Black Hole as a Window to HigherDimensional Gravity

1
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
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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

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Black Holes in Four Dimensions
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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)

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Rigidity Theorems

• Rigidity
• Some symmetry requirement ? higher symmetries
• Static black holes
• Israel 1967 Bunting, Masood-ul-alam 1987
• Cf. Birkhoffs theorem spherically symmetric ?
  static
• Cf. 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.
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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
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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.
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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

Singularities formed by gravitational collapse
will be hidden inside horizon. Penrose 1969
Black holes in accretion disks and at galactic
centres will be well described by the
Kerr(-Newman) solution.
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Horizon and singularity of the TS2
Kodama Hikida, Class.Quant.Grav.205121-5140,200
3
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Classification of Regular AF BHs in Four
Dimensions
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Black Holes in Higher Dimensions
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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.

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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)

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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
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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

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Rotating Regular Black Ring Solution
The membrane singularity of a black ring can be
removed by rotation. Emparen, Reall 2002

• 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 0lt?lt1.
• Non-unique the parameter ? can not be uniquely
  determined only by the asymptotic conserved
  charges M and J.

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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

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Infinite Non-uniqueness

• Black Rings with Dipole Charges
• For 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

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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 J??0 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

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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
  J??0 was constructed by this method. Pomeranski,
  Senkov 2006
• The regularity of this solution has not been
  exactly shown yet.

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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.

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Instabilities
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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.

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Gregory-Laflamme Instability

• Black branes are unstable against S-mode
  perturbations
• with Gregory
  Laflamme 1993

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• The effective potential V has a negative region
  for

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Implication of Gregory-Laflamme Instability

• Non-uniqueness of black holes in spacetimes SxM
• Kudoh Wiseman 2003, 2004

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Fate of Instability

• Naked 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 Bubbles
• In 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 2003Elvang, Harmark Obers 2005

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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 Dgt5. 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

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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.

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Braneworld Black Hole

• 4-dim Braneworld Model
• SO(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 Model
• SO(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 2000Kodama 2002
• The existence of an tublar horizon extending to
  infinity is quite likely. This suggests the
  instability of the solution. Kodama 2007

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Discussions
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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.

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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 Dgt5 ?
• Is there an asymptotically AdS black ring?
• Black Hole/Brane Stability
• What 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