Black Hole as a Window to Higher-Dimensional Gravity

Cosmophysics Group, IPNS, KEK

Hideo Kodama

Black Hole and Singularity Workshop at TIFR, 3 – 10 March 2006

Contents

4D black holes Rigidity, uniqueness, stability Cosmic censorship, singularities

Black holes in higher dimensions Static black holes, generalised Weyl formulation Black ring, non-uniqueness SUGRA version, soliton method, black saturn

Instabilities Gregory-Laflamme instability Instability of higher-dimensional black holes Brane world black hole

Discussions

Black Holes in Four Dimensions

What Is A Black Hole?

Definition(?) Mathematical: the outside of the causal past of a

global hyperbolic domain of outer-communication. Practical: a spacetime region whose boundary is a

stationary null surface (=a Killing horizon).

Why does it exist?

Examples Schwarzschild bh (1916) Reissner-Nordstrom bh (1916) Kerr bh (1963) Kerr-Newman bh (1965)

Rigidity Theorems Rigidity

Some symmetry requirement higher symmetries

Static black holes

[Israel 1967; Bunting, Masood-ul-alam 1987]

Cf. Birkhoff’s theorem: spherically symmetric staticCf. The corresponding statement about a normal star has not been

proved in general relativity yet.

Rotating black holes

[Hawking 1973; Chrusciel 1996]

A regular static non-degenerate black hole in (electro-)vacuum is spherically symmetric.

A regular stationary rotating black hole in (electro-)vacuum is axisymmetric.

Topology of Black Holes

Positive Energy Theorem

The horizon of a non-degenerate static black hole is connected.

Topological Censorship Theorem

Each connected component of the horizon of a 4D black hole is a sphere.

An asymptotically flat regular (black hole) spacetime has a non-negative mass if the dominant energy condition is satisfied. In particular, it is flat if the ADM mass on an initial surface is zero. [Schoen, Yau 1979]

The domain of outer communication (the region outside a black hole) is simply connected if the strong energy condition is satisfied and the spacetime is asymptotically flat. [Friedman, Schleich, Witt 1993]

Uniqueness Theorems

Static Black Holes

[Israel 1967, Bunting-Masood-ul-Alam 1987]

Rotating Black Holes

[Hawking, Carter 1972; Mazur 1982, Chrusciel 1996]

An asymptotically flat, regular, non-degenerate and static black hole in electrovacu spacetime is spherically symmetric and uniquely determined by mass and charge (Reissner-Nordstrom solution)..

An asymptotically flat, regular analytic, stationary and rotating black hole in electrovacu spacetime is axisymmetric and uniquely determined by mass, charge and angular momentum (Kerr-Newman solution), if the horizon is connected.

Physical Implications

Stability Schwarzschild/Reissner-Nordstrom black holes

[Vishveshwara 1970; Chandrasekar 1983] Kerr black hole [Whiting 1989]

Weak Cosmic Censorship Hypothesis

Cf. Singularity Theorem [Penrose, Hawking 1965-70]

Predictability in astrophysics

Black holes in accretion disks and at galactic centres will be well described by the Kerr(-Newman) solution.

Singularities formed by gravitational collapse will be hidden inside horizon. [ Penrose 1969]

Horizon and singularity of the TS2

Kodama & Hikida, Class.Quant.Grav.20:5121-5140,2003

Classification of Regular AF BHs in Four Dimensions

Static Rotating

Non-deg. Degen. Connected horizon

Multiple-horizons

Vacuum ○ S None ○ K ?Weinstein fam.

EM ○ RN △ MP ○ KN ?Weinstein fam.

EM+Dilaton ○ GM ? ? ?

EM+harm. scalar ○ S ? ○ K ?

EM+Dirac ○ RN ? ? ?

YM ✕ 3 families ? ? ?

Skyrme ✕ 2 families ? ? ?

Black Holes in Higher Dimensions

What are Different in Higher Dimensions? Gravitational potential

No stable Kepler orbit (and no stable atom) if the spacetime dimension is higher than 4.

Topological properties Topological censorship theorem holds in higher dimensions

as well. However, there are a veriety of closed manifolds of

dimesion 3 or higher that are cobordant to a sphere by a simply connected manifold.

Static AF Black Holes are Unique and Stable

Vacuum: unique (Tangherlini-Schwarzschild) [S. Hwang(1998), Rogatko(2003)]

Tangherlini-Schwarzschild bh: stable

[Ishibashi & Kodama 2003]

Einstein-Maxwell: unique (HD RN or Majumdar-Papapetrou) [Gibbons, Ida & Shiromizu(2002), Rogatko(2003)]

Einstein-Maxwell-Dilaton system (non-degenerate):

unique (Gibbons-Maeda sol) [Gibbons, Ida & Shiromizu(2002)]

Einstein-Harmonic scalar system (non-degenerate): unique (Tangherlini-Schwarzschild) [Rogatko (2002)]

Generalised Weyl Formulation

For RD-2 symmetric spacetime of dimension D,

the Einstein equations reduce to a linear PDE system:

Utilising this formulation in four dimensions, we can construct the Israel-Kahn solutions that represent chains of black holes supported by struts or strings, as superpositions of Schwarzschild black holes.

strut

Static Black Ring Solution

In five dimensions, utilising the generalised Weyl formulation, we can construct a static asymptotically flat black hole solution whose horizon has

non-trivial topology S1£ S2 : [Emparan, Reall 2002]

Rotating Regular Black Ring Solution

Asymptotically flat regular solution with two parameters: R,

Non-trivial horizon topology: S1£ S2

Rotating in a special 2-plane (in the S1 direction).

where 0<<1. Non-unique: the parameter can not be

uniquely determined only by the asymptotic conserved ‘charges’ M and J.

The membrane singularity of a black ring can be removed by

rotation. [Emparen, Reall 2002]

Rotating Black Holes Are Not Unique

For the 5-dim vacuum system, there exist two families of stationary 'axisymmetric' regular solutions:

Myers-Perry solution (1986): 3 params, horizon Emparan-Reall solution (2002): 2 params, horizon

Infinite Non-uniqueness

Black Rings with Dipole ChargesFor the Einstein-Maxwell(-Dilaton) system, there exists a continuous family of regular black ring solutions parametrized by a dipole charge Q for fixed mass and angular momenta [Emparan (2004)]

The dipole charge Q appears in the thermodynamic formula:

Supersymmetric Black Rings Rigidity theorem

Rigidity theorem still holds in higher dimensions, but only guarantees the existence of one spatial U(1) symmetry. [Hollands, Ishibashi, Wald 2006]

Since the ER solution and the 5D MP solution have the spatial U(1)x U(1) symmetry, it was conjectured that there would be a less symmetric new solution. [Reall 2002]

Reduction to a linear system for BPS solutions General supersymmetric solutions to the minimal and extended 5-dim

SUGRA were completely classified. [Gauntlett et al 2003] A subfamily of these solutions can be described by a set of harmonic

functions. Superpositions of black rings and holes

Utilising this formulation; a supersymmetric black ring solution in 5D with J 0 and J0 was found.

[Elvang, Emparan, Mateos, Reall 2004] Solutions with only one spatial U(1) symmetry were constructed by

superpositions of black rings solutions.[Gauntlett, Gutowski 2004]

General Vacuum Black Ring

Belinsky-Sakharov method A systematic method to derive a new solution from a given

solution by adding solitons utilising the inverse scattering type formulation; this method can be applied to spacetimes with R D-2 symmetry.

In four dimensions, this method was not so useful to obtain a new regular black hole solution because of the uniqueness theorem.

In five dimensions, we can use this method to obtain new regular black hole/ring solutions. [Mishima, Iguchi, Tomizawa 2006]

Pomeranski-Senkov solution A rotating black ring solution with J 0 and J0 was

constructed by this method. [Pomeranski, Senkov 2006] The regularity of this solution has not been exactly shown yet.

Black Saturn

A superposition of a black hole and a black ring can be constructed by the Belinsky-Sakharov method. [ Elvang, Figueras 2007] A family of regular asymptotically flat vacuum solutions with 4 independent parameters

in 5 dimensions.

The horizon is a disjoint sum of S3 and S2£ S1 .

There exists a non-static subfamily with vanishing total angular momentum and one extra parameter in addition to mass. For these solutions, the central black hole and the black ring are counter rotating.

Instabilities

Black Brane

Direct-product-type spacetime

Vacuum Einstein equations

For D ≤ 4, possible solutions are locally

For D ≥ 5, there are infinitely many solutions if m ≥ 4:e.g.

Gregory-Laflamme Instability

Black branes are unstable against S-mode perturbations

with [Gregory & Laflamme 1993]

The effective potential V has a negative region for

Implication of Gregory-Laflamme Instability

Non-uniqueness of black holes in spacetimes SxM[Kudoh & Wiseman 2003, 2004]

Fate of InstabilityNaked Singularities

Due to the famous theorem by Hawking and Ellis, a black hole horizon cannot bifurcate without formation of naked singularities.Further, it was shown that even if naked singularities are allowed, a black string cannot be pinched off to localised black holes within a finite affine time. [Horowitz & Maeda 2001]Nevertheless, some people argue that such a pinching off can be realised in a finite time with respect to some observers.

Kaluza-Klein BubblesIn addition to the black string, non-uniform black string and caged black holes, there is a large family of solutions consisting of black holes and static Kaluza-Klein bubbles [Elvang & Horowitz 2003;Elvang, Harmark & Obers 2005]

Instability of Rotating BH and BR Rapidly rotating black holes may be unstable in higher

dimensions. The metric of the MP solution rotating in a 2-dim plane

approaches a black membrane solution near the ‘rotation axis’ in the high rotation limit for D>5. [Emparan, Myers 2003]

An asymptotically AdS black hole rotating in a special way is unstable when the angular momentum is sufficiently large. [Kunduri, Lucietti, Reall 2006]

Cf. An asymptotically AdS black hole rotating in a 2-dim plane is stable for the same type of perturbations when || is sufficiently large. [HK 2007]

Sufficiently thin black rings will be unstable. In the thin limit, the Emparan-Reall solution approaches a

boosted black string solution. [Emparan, Reall 2002]

Braneworld Model

A braneworld model provides another method of dimensional reduction Our universe is realised as a hypersurface called a brane in a bulk

spacetime. Low energy matter lives only In the brane, while gravity lives in the bulk. In the Randal-Sundrum model, the bulk is an anti-de Sitter spacetime with Z2

symmetry, and the brane is the fixed hypersurface of this symmetry.

Braneworld Black Hole

4-dim Braneworld ModelSO(2) symmetric static regular bh solution is obtained from a half of the C-metric. The conic singularity associated with a string is hidden behind the brane.[Empran,Gregory,Santos 2001]

5-dim Braneworld ModelSO(3) symmetric static regular bh solution yet to be found should have naked singularity or non-compact horizon back behind the brane, provided that a regular static AdS bh is unique. [Chamblin,Hawking,Reall 2000;Kodama 2002]

The existence of an tublar horizon extending to infinity is quite likely. This suggests the instability of the solution. [Kodama 2007]

Discussions

Implications

In higher dimensions, black holes are far from unique and often unstable.

⇒ Higher-dimensional classical gravity is quite rich and

fascinating. There may not exist ‘a final state’ for classical

gravitational collapse in higher dimensions (at least if supersymmetry is broken). This feature together with quantum physics may explain the

four-dimensionality of the low energy world. In the AdS/CFT perspective, this implies the non-existence

of thermal equilibrium states in CFTs or the severe break down of the AdS/CFT correspondence when SUSY is violated.

Open Problems

Black Hole Classification For each horizon topology, is there a single continuous family of black holes?How large is the maximum number of parameters characterising a black hole/ring family ?Is there a black ring solution for D>5 ?Is there an asymptotically AdS black ring?

Black Hole/Brane StabilityWhat is the fate of the Gregory-Laflamme instability?Are Myers-Perry solutions and black ring solutions stable?Does the horizon area really provide a criteria for stability? Develop a tractible formulation for perturbations of a rotating black hole/ring in higher dimensions