=1,\ldots,N$ be the connected components of
$\partial \hyp $. Let $\xflat=g_{\mu\nu}X^\mu dx^\nu$, where
$X^\mu$ is the Killing vector field which asymptotically
approaches the unit normal to $\Sext$. Similarly set
$\yflat=g_{\mu\nu} Y^\mu dx^\nu$, $Y^\mu$ being the Killing vector
field associated with rotations. On each $\partial\hyp _a$ there
exists a constant $\OOmega _a$  such that the vector $X+\OOmega _a
Y$ is tangent to the generators of the Killing horizon
intersecting $\partial \hyp_a$. The constant $\OOmega _a$ is
called the angular velocity of the associated Killing horizon.
Define
\begin{eqnarray}%
%& q_a=-\frac{1}{4\pi}\int_{\partial\hyp _a}*F\ , & \label{qa}
%\\
& m_a =-\frac{1}{8\pi}\int_{\partial\hyp _a}*d\xflat \ , &
\label{ma}
%\begin{equation}
\\
\label{lint} & L_a=-\frac{1}{4\pi}\int_{\partial\hyp _a}*d\yflat \
. &
%\end{equation}
\end{eqnarray}
Such integrals are called \emph{Komar integrals}. One usually
thinks of $L_a$ as the angular momentum of each connected
component of the black hole. Set
\begin{equation}
  \label{mua}
\mu_a=m_a -2 \OOmega _a L_a\;.
  \end{equation}
  Weinstein shows that one necessarily has $\mu_a >0$. The problem at hand can be reduced
  to a \emph{harmonic map} equation, also known as the \emph{Ernst equation},
   involving a singular map from $\R3$ with Euclidean metric $\delta$ to the two-dimensional hyperbolic space.
    Let $r_a >0$,
  $a=1,\ldots,N-1$, be the distance in $\R3$  along the axis between
  neighboring black holes as measured with respect to the
  (unphysical) metric $\delta$. Weinstein proved that for \emph{non-degenerate} regular black holes
   the inequality \eq{acondition} holds, and that
  the metric on $\doc$ is determined up to isometry
  by the $3N-1$ parameters
  \begin{equation}
    \label{par}
    (\mu_1,\ldots,\mu_N,L_1,\ldots,L_N,r_1,\ldots,
r_{N-1})
  \end{equation}
just described, with $r_a,\mu_a>0$. These results by Weinstein
contain the no-hair theorem of Carter and Robinson as a special
case.  Weinstein also shows that for every $N\ge 2$ and for every
set of parameters \eq{par} with $\mu_a,r_a>0$, there exists a
solution of the problem at hand. It is known that for some sets of
parameters \eq{par} the solutions will have ``strut
singularities'' between some pairs of neighboring black holes, but
the existence of the ``struts'' for all sets of parameters as
above is not known, and is one of the main open problems in our
understanding of stationary--axisymmetric electro--vacuum black
holes.  The existence and uniqueness results of Weinstein remain
valid when strut singularities are allowed in the metric at the
outset, though such solutions do not fall into the category of
regular
black holes discussed here. %Weinstein's proof of \eq{acondition},
%together with the results of Mazur and (independently)
%Bunting,\ptc{Mazur and Bunting not needed here} give a complete
%description of \emph{non-degenerate, connected,
%stationary-axisymmetric} black holes.

\bigskip

\noindent {\bf See also:} Asymptotic  Structure and Conformal
Infinity. Black Hole Thermodynamics. Initial Value problem for
Einstein Equations. Positive energy Theorem and other inequalities
in General Relativity. Spacetime Topology, Causal Structure and
Singularities.


\bigskip


\noindent {\bf Suggestions for Further Reading:}
\cite{BONeillKerr,Waldbook,CarterlesHouches,HE,Chrusciel:2002mi,Heusler:book,Heusler:living,VolkovGaltsov}


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\begin{thebibliography}{1}

\bibitem{CarterlesHouches}
B.~Carter, \emph{Black hole equilibrium states}, Black Holes (C.\
de~Witt and
  B.\ de~Witt, eds.), Gordon \& Breach, New York, London, Paris, 1973,
  Proceedings of the Les Houches Summer School.

\bibitem{Chrusciel:2002mi}
P.T. Chru\'{s}ciel, \emph{Black holes}, Proceedings of the
T{\"u}bingen
  Workshop on the Conformal Structure of Space-times, H. Friedrich and J.
  Frauendiener, Eds., Springer Lecture Notes in Physics {\bf 604}, 61--102
  (2002), gr-qc/0201053.

\bibitem{HE}
S.W. Hawking and G.F.R. Ellis, \emph{The large scale structure of
space-time},
  Cambridge University Press, Cambridge, 1973.

\bibitem{Heusler:book}
M.~Heusler, \emph{Black hole uniqueness theorems}, Cambridge
University Press,
  Cambridge, 1996.

\bibitem{Heusler:living}
\bysame, \emph{Stationary black holes: uniqueness and beyond},
Living Reviews
  \textbf{1} (1998), \url{http://www.livingreviews.org}.

\bibitem{BONeillKerr}
B.~O'Neill, \emph{The geometry of {K}err black holes},
A.K.~Peters, Wellesley,
  Mass., 1995.

\bibitem{VolkovGaltsov}
M.S. Volkov and D.V. Gal'tsov, \emph{Gravitating non-{Abelian
solitons and
  black holes with Yang--Mills} fields}, Phys.\ Rep. \textbf{319} (1999),
  1--83, hep-th/9810070.

\bibitem{Waldbook}
R.M. Wald, \emph{General relativity}, University of Chicago Press,
Chicago,
  1984.

\end{thebibliography}

\end{document}

gr-qc0502041.extracted_bib
\begin{thebibliography}{1}

\bibitem{CarterlesHouches}
B.~Carter, \emph{Black hole equilibrium states}, Black Holes (C.\
de~Witt and
  B.\ de~Witt, eds.), Gordon \& Breach, New York, London, Paris, 1973,
  Proceedings of the Les Houches Summer School.

\bibitem{Chrusciel:2002mi}
P.T. Chru\'{s}ciel, \emph{Black holes}, Proceedings of the
T{\"u}bingen
  Workshop on the Conformal Structure of Space-times, H. Friedrich and J.
  Frauendiener, Eds., Springer Lecture Notes in Physics {\bf 604}, 61--102
  (2002), gr-qc/0201053.

\bibitem{HE}
S.W. Hawking and G.F.R. Ellis, \emph{The large scale structure of
space-time},
  Cambridge University Press, Cambridge, 1973.

\bibitem{Heusler:book}
M.~Heusler, \emph{Black hole uniqueness theorems}, Cambridge
University Press,
  Cambridge, 1996.

\bibitem{Heusler:living}
\bysame, \emph{Stationary black holes: uniqueness and beyond},
Living Reviews
  \textbf{1} (1998), \url{http://www.livingreviews.org}.

\bibitem{BONeillKerr}
B.~O'Neill, \emph{The geometry of {K}err black holes},
A.K.~Peters, Wellesley,
  Mass., 1995.

\bibitem{VolkovGaltsov}
M.S. Volkov and D.V. Gal'tsov, \emph{Gravitating non-{Abelian
solitons and
  black holes with Yang--Mills} fields}, Phys.\ Rep. \textbf{319} (1999),
  1--83, hep-th/9810070.

\bibitem{Waldbook}
R.M. Wald, \emph{General relativity}, University of Chicago Press,
Chicago,
  1984.

\end{thebibliography}