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

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
x 5
x 5
x 5-circles Hopf-fibered on orbital S 2

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

5. KK monopole: 4D-5D connection magnetic KK black hole 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. 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

8. 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

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

10. 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 ( G M ) J

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

12. 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

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. 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 8 G A + J 
(Smarr) M = X i 8 G A + J 
First Law

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

19. 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 ;

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. 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!