Phases of 5D Black HolesPhases of 5D Black Holes

Roberto EmparanRoberto Emparan

ICREA & U. BarcelonaICREA & U. Barcelona

w/ H. Elvang & P. Figueras

hep-th/0702111

Phases of black holes

• Find all stationary solutions that – are non-singular on and outside event horizons– satisfying Einstein's equations – with specified boundary conditions

• What does the phase diagram look like?

• Which solutions maximize the total horizon area (ie entropy)?

Boundary conditions

• In 5D there are three natural boundary conditions (with :

Asymptotically flat

Kaluza-Klein vacuum

Kaluza-Klein monopole

x 5

x 5

x 5-circles Hopf-fibered on orbital S

2

• KK vacuum:

(to be discussed by others)

• KK monopole:

4D-5D connectionItzhaki

magnetic KK black hole

~5D neutral black hole4D KK black hole

• KK dyonic black holes and D0-D6 systems: KK gauge potential=RR 1-form

D0 electric charge: self-dual rotation of black holeD6 magnetic charge: Nut charge (degree of Hopf fibration)

Most results from asymp. flat 5D can be mapped into Taub-NUT

This allows for a stringy microscopic description of neutral 5D black holes RE+Horowitz

Phases of 4D black holes• No KK monopole, no bh's with KK asymptotics • Asymp flat: just the Kerr black hole

• End of the story!Multi-bhs not rigorously ruled out, but physically unlikely (eg multi-Kerr can't be balanced)

J

(fix scale: M=1)

1

extremal Kerr

Area

5D: one-black hole phases

thin black ring

1

fat black ring

Myers-Perry black hole

(naked singularity)

(slowest ring)

q2732

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

A

J 2

(fix scale: M=1)

Multi-black holes

• Phasing in black Saturn:

• Exact solutions available• Co- & counter-rotating, rotational dragging…

Elvang+Figueras

Top achievers for and J• Which black object can more efficiently (ie with

minimal mass) carry area or spin?

For fixed mass, spin reduces area

J

maximum for given mass: static black hole given J, minimum mass:

infinitely thin and long black ring

Amax =323

r2¼3

(GM )3=2

Maximizing the area

Put spin with as small mass as possible

then put mass into maximal area

=max

A simple model• If the ring radius black hole radius, their interactions are

negligible

• Agrees very well with exact results for very thin and long rings

• Allows better analysis of corners of parameter space: confirms maximal area configuration

A = Ah + A r

M = 1= Mh + M r

J = J h + J r

Filling the phase diagram• Black Saturns cover a semi-infinite strip

there is a 1-parameter family at each point!(double continuous non-uniqueness)

Multi-rings are also possible

• Di-rings explicitly constructed• Systematic method available (but messy)

• Each new ring 2 more continuous parameters

Iguchi+Mishima

Infinite-dimensional phase space

at each point there is an infinite number of continuous families of multi-ring solutions!

and even more:• Include second independent spin:

– general Myers-Perry bh– doubly-spinning ring Pomeransky+Senkov

then cancel against each other to leave only one nonzero spin another continuous parameter

• Yet-to-be-found solutions?– black holes with only one axial symmetry? Reall

– bubbly black holes? RE+Reall, Elvang+Harmark+Obers

• All these would give even more families of solutions

The first law of multi-black hole mechanics• Each connected component of the horizon Hi is

generated by a different Killing vectork(i ) = @t + i @Ã

M =32

X

i

³ · i

8¼GA i + i J i

´(Smarr)

±M =X

i

³ · i

8¼G±A i + i ±J i

´First Law

• With N black objects, phases are determined (up to discrete degeneracies) by energy function

• 2N-dimensional phase space

M (A i ; J i ) i = 1;:: : ;N

Thermodynamical equilibrium

is not in thermo-equil

• Maximal entropy = thermal equilibrium ??!!

• Beware: bh thermodynamics makes sense only with Hawking radiation

• Radiation can't be in equilibrium if Ti 6= Tj ; i 6= j

Tr À Th ; r 6= h

• Radiation will couple different black objects and drive towards thermal equilibrium– otherwise, they act as separate thermodynamical

systems

• (further: dynamical instabilities)

• Black Saturns in thermodynamical equilibrium:

Ti=Tj , i=j

• This fixes 2(N-1) parameters

Continuous degeneracies are completely removed

Phases in thermal equilibrium

black Saturns form a curve in (J,) plane

• Multi-rings in thermodynamical equilibrium are unlikely!(pile up rings on top of each other)

J

• Phase space becomes two-dimensional again:

• Just a few families of solutions:

– Myers-Perry black holes

– black rings

– single-ring black Saturns

– (exotica?)

given by functions M(J, )

(by M(J1, J2, ) in general)

An instability of all rotating bh's?

• Black holes can (in principle) evolve to increase horizon area

• Sometimes this has signalled a classical instability: Gregory-Laflamme, ultra-spinning…

• are all rotating bhs unstable?

• Not clear: it is unlikely that classical evolution (possibly through singularity) drives to maximal area

• however, it may still be possible to have

while increasing the total area

Outlook

• Other infinite-dimensional phase spaces:– Caged black holes in KK circles

but single-bh phases dominate entropy (even away from thermal eq.)

• Many black Saturns are unstable– GL-instability of thin black rings

• Dynamically and thermodynamically stable phase with maximal entropy?– probably MP black hole + spinning radiation

• Dipole black Saturns: – MP black hole + dipole black ring

– Can be dynamically stable

• Supersymmetric black Saturns– constructed right after susy rings Gauntlett+Gutowski

– 9 susy multi-rings with higher entropy than BMPV

• Black Saturns at the LHC? – quicker, hotter spin-down

• 4D-5D connection

maps into D0-D6 dynamics (in progress)

• D>5:

– thin black rings argued to exist

black Saturns will also be possible

expect a similar story

+ probably more!