Phases of Phases of Higher-Dimensional Higher-Dimensional

Black HolesBlack Holes

Roberto EmparanRoberto Emparan

ICREA & U. BarcelonaICREA & U. Barcelona

• Earlier work w/ R.Myers, H.Reall, H.Elvang, P.Figueras

• To appear, w/ T.Harmark, N.Obers, V.Niarchos, M.J.Rodríguez

Strings 07Strings 07

Phases of black holes• Find all stationary solutions

– of Einstein's equations R= 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…

• Just the Kerr black hole: Uniqueness thm

No multi-bhs (eg multi-Kerr can't be stationary)

J

(fix scale: M=1)

1

extremal Kerr

Area

Phases of 4D black holes

Myers-Perry black holes in D dimensions

• Spherical topology

• Consider a single spin:

JD = 5

no upper limit on Jfor fixed M

SD ¡ 2

J

D 6

JD 6

J

D = 5

extremal, singular

5D: one-black hole phases

1

thin black ring

q2732

3 different black holes with the same value of M,J

A

J 2

(fix scale: M=1)

fat black ring

Multi-black holes

• Black Saturn:

• Exact solutions available• Double continuous non-uniqueness:

– eg, fix total M, J, vary Mring, Jring

Elvang+Figueras

5D phases in thermal equilibrium

J

Tbh=Tbr , bh=br

is there anything else?

Towards a complete classification of 5D black holes

• Topology: S 3, S

1x S 2

(+ finite connected sums) Galloway+Schoen

• Rigidity: stationarity ) one axial U(1), but not (yet?) necessarily two

• U(1)t xU(1)xU(1) ) complete integrability

– All bh solutions may have been found already:

MP, black rings, multi-bhs (saturns & multi-rings)

(what about bubbly black holes?)

(including also with two spins Pomeransky+Sen'kov)

Hollands et al

D6: Black Holes galore Why is D6 different?1. Centrifugal repulsion wins over gravitational

attraction at large distance

2. Black p-branes w/ p2

may give much richer blackfold topologies

eg: 2xS 2 ) S

1xS 1xS

2, S 2xS

2, (2)gx S 2

) ultra-spinning black holes¡

GMrD ¡ 3

+J 2

M 2r2

So, let's go find them?

• Very limited success in extending 4D & 5D approaches to D6– 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

Lumpy black holes in D6

…more axisymmetric dimples

J

RE+Myers

Dual pinched plasma balls recently found by Lahiri+Minwalla!

Gregory-Laflamme-like axisymmetric zero-mode

Ultra-spinning membrane-like

Thin black rings in D6

• Heuristic: seems plausible

• Thin black rings 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?

SD ¡ 3

Matched asymptotic expansion

2- perturbations of a boosted black string

1- linearized solution around flat space

r0

3- match in overlap zone

r0 ¿ r ¿ R

Requivalent delta-source

need bdry conditions to fix integration constants

very thin ring

HarmarkGorbonos+Kol

r0

r¿ 1

rR

¿ 1

1/R corrections are singular unless Tzz=0

(M,J,R) ) (M,J)

(r ¹ T ¹ º = 0)

) R =D ¡ 2

pD ¡ 3

JM

curving a black string

D6 phase diagram

(M,J) from linearized analysis of equilibrium

J

(fix M)

Black rings dominate the entropy at large J

J

D6 phase diagram: a conjecture

…infinite sequence of multi-lumps + multi-rings

(i)

(ii)

(iii)

(iv)

6D13

J

D6 phase diagram: a conjecture

…infinite sequence of multi-lumps + multi-rings

D13

Conclusion: More is differentVacuum gravity R= 0 in

• D=3 has no black holes– GM is dimensionless can't construct a length scale

(, or h, provide length scale)

• D=4 has one black hole– but no 3D bh no 4D black strings no 4D black rings

• D=5 has three black holes (two topologies); – black strings black rings, infinitely many multi-bhs…

• D6 seem to have infinitely many black holes – many topologies & lumpy horizons – black branes blackfolds, infinitely many multi-bhs…

and we've just begun…