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

The forging of the ring (in D=5)

spin it upbend into a circle

black string

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

"Ring coordinates"

5),#*

1 5),#*

1 1&

ψ

φ

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

Plausible instabilities of black rings

Radial

“Longitudinal”

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

Evolution: rolling down the hill

K

7

7

area increases during evolution

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

Probing radial equilibrium off-shell

Conical defect membrane

> ? @

> A @

!BB& *B,#), 3B##$B> *)CBB3 ,D #*+*5+* E

E A @

F$# *B,D +&$# G+DB++A H)I B9$G"$I

STABLE

UNSTABLE