Roberto Emparan ICREA & U. Barcelona Phases of 5D Black Holes Roberto Emparan ICREA & 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 L=0): 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 connection magnetic KK black hole Itzhaki ~5D neutral black hole 4D 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 hole D6 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 End of the story! J A r e a 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) A r e a (fix scale: M=1) extremal Kerr 1 J

5D: one-black hole phases Myers-Perry black hole (fix scale: M=1) A (slowest ring) thin black ring fat black ring (naked singularity) J 2 q 2 7 3 1 3 different black holes with the same value of M,J

Multi-black holes Phasing in black Saturn: Exact solutions available Co- & counter-rotating, rotational dragging… Elvang+Figueras

Top achievers for A and J Which black object can more efficiently (ie with minimal mass) carry area or spin? For fixed mass, spin reduces area A maximum A for given mass: static black hole given J, minimum mass: infinitely thin and long black ring A m a x = 3 2 r ¼ ( G M ) J

Maximizing the area Put spin with as small mass as possible then put mass into maximal area A=Amax

A simple model If the ring radius p 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 = h + r M 1 J

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 vector k ( i ) = @ t + ­ Ã M = 3 2 X i ³ · 8 ¼ G A + ­ J ´  (Smarr) ± M = X i ³ · 8 ¼ G A + ­ J ´  First Law

With N black objects, phases are determined (up to discrete degeneracies) by energy function 2N-dimensional phase space M ( A i ; J ) = 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 T r À h ; ­ 6 = T i 6 = j ; ­

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 , Wi=Wj This fixes 2(N-1) parameters  Continuous degeneracies are completely removed

Phases in thermal equilibrium  black Saturns form a curve in (J,A) plane Multi-rings in thermodynamical equilibrium are unlikely! (pile up rings on top of each other) A 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, A) (by M(J1, J2, A) 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

D>5: 4D-5D connection maps into D0-D6 dynamics (in progress) thin black rings argued to exist  black Saturns will also be possible expect a similar story + probably more!