phases of 5d black holes roberto emparan icrea & u. barcelona w/ h. elvang & p. figueras...
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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!