Phases of HigherDimensional Black Holes

1
Phases of Higher-Dimensional Black Holes

• Roberto Emparan
• ICREA U. Barcelona

Strings 07

• Earlier work w/ R.Myers, H.Reall, H.Elvang,
  P.Figueras
• To appear, w/ T.Harmark, N.Obers, V.Niarchos,
  M.J.Rodríguez

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Phases of black holes

• Find all stationary solutions
• of Einstein's equations ? Rmn 0
• with specified boundary conditions ? Asymp Flat
• non-singular on and outside event horizons
• What does the phase diagram look like?
• Which solutions maximize the total horizon area
  (ie entropy)?
• NB Classical stability? Not today

3
Phases of 4D black holes

• Just the Kerr black hole Uniqueness thm
• No multi-bhs (eg multi-Kerr can't be stationary)

(fix scale M1)
extremal Kerr
1
J
4
Myers-Perry black holes in D dimensions

• Spherical topology
• Consider a single spin

J
D 5
no upper limit on J for fixed M
5
5D one-black hole phases
(fix scale M1)
thin black ring
fat black ring
3 different black holes with the same value of
M,J
6
Multi-black holes

• Black Saturn
• Exact solutions available
• Double continuous non-uniqueness
• eg, fix total M, J, vary Mring, Jring

ElvangFigueras
7
5D phases in thermal equilibrium
TbhTbr , WbhWbr
A
J
is there anything else?
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Towards a complete classification of 5D black
holes

• Topology S 3, S 1x S 2 ( finite connected
  sums) GallowaySchoen
• Rigidity stationarity ) one axial U(1), but not
  (yet?) necessarily two
• U(1)t xU(1)f xU(1)y ) complete integrability
• All bh solutions may have been found already
• MP, black rings, multi-bhs (saturns
  multi-rings)
• (what about bubbly black holes?)

Hollands et al
(including also with two spins
PomeranskySen'kov)
9
Dr6 Black Holes galore

• Why is Dr6 different?
• Centrifugal repulsion wins over gravitational
  attraction at large distance
• Black p-branes w/ pr2
• ? may give much richer blackfold topologies
• eg R2xS 2 ) S 1xS 1xS 2, S 2xS 2, (H2/G)gx S 2

) ultra-spinning black holes
So, let's go find them?
10

• Very limited success in extending 4D 5D
  approaches to Dr6
• Known explicit construction techniques don't
  apply
• Higher-D topologies are poorly understood
• ) Need new approaches
• more qualitative less rigorous methods
  (physics-guided)
• may guide later numerical attacks

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Lumpy black holes in Dr6
REMyers
Ultra-spinning ? membrane-like
Gregory-Laflamme-like axisymmetric zero-mode
J
more axisymmetric dimples
Dual pinched plasma balls recently found by
LahiriMinwalla!
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Thin black rings in Dr6

• Heuristic seems plausible
• Thin black rings gt circular boosted black strings
• Equilibrium can be analyzed in linearized
  gravity
• balance between tension and centrifugal
  repulsion gravitational self-attraction is
  subdominant
• Can we construct approximate thin ring solutions?

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Matched asymptotic expansion
Harmark GorbonosKol
1- linearized solution around flat space
R
very thin ring
equivalent delta-source
2- perturbations of a boosted black string
curving a black string
need bdry conditions to fix integration constants
3- match in overlap zone
1/R corrections are singular unless Tzz0
A(M,J,R) ) A(M,J)
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Dr6 phase diagram
(fix M)
A
A(M,J) from linearized analysis of
equilibrium
J
? Black rings dominate the entropy at large J
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Dr6 phase diagram a conjecture
6bDd13
A
(i)
(iii)
(ii)
(iv)
J
infinite sequence of multi-lumps multi-rings
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Dr6 phase diagram a conjecture
Dt13
A
J
infinite sequence of multi-lumps multi-rings
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Conclusion More is different

• Vacuum gravity Rmn 0 in
• D3 has no black holes
• GM is dimensionless ? can't construct a length
  scale
• (L, or h, provide length scale)
• D4 has one black hole
• but no 3D bh ? no 4D black strings ? no 4D black
  rings
• D5 has three black holes (two topologies)
• black strings ? black rings, infinitely many
  multi-bhs
• Dr6 seem to have infinitely many black holes
• many topologies lumpy horizons
• black branes ? blackfolds, infinitely many
  multi-bhs
• and we've just begun