Chapter 3. Topology and Uniqueness of Higher Dimensional Black Holes
ABSTRACT We review recent results concerning general properties of higher dimensional black holes. The topics selected with particular focus are those concerning topology, symmetry, and uniqueness properties of asymptotically flat vacuum black holes in higher dimensional general relativity.
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ABSTRACT: In the Einstein gravity, it is well-known that strictly stationary and vacuum regular spacetime should be the Minkowski spacetime. In the Einstein-Gauss-Bonnet theory, we shall show the similar statement, that is, strictly static(no event horizon), vacuum and asymptotically flat spacetimes with conformally static slices are the Minkowski spacetime when the curvature corrections are small.Physical review D: Particles and fields 02/2013; 87(8). - [Show abstract] [Hide abstract]
ABSTRACT: We show that the static and asymptotically flat black hole spacetime is unique to be Schwarzschild spacetime in the dynamical Chern-Simons gravity. In addition, we show that the strictly static spacetimes should be the Minkowski spacetime.Physical review D: Particles and fields 04/2013; 87(8). - [Show abstract] [Hide abstract]
ABSTRACT: We proved that strictly stationary Einstein-Maxwell-axion-dilaton spacetime with negative cosmological constant could not support a nontrivial configuration of complex scalar fields. We considered the general case of the arbitrary number of U(1) gauge fields in the theory under consideration.Physical review D: Particles and fields 05/2013; 87(8).
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arXiv:1105.3491v2 [hep-th] 2 Jul 2011
1
Topology and Uniqueness of Higher Dimensional Black Holes
Daisuke Idaa, Akihiro Ishibashibcand Tetsuya Shiromizud
aDepartment of Physics, Gakushuin University,
Tokyo 171-8588, Japan
bTheory Center, Institute of Particle and Nuclear Studies,
High Energy Accelerator Research Organization (KEK),
Tsukuba, 305-0801, Japan
cDepartment of Physics, Kinki University,
Higashi-Osaka 577-8502, Japan
and
dDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan
We review recent results concerning general properties of higher dimensional black holes.
The topics selected with particular focus are those concerning topology, symmetry, and
uniqueness properties of asymptotically flat vacuum black holes in higher dimensional general
relativity.
Contents
1. Introduction
2. General properties of higher dimensional black holes
2.1. Global structure and area theorem . . . . . . . . . . . . . . . . . . . .
2.2. Apparent horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Topological structure of dynamical black hole horizons . . . . . . . . .
3. Stationary black holes in higher dimensions
3.1. Topology theorems in higher dimensions . . . . . . . . . . . . . . . . . 11
3.2. Rigidity theorem in higher dimensions . . . . . . . . . . . . . . . . . . 13
4. Uniqueness of static black holes
4.1. Basic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2. Outline of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3. Vacuum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4. Electro-vacuum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.5. Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5. Uniqueness of stationary, rotating black holes
5.1. Basic strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2. Rotating black holes in 5-dimensions . . . . . . . . . . . . . . . . . . . 32
6. Summary
1
3
3
5
7
10
19
31
38
§1.Introduction
The black hole uniqueness theorem in 4-dimensions is a triumph of classical
general relativity, implying that a tremendous number of black holes existing in our
typeset using PTPTEX.cls ?Ver.0.9?
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2D. Ida, A. Ishibashi and T. Shiromizu
observable universe can be described accurately by the Kerr metric, which possesses
only two parameters. In the course of complete proof of the uniqueness theorem,
there have appeared a number of remarkable results, each of which itself reveals
physically an important property of black holes, such as those concerning topology
and symmetry.1)These results also give us deep insights into thermodynamic aspects
of black holes.2)
It is of great interest to consider generalizations of a number of theorems estab-
lished for 4-dimensional black holes to higher dimensional case. One might expect
that such a generalization could straightforwardly be done by merely replacing “4”
with a general number “D.” However, as is by now well-known, the discovery of the
black ring solution in 5-dimensions3)(as well as a large variety of exact solutions dis-
cussed in Chapter 1 and 4) has drastically changed our view of the issue, highlighting
that the uniqueness theorem no longer holds as it stands in higher dimensions.
In this chapter we shall review general properties of higher dimensional black
holes, attempting to clarify which properties of 4-dimensional black holes can be
straightforwardly generalized to higher dimensions and which properties hold only
in 4-dimensions. There have already been a number of theorems for higher dimen-
sional black holes. This review is, however, not intended to cover the whole relevant
subjects or supply a complete list of existing literature. Rather, we will focus on
some specific topics relevant to topology and uniqueness/non-uniqueness feature of
asymptotically flat vacuum black holes and describe some key ideas and methods for
obtaining the results in higher dimensions.
In the next section we shall first recapitulate how to describe global structure
of black hole spacetimes in general dimensions. Then we discuss topological aspects
of apparent and the event horizon of dynamical black holes. In section 3, we con-
sider stationary black holes in higher dimensional general relativity. After briefly
summarizing a few critical steps in the proof of black hole uniqueness theorems in
4-dimensions, we shall discuss how and to what extent each of the steps can be
generalized to higher dimensional setting. In particular, we review recent results
concerning possible restrictions on the horizon topology and also an enhancement of
Killing symmetries, called the rigidity theorem, in higher dimensions. In section 4,
we review uniqueness theorems for static black holes in higher dimensions. Unique-
ness theorems for asymptotically flat, stationary rotating, vacuum black holes are
discussed in detail in section 5. Section 6 is devoted to summary.
Notations and Conventions
In this chapter, we mainly use the abstract index notation for tensor fields on
a spacetime, where each slot for tangent or cotangent field is denoted by a lower-
case latin index: a,b,c,···. In section 4, a tensor field is written in terms of its
components, where upper-case latin indices M,N,··· run from 0 to D − 1, and
lower-case latin indices i,j,··· run from 1 to D − 1. In section 5, we mainly treat
the 5-dimensional spacetimes. There, a tensor field is mainly written in terms of its
components, where lower-case latin indices i,j,··· run from 1 to 2 and upper-case
latin indices I,J,··· run from 3 to 5.
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‘Topology and Uniqueness in Higher Dimensional Black Holes’3
We use the natural unit where the speed of light c and the reduced Planck
constant ? are set to unity.
§2. General properties of higher dimensional black holes
We begin with noting that the uniqueness feature of a stationary black hole is
related to its thermodynamic aspects in the sense that a thermodynamically equi-
librium system can completely be characterized by a small number of state param-
eters.2)The idea of black hole thermodynamics has originated from Bekenstein’s
interpretation4)of Hawking’s area theorem5)as the 2nd law of thermodynamics,
as well as from the black hole mechanics due to Bardeen, Carter and Hawking.6)
While the latter concerns a stationary equilibrium configuration, the former involves
a dynamical process concerning the total area of all black holes in the universe.
Therefore, before going into discussion of stationary black holes, in this section we
shall discuss general circumstances that can include some dynamical processes such
as a formation and evolution of black holes, to which the area theorem becomes
relevant.
2.1. Global structure and area theorem
First of all, in order to define an isolated black hole in general context, one needs
to introduce a suitable notion of “infinity” and associated asymptotic structure.
In 4-dimensions this is usually, and elegantly, done in the conformal framework,
in which an unphysical spacetime (˜M,˜ gab), conformally isometric to our physical
spacetime (M,gab) in M∩˜ M, plays a role. A desired notion of infinity and asymptotic
flatness can be defined by specifying the behavior of the conformal metric ˜ gabnear
a conformal null boundary I = ∂˜
M. If one further imposes an additional condition
that every maximally extended null geodesic in M has past and future endpoints
on null boundary I in ˜M, then I is divided into disjoint sets of the future and
past null infinity, I+and I−. Such a spacetime is called asymptotically simple.
A spacetime (M,gab) is said to be weakly asymptotically simple at null infinity if
(˜M,˜ gab) has a neighborhood of I which is isometric to a neighborhood of I for some
asymptotically simple spacetime. The notion of strong asymptotic predictability
is then defined such that the closure of M ∩ J−(I+) is contained in a globally
hyperbolic open subset of˜M. A black hole region B is defined as the complement
of J−(I+) and the future event horizon H as the boundary of B in M. As such,
H is a null hypersurface ruled by null geodesics. Since B is a future set, every null
geodesic generator of H is future inextendible, but in general admits a past end
point. With the set of these definitions, general properties of 4-dimensional black
holes are studied by using the global method, which consists of a number of general
results concerning causal structure, behavior of causal geodesic congruence, etc, as
in Refs. 7),8).
A key equation for the global method is the Raychaudhuri equation for causal
geodesics, which together with certain energy condition, governs the occurrence of (a
pair of) conjugate points. The structure of the Raychaudhuri equation is unchanged
in higher dimensions as far as (higher dimensional version of) general relativity is
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4D. Ida, A. Ishibashi and T. Shiromizu
considered; it takes the form for, e.g., a surface orthogonal null geodesic congruence
in D-dimensions,
d
dλθ = −
with ka= (d/dλ)a, θ, ˆ σabbeing the tangent of a null geodesic with affine parameter
λ of the congruence, its expansion and the shear. Therefore, once well-defined no-
tions of asymptotic flatness at null infinity and strong asymptotic predictability are
formulated in higher dimensions, one can apply general results in 4-dimensions–more
specifically, Propositions and Theorems in Section 12.2 of Wald8)–to higher dimen-
sions. Note that the predictability is needed, in particular to show the area theorem
without demanding that null geodesic generators of the event horizon be complete.
Note also that in order for a weakly asymptotically simple spacetime in D-dimensions
to be consistent with the asymptotic simplicity under the additional condition on
maximally extended null geodesics mentioned above, each component of I has to be
topologically R × SD−2. This, combined with the topological censorship,9)ensures
that the domain of outer communication is simply connected.
For D = even-spacetime dimensions, there exists a stable notion of conformal
null infinity and weak asymptotic simplicity [see Ref. 10) for definition], and therefore
there is no obstruction to apply general results [i.e., those in Section 12.2 of Ref. 8)]
to higher-even-dimensional spacetimes.
However, when spacetime dimension is odd, one needs to be more careful; The
conformal method for defining null infinity would not in general work since the
unphysical metric fails to be smooth at conformal null infinity I when radiation
is present around I.11)This is essentially due to the fact that the leading fall-off
behavior of gravitational radiation near null infinity is in proportion to a half-integer
power of the conformal factor, Ω, when D is odd. Therefore, for the case of odd
spacetime dimensions, we need to formulate a sensible definition of null infinity that
can be used to define asymptotic flatness and some equivalent notion of the strong
asymptotic predictability [see Ref. 12) for such an attempt to define asymptotic
flatness in 5-dimensions without using the conformal method]∗).
In the following when we discuss a black hole in a dynamical, non-stationary
spacetime, we simply assume that a sensible definition of infinity and asymptotic
flatness, equivalent to I and weak asymptotic simplicity above, are formulated, and
we just use the same symbol I to denote thus defined infinity, even if the spacetime
dimension is odd and gravitational radiation is present near infinity. This should be
kept in mind when, e.g., a topology changing process of cross-sections of the event
horizon is considered, since such a phenomenon can only occur in non-stationary,
dynamical spacetime where gravitational radiation is likely to generate.
With the above caveat concerning definitions of null infinity and the predictabil-
1
D − 2θ2− ˆ σabˆ σab− Rabkakb, (2.1)
∗)One may also want to consider non-asymptotically flat spacetimes, such as asymptotically
Kaluza-Klein spacetimes. For that case, we would not be able to use the standard conformal
approach to defining null infinity since the compactified dimensions shrink to a single point by the
conformal transformation. We again need to formulate a suitable definition of infinity, presumably
by dealing with the physical spacetime metric and its asymptotic expansion. Note however the
stationary case discussed below.
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‘Topology and Uniqueness in Higher Dimensional Black Holes’5
ity, the standard proof of the theorem [Prop. 9.2.8 in Ref. 7), Prop. 12.2.2 in Ref. 8)]
that under the null convergence condition, an apparent horizon (see below) is con-
tained in the black hole region is straightforwardly generalized to arbitrary dimen-
sions. In particular, the standard proof of the Hawking’s black hole area theorem7)
[Theorem 12.2.6 in Ref. 8)] is generalized to arbitrary higher dimensions.
2.2. Apparent horizon
For some purposes, instead of dealing with the event horizon, one is more inter-
ested in an apparent horizon which, in a sense, defines a black hole region in a local
manner and plays a role in particular in numerical studies [See Chapter 9]. This
notion can be straightforwardly generalized to higher dimensional case as discussed
below.
Let us consider a partial Cauchy surface, S, in D-dimensional spacetime M,
which is assumed to be an (D − 1)-dimensional connected hypersurface smoothly
embedded in M. A closed connected (D − 2)-surface, T, smoothly embedded in
S is called a trapped surface if the expansion of the congruence of outgoing light
rays orthogonal to T is non-positive at each point of T.
trapped surface is well-defined only when the notion of “outgoing” has a definite
meaning. The asymptotic flatness and the orientability of S and T do not give the
unique definition of ‘out direction’ on T. In many cases, it is assumed that S is
also simply connected. In this case, the out direction of T can be defined in terms
of the Z2-intersection numbers of curves from T to the spatial infinity. However,
since the simple connectedness of S might be too restrictive, it is worth giving
another example of a condition that determines the out direction of T without an
ambiguity. Possible such conditions are that S is orientable and that T separates S
into two disconnected parts. In other words, it is required that S\T = Sin∪ Sout
and Sin∩ Sout= ∅ hold, where Soutis determined by the property that it contains
a neighborhood of the spatial infinity. This clearly ensures that T is orientable and
defines the out direction of T in an obvious way. This condition also applies even
when S is not simply connected.
Under the above conditions, the inner region Sinwith respect to a trapped surface
T is called the inside region Sin(T) of T. The inside region of T will be a closed
subregion of S, whose topological boundary in S consists of T. Then, the trapped
region of S is defined to be the topological sum of Sin(T) over all possible T. The
trapped region might not be a closed region or a smooth region of S. For simplicity,
we however only consider the case where the trapped region is a smooth closed region
of S. Then, the apparent horizon on S is defined to be the topological boundary
of the trapped region. It turns out that the apparent horizon on S is a marginally
outer trapped surface, or in other words, that the expansion of the congruence of the
outgoing light rays orthogonal to the apparent horizon is zero everywhere on it.7)
Now let us consider topological aspects of apparent horizons. Hawking13)has
shown under the dominant energy condition that the apparent horizon in 4-dimensional
spacetime must be diffeomorphic to a 2-sphere or possibly to a 2-torus. Hawking’s
proof of the horizon topology theorem takes two steps: First (i) it is derived under
This definition of the