Black Holes in Higher Dimensions (V)

Roberto EmparanICREA & U. Barcelona

3rd Asian Winter School, Beijing 7-17 Jan 2009

• Charges & Dipoles• Λ(<0)

• Extremal black holes• Some general results on hi-d

black holes

Charges & Dipoles

Charged black holes• Black holes in D dimensions can carry a conserved

electric charge for a 2-form field strength Fµν

(or, dually, magnetic charge w.r.t. D−2-form)

• Hi-D Reissner-Nordstrom in Einstein-Maxwell is easy:

Q =

∫

SD−2∗F(2)

Dipoles• Conserved gauge charge for asymp flat solns can only be electric

w.r.t. a 2-form field strength Fµν (or dual magnetic w.r.t. D-2-form)

• But higher p-forms can also be excited – although they have no net charge associated

• Simplest: black rings as dipoles of Hµνρ

Btψ

String:

H(3)=dB(2)

Ring:ψ

SD-3Now q is not conserved:can shrink ring to zero

zSD-3

q ∝∫

SD−3

∗H(3) Btz

Simplest set up: minimal 5D sugra

Einstein+Maxwell+Chern-Simons

• Ring couples electrically to F (charge Q ) and magnetically to its magnetic dual 3-form *F (dipole q)

• Exact solutions available:• Non-extremal charged&rotating bh (Susy BMPV) • Black rings:

• non-susy, w/ dipole, and w/ or w/out charge (but not most general)

• supersymmetric (w/ charge and dipole)• Most results extend to sugra w/ vectors

I =1

16πG

∫ (R ∗ 1− 2F ∧ ∗F − 8

3√3F ∧ F ∧A

)

Q =

∫

S3∗F(2) q =

∫

S2F(2)

Dr6 Sugra (ungauged)

• No known charged rotating bh solutions –mere technical problem: they must exist! (presumably numerically)

• No susy bhs known – static seem unlikely (but no simple argument)– stationary? 6D susy rings?

• (Small susy bhs do exist: e.g. from FP strings)

Stability of 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 radially stable

– GL instability expected to switch-off close to extremality (even if not close to susy)

larger stability window than for neutral rings

Λ(<0)

BHs with Cosmo Constant: Neutral • Schw-(A)dS Kottler 1918, Witten 1998

• Rotating (A)dS black hole w/ all spins Carter 1968, Hawking et al 1998, Gibbons et al 2004

• Def of mass & spins subtle – but clarified: satisfy 1st law (but not Smarr)

• Phase space:

• BPS bound: MLAdS r Σi|Ji| Chrusciel et al (Witten-type proof)

not saturated by these bhs – only in Dr6 w/ one spin

5D 6D

M

J1

J2

solutions exist inside the "deformed inverted pyramids"

• ΩiLAdS b1: Hartle-Hawking state exists: matter at infty

can corotate with horizon

• ΩiLAdS >1: Ergoregion instability (superradiant black

hole bomb) – but endpoint unknown. Dual CFT matter would rotate superluminally Hawking+Hunter+Taylor, Hawking+Reall

• Extremal bhs lie in unstable region – but instability may be very slow for small bhs

• Black rings in AdS: blackfold approach (thin rings)

• Multi-bh configurations presumably can (asymptotically) saturate the BPS bound in any D

D=5

AdS phase diagram w/ one spin

A

J

fixed MJ=MLAdS

Dr6

A

J

"compressed" versions of Λ=0

J=MLAdS

(Thin AdS rings and Saturns in any Dr5 can be studied w/ approximate techniques (cf. lecture 6))

• Static charged bhs – extremal is not BPS• Rotating BPS bhs in 5D sugra (minimal+vectors)

Gutowski+ReallKunduri et al

• Non-extremal rotating charged in U(1)3 – not the most general yet Pope et al

• Non-extremal and susy rotating bhs in 6Dgauged sugra Chow

BHs in Gauged Sugras

Extremal black holes

Extremal = zero surface gravity = zero temperature

• BPS ⇒Extremal but Extremal BPS

• Extremal entropy often simplifies, e.g.:extremal Kerr: S=2π|J| (quantized & indep of G !)

extremal 5D Myers-Perry: S=2π|J1J2|1/2

• Extremal Near-Horizon Geometry simplifies:

– Typically exhibits attractor mechanism (independence from asymptotic moduli) entropy can be extrapolated from strong to weak coupling

– Typically has conformal symmetry SL(2,R) (from AdS2 factor)

dual CFT entropy counting via Cardy formula (eg Kerr/CFT)

Near-horizon geometry of extremal black holes• Choose Gaussian null

(Eddington-Finkelstein) coordinates (v, r, x a)

• Assume extremal horizon:

• Near-horizon limit: r → εr , v → v/ε , ε → 0

⇒ r -dependence is fixed (& enhanced symmetry)

⇒ ODEs: May solve for H geometry gab(x)

v

rH

H+

ReallKunduri,Lucietti,Figueras,Rangamani

ds2 = r2F (x)dv2 + 2dvdr + 2rha(x)dvdxa + gab(x)dx

adxb

ds2 = r2F (x, r)dv2 + 2dvdr + 2rha(x, r)dvdxa + gab(x, r)dx

adxb

• Vacuum Λ=0: assume two U(1)– S 3 near-horizon MP, KK bhs (ergo & ergo-

free),

– S 1x S 2 extremal ring, Kerr string – Solve w/ single U(1)?

• Λ<0: reduces to 6-th order ODE

• Susy: complete classification of near-horizon geometries in 5D sugra w/ vectors– Near-horizon BMPV, AdS3xS 2, flat

Reall, Gutowski

Pros & Cons• Pros: simplified fully nonlinear eqs

• Cons: – near-horizon can't distinguish strings vs rings – misses many classes of bhs that don't have

regular extremal limit – can't identify asymptotic magnitudes– eqns still hard to solve

Some general results on hi-d black holes

• Hawking's 4D theorem relies on Gauss-Bonnet thm:

• D=5: Galloway+Schoen: +ve Yamabe R(D-2) > 0 S 3, S 1x S 2

• D=6: Helfgott et al: S 4, S 2x S 2, S 1x S 3, Σgx S 2

so far: S 4 exactly (MP, but possibly others too)

S 1x S 3, T 2x S 2 approximately*

• D>6: essentially unknown

so far: S D-2 exactly (MP, but possibly others too)

S 1x S D-3, Tp x S q (p b q+1) plus many others approximately* *cf lecture 6

Horizon topology

∫HR(2) > 0⇒ H = S2

Uniqueness & Classification

• SchwarschildD is unique among static AF black holes

(proof extends to charged bhs)

(Note that ∃ non-static solutions with zero

angular momentum, eg black saturns)

• STATIC classification solved

Gibbons+Ida+Shiromizu

Uniqueness & Classification

• STATIONARY bh's must admit one spacelike Killing that generates rotations

• But there may be as many as ⌊(D−1)/2⌋ such Killings

• Are there solutions with less than this symmetry?Where? How?

• Also: Tools to classify pinched bh's still to be developed

1. What is the simplest and most convenient set of parameters that fully specify a bh?

– In 5D: M, J, + "rod structure": more physical parametrization? Higher D??

2. How many bh's with given charges are relevant to a given physical situation?

– Conserved charges + additional conditions:• Horizon topology alone is not enough• Dynamic linear stability (not an issue in 4D classification)

may be (just may be) enough • But stability does not per se rule out a solution – must

compare timescales• Dipoles introduce more non-uniqueness and enhance

stability

Laws of black hole mechanics

• Generally valid indep of dimension• Dipoles introduce additional terms:

even if dipoles are not conserved chargescan't define globally the dipole potential φ

extra surface term

dM =κ

8πdAH +ΩdJ +ΦdQ+ φdq

RECopsey+Horowitz

Multi-black hole mechanics

• Each connected component of the horizon Hi is generated by a different Killing vector

k(i) = ∂t +Ωi∂ψ

M =3

2

∑

i

( κi

8πGAi + ΩiJi

)(Smarr)

δM =∑

i

( κi

8πGδAi +Ωi δJi

)First Law

Hawking radiation

• Technical analysis complicated, but physics should remain the same: bh's emit radiation at temperature T=κ/2π and "chemical potentials" Ωi, Φ…

• Multi-bhs will emit multiple components – thermal only if all Ti, Ωi etc are equal

• Euclidean thermodynamics: much like in 4D– real Euclidean sections may not exist (do not exist for black ring!)

– convenient to work with complex sections that have real actions