Phases of 5D Black Holes

1. Phases of 5D Black Holes Roberto Emparan ICREA & U. Barcelona w/ H. Elvang & P. Figueras hep-th/0702111

2. 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)?

3. Boundary conditions • In 5D there are three natural boundary conditions (with L=0): Asymptotically flat Kaluza-Klein vacuum Kaluza-Klein monopole x5 x5 x5-circles Hopf-fibered on orbital S2

4. KK vacuum: (to be discussed by others)

5. magnetic KK black hole • KK monopole: 4D-5D connection Itzhaki ~5D neutral black hole 4D KK black hole

6. 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

7. A r e a 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) (fix scale: M=1) extremal Kerr 1 J

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

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

10. r 3 2 2 ¼ = 3 2 ( ) A G M = m a x 3 3 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 J

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

12. A A A + = h r M M M 1 + = = h r J J J + = h r 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

13. Filling the phase diagram • Black Saturns cover a semi-infinite strip there is a 1-parameter family at each point! (double continuous non-uniqueness)

14. Multi-rings are also possible • Di-rings explicitly constructed • Systematic method available (but messy) • Each new ring  2 more continuous parameters Iguchi+Mishima

15. Infinite-dimensional phase space • at each point there is an infinite number of continuous families of multi-ring solutions!

16. 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

17. 3 k @ @ ­ ³ ´ + ³ ´ · · X X i i = ( ) Ã i i t ± ± ± M A ­ J M A ­ J + + = = i i i i i i G G 2 8 8 ¼ ¼ i i The first law of multi-black hole mechanics • Each connected component of the horizon Hi is generated by a different Killing vector  (Smarr)  First Law

18. ( ) M A J N i 1 = i i ; ; : : : ; • With N black objects, phases are determined (up to discrete degeneracies) by energy function • 2N-dimensional phase space

19. 6 6 6 T T T T ­ ­ ­ ­ À = = = h h i j i j r r ; ; 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

20. 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

21. 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

22. 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)

23. 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?

24. 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

25. 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

26. 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

27. 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!