Brane Black Holes

Takahiro Tanaka (YITP, Kyoto university)

in collaboration with N. Tanahashi, K. Kashiyama, A. Flachi

Prog. Theor. Phys. 121 1133 (2009) (arXiv:0709.3674) TT

arXiv:0910.5376 KK, NT, AF, TTarXiv:0910.5303 NT, TT

• Extension is infinite, but 4-D GR seems to be recovered!

x

Bran

e

??

z

AdSBulk

dxdxdz

zds 2

2

22

z

Volume of the bulk is finite due to warped geometry although its extension is infinite.

5

2

3

6

AdS curvature radius :

4G

Negative cosmological constant

Brane tension

Infinite extra-dimension: Randall-Sundrum II model

No Schwarzshild-like BH solution???? BUT

z

dxdxgdzz

ds Sch 22

22

xg

Black string solution

Metric induced on the braneis exactly Schwarzschild solution.

However, this solution is singular. CC∝ z 4

behavior of zero mode

Moreover, this solution is unstable.

Gregory Laflamme instability

( Chamblin, Hawking, Reall (’00) )

“length width” ≳

Numerical brane BH• Static and spherical symmetric configuration

T, R and C are functions of z and r.

Kudoh, Nakamura & T.T. (‘03)Kudoh (’04) Yoshino(‘08)

Comparison of 4D areas with 4D and 5D Schwarzschild sols.

4A

0 1 2 3 4 5 6Log @L kD

1

1.2

1.4

1.6

1.8

2

2.2

k

!!!!!!

!!!!!

A4

p

100*1000

5D Sch.

4D Sch.

log

4D Sch.

5D Sch.

is surface gravity

It becomes more and more difficult to construct brane BH solutions numerically for larger BHs.

Small BH case (–1 < ℓ ) is beyond the range of validity of the AdS/CFT correspondence.

brane tension

Z[q]=∫d[] exp(-SCFT[,q])

=∫d[gbulk] exp(- SHE- SGH+S1+S2+S3)≡ exp(-WCFT[q])

z0→ 0 limit is well defined with the counter terms.

∫d[g] exp(- SRS) = ∫d[g] exp(- 2(SEH+ SGH) + 2S1- Smatter )

= exp(- 2S2 -Smatter- 2(WCFT+ S3))

AdS/CFT correspondence

Boundary metric

Counter terms

255

25

12

2

1

RgxdSEH

KqxdSGH4

25

1

qxdS 425

1

3

RqxdS 44

25

2 4

3S

Brane position z0 ⇔ cutoff scale parameter

4D Einstein-Hilbert action

( Hawking, Hertog, Reall (’00) )( Gubser (’01) )( Maldacena (’98) )

Evidences for AdS/CFT correspondence

Linear perturbation around flat background (Duff & Liu (’00))

Friedmann cosmology ( Shiromizu & Ida (’01) )

Localized Black hole solution in 3+1 dimensions

( Emparan, Horowitz, Myers (’00) )

Tensor perturbation around Friedmann ( Tanaka )

4D Einstein+CFT with the lowest order

quantum correction

Classical black hole evaporation conjecture

5D BH on brane4D BH with CFT

equivalent

equivalent

Classical 5D dynamics in RS II model

22 plMnumber of field of CFT

Hawking radiation in 4D Einstein+CFT picture equivalent

Classical evaporation

of 5D BH

AdS/CFT correspondence

(T.T. (’02), Emparan et al (’02))

8

9

Time scale of BH evaporation

32

32

1species

ofNumber

MGMGM

M

NN

year120

mm123

SolarM

M

X ray binary: orbital period derivative A0620-00: 10±5M BH+K-star　　　　　　　　　　　ℓ < 0.132mm (assuming 10M BH )

(Johannsen, Psaltis, McClintock (2009))

J1118+480: 6.8±0.4M BH+K~M-star　　 ℓ < 0.97mm (assuming 6.8M BH )

(Johannsen (2009))

In globular cluster RZ2109 located in an elliptical galaxy NGC4472 in the Virgo cluster

Compact object with rapid variability and broad emission line ~2000km/s ~10M BH

Assuming that the association with the globular cluster means old age of BH.

~10±3Gyr.

(Gnedin, Maccarone, Psaltis, Zepf (2009))

ℓ < 0.003mm

(Zepf, Stern, Maccarone, Kundu, Kamionkowski, Rhode, Salzer, Ciardullo & Gronwall (2008))

BH accreting at around Eddington limit From the luminosity

Rzb/z ~ l

Black stringregion

BH cap

most-probable shape of a large BH

Droplet escaping to the bulk

R

l

dt

dA 3

3

211

R

l

dt

dA

Adt

dM

M

Droplet formation Local proper time scale: l R on the brane due to redshift factor Area of a droplet: l3

Area of the black hole: A ～ lR2

R

Structure near the cap region will be almost independent of the size of the black hole. ~discrete self-similarity

Assume Gregory-Laflamme instability at the cap region

brane

bulkbulk

1) Classical BH evaporation conjecture is correct.

Let’s assume that the followings are all true,

Namely, there is no static large localized BH solution.

then, what kind of scenario is possible?

2) Static small localized BH solutions exist.

3) A sequence of solutions does not disappear suddenly.

In generalized framework, we seek for consistent phase diagram of sequences of static black objects.

RS-I (two branes)Karch-Randall (AdS-brane)

Un-warped two-brane model

M

deformation degree uni

form

bla

ck s

trin

g x

localized BH

non-uniform

black stringx

floating BH

We do not consider the sequences which produce BH localized on the IR(right) brane.

(Kudoh & Wiseman (2005))

= 0 & = 0

Warped two-brane model (RS-I)In the warped case the stable position of a floating black hole shifts toward the UV (+ve tention) brane.

22/222 xddtedyds y Acceleration acting on a test particle in AdS bulk is

Compensating force toward the UV brane is necessary.

Self-gravity due to the mirror images on the other side of the branes

UV IR

When RBH > ℓ, self-gravity (of O(1/RBH) at most), cannot be as large as 1/ℓ.

Large floating BHs become large localized BHs.

Pair annihilation of two sequences of localized BH,which is necessary to be consistent with AdS/CFT.

≠ 0 & is fine-tuned

1log ,

yy

ytt

g

gaUV (+ve tention) IR (-ve tention)

1/ℓ

mirrorimage

mirrorimage

Phase diagram for warped two-brane model

M

deformation degree

uni

form

bla

ck s

trin

g x

non-uniform

black stringx

floating BH

Deformation of non-uniform BS occurs mainly near the IR brane. (Gregory(2000))

localized small BH

localized BHas large as

brane separation

Model with detuned brane tensionKarch-Randall model JHEP0105.008(2001)

22222

4/cosh AdSdsydyds

Brane placed at a fixed y.

single brane

RS limit

y 0

warp factor

xdgRg

xdS 44

5

5 22

Background configuration:

tension-less limit

RS

Effectively four-dimensional negative cosmological constant

y→∞

Effective potential for a test particle (=no self-gravity).

There are stable and unstable floating positions.

Ueff=log(g00)

y

brane

necessarily touch the brane.

y

finite distance Very large BHs cannot float,

size

distance from the UV brane

large localized

BH

stable floating BH

floatingBH

small localized BH

Phase diagram for detuned tension model

critical configuration

Large localized BHs above the critical size are consistent with AdS/CFT?

Why doesn’t static BHs exist in asymptotically flat spacetime?

In AdS, temperature drops at infinity owing to the red-shift factor.

Hartle-Hawking (finite temperature) state has regular T on the BH horizon, but its fall-off at large distance is too slow to be compatible with asymptotic flatness.

2100 /1/1/1 LrrgT

Quantum state consistent with static BHs will exist if the BH mass is large enough:

mBH > mpl(ℓL)1/2. (Hawking & Page ’83)

4D AdS curvature scale

2

CFT star in 4D GR as counter part of floating BHFloating BH in 5D

The case for radiation fluid has been studied by Page & Phillips (1985)

4-dimensional static asymptotically AdS star made of thermal CFT

200

4 /13 gaTP loc

Sequence of static solutions does not disappear until the central density diverges.

g00 → 0 → ∞

In 5D picture, BH horizon will be going to touch the brane

L2c

M/LT(lL)1/2

S L-3/2l-1/2

10-2 102 106

2

1 L

rTT circ

locr

lim

(central density)

Sequence of sols with a BH in 4D CFT picture

Naively, energy density of radiation fluid diverges on the horizon:

4-dimensional asymptotically AdS space with radiation fluid+BH

200

4 /1 gaTloc

BH

radiation fluid radiation fluid with with

empty zone with thicknessr~rh

-1 -0.5 0.5 1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

log10M/L

log10T(Ll)1/2

00/ gTT BHloc

pure gravity without back reaction

(plot for l=L/40)BH+radiation

radiation star

Temperature for the Killing vector

　∂ t normalized at infinity,

diverge even in the limit rh 0.

L

rTT circ

locr

lim , does not

Stability changing critical points

Floating BHs in 5D AdS picture

However, it seems difficult to resolve two different curvature scales l and L simultaneously. We are interested in the case with l << L.

bran

e

Numerical construction of static BH solutions is necessary.

We study time-symmetric initial data just solving

the Hamiltonian constraint,

extrinsic curvature of t-const. surface K=0.

We use 5-dimensional Schwarzschild AdS space as a bulk solution. Hamiltonian constraint is automatically satisfied in the bulk.

Time-symmetric initial data for floating BHs N. Tanahashi & T.T.

Bulk brane

222

22 drrU

drdtrUds

Brane:=3 surface in 4-dimensional space. t=constant slice

Trace of extrinsic curvature of this 3 surface 22

113

Ll

Hamiltonian constraint on the brane

r0

5D Schwarzschild AdS bulk:

Then, we just need to determine the brane trajectory to satisfy the Hamiltonian constraint across the brane.

Critical value is close to

lG

AreaS

44

2

Abo

tt-D

esse

r m

ass

100/1/ lL

3.4/ 3 lLScrit

Critical value where mass minimum (diss)appears is approximately read as

6.6/ 3 lLS

expected from the 4dim calculation,

M/L

Massminimum

Massminimum

Massmaxmum

Massmaxmum

Asymptoticallystatic

Asymptoticallystatic

M/L

T(l

L)1/

2

SL-3

/2l-1

/2

SL-3/2l-1/2

radius/L

L

2Comparison of the four-dim effective energy density for the mass-minimum initial data with four-dim CFT star.

3.4/ 3 lLS

89.0/ 3 lLS

Summary• AdS/CFT correspondence suggests that there is no static large

(–1≫ℓ ) brane BH solution in RS-II brane world.

– This correspondence has been tested in various cases.

• Small localized BHs were constructed numerically. – The sequence of solutions does not seem to terminate suddenly, – but bigger BH solutions are hard to obtain.

• We presented a scenario for the phase diagram of black objects including Karch-Randall detuned tension model,

which is consistent with AdS/CFT correspondence.

• Partial support for this scenario was obtained by comparing the 4dim asymptotic AdS isothermal star and the 5dim time-symmetric initial data for floating black holes.

As a result, we predicted new sequences of black objects. 1) floating stable and unstable BHs 2) large BHs localized on AdS brane