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Einstein in MordorEinstein in Mordor
Black Rings in HiBlack Rings in Hi--D General RelativityD General Relativity
Roberto EmparanRoberto EmparanICREA & U. BarcelonaICREA & U. Barcelona
Andrew Chamblin Memorial Conference, Cambridge 14 Oct 2006
Black hole uniqueness• Uniqueness is one of the most remarkable properties of
classical 4D black holes
ALL black holes in the Universe are exactly described by the Kerr solution
“The most shattering experience in my entire scientific life”(S. Chandrasekhar)
M, J
(a) (b) (c)
Into the higher-dimensional world
Motivations• String / M theory• Large Extra Dimensions• Understand gravity and black holes better:
= # of dimensions as a tunable parameter
What properties are dep/indep of ?
• Natural theoretical curiosity / Mathematical Physics
(c)
(a)
(b)
Black holes in D>4 Tangherlini 1963Myers+Perry 1986
• Schwarzschild is easy
• Kerr is harder. In there are several independent rotation planes:
• Myers+Perry found rotating black hole solutions with angular momentum in an arbitrary number of planes
…
Myers-Perry black holes • They all have spherical topology
• Consider a single spin:
J
JD = 5
D r 6
upper (extremal) limiton is singular
no upper limit on for fixed
(expect ultra-spinning instability dynamical upper limit on )
JD = 4 upper (extremal) limiton is regular
Is this all?• Is uniqueness a generic property of black holes r4?
• With Kaluza-Klein boundary conditions, it is readily clear that not:
• But, if asymptotically flat?
localized black holeblack string !"#$% & x
Metric '()*+*),
,- - .-
)- /-
)1
)
2 1
)1'
!
1)
3
,1
)1'
1
, 1
,,
)
4
Parameters λ, ν, 0equilibrium λ(ν)
RE+Reall 2001from Wick-rotated form inChamblin+REDowker et al
Phases and Shapes Horizon area vs spin, for fixed mass
thin black ring
fat black ring
MP black hole
(naked singularity)
(slowest ring)
6
7
8&
3 different black holes with the same value of
9
“Holey trinity”
Other properties
9
:thin
thin
fat
fat
MP bh
MP bh
fat rings ~ ("drilled-through") MP black holesthin rings ~ (circular) black strings
Ω
9
"Compact" vs Ultra-Spinning Black HolesA striking similarity:
Ω:$ &
;
9
Ω:$ &
r<
9Ultra-spinning, black brane-like (GL-unstable)
Moderately-spinning, compact (stable)
Probably relevant for microscopic description
Are they dynamically stable?• Tensor perturbation theory in = is largely to be
developed yet– No Newman-Penrose formalism known– Many more degrees of freedom– Black rings don’t possess Stachel-Killing tensors (no
separation of variables) -- MP bh’s do.
• Proceed heuristically– Hope to guide future analytical / numerical studies
Elvang+RE+Virmani 2006
Probing radial equilibrium off-shellF$# *B,D +&$# G+DB++A H)I B9$G"$I
> ? @
Conical defect membrane
STABLE
UNSTABLE
!BB& *B,#), 3B##$B> *)CBB3 ,D #*+*5+* E
E A @
> A @
Black Ring mechanics
Very thin black rings satisfy
> J
) Membrane tension(centripetal) Centrifugal force
J
String-like tension
This is the Newtonian balance of forces!
K J > &
&Radial potential
B9$G"$I +*K & @
thin ringfat ring
slowest ring
UNSTABLE STABLE
(fixed spin and mass)
UNSTABLE
STABLEthin
fat
7
7
K
Fits well with Arcioni+Lozano-Tellechea
Gregory-Laflamme instability/, N O PP
Expect instability for thin enough rings 9tLM/
RE+ReallHovdebo+MyersElvang+RE+Virmani
Breakdown of the ring• GL instability is much faster
than gravitational wave emission
• GL evolution is still uncertain, but fragmentationis plausible dominated by fastest unstable mode
, N
LGD,
N
N.B.: fragmentation will increase but not maximize the final entropy
Stability: putting results together
unstable
stable
evidence
evidence
possible/conjectured
possible/conjectured
Gregory-Laflamme
radial ?
Ishibashi+Kodama
8&
?
7
On to D>5• Do black rings x Q exist?
– Very likely YES:Balance for thin rings is achieved between centrifugal rotation and string-like tension
confined to ring plane indep of total D(grav force is D-dependent but falls off quickly)
Would violate uniqueness, since for D>6 RMP bhs
MP bh
black ring
$
• Rblack branes w/ horizons RHx QHQS
• Do black : Hx QHQS, Hx x #, …exist?– Much less clear…– Few restrictions on topology Helfgott+Oz+Yanai,
Galloway+Schoen
– String theory microscopics may be of help
Concluding remarks• Db3: no gravity dynamics, no (AF) black holes• D=4: gravity has dynamical degrees of freedom & bh's,
but dynamics is so constrained that bh uniqueness holds
• D=5: richer dynamics, uniqueness fails,but axial symmetry U(1)2 allows detailed study (complete integrability)
• Dr6: new qualitative behavior (black ?, bumpy black holes?), largely terra incognita
Black rings w/ charges and dipoles
• Charge increases stability
• Supersymmetric black rings expected to be linearly stable
• Near-susy are expected stable too
• Dipole rings (w/out conserved charges):
– fat rings radially unstable / thin rings stable
– GL instability expected to switch-off close to extremality (even if not close to susy)
larger stability window than for neutral rings
Black rings w/ two spinsψ
φ
• Yet to be found (ψ ;Tφ Tis known Figueras, Mishima+Iguchi )
• Non-zero φ may change behavior of some instabilities, but likely to introduce superradiant ones Dias