Brane Localized Black HolesClassical BH evaporation conjecture

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

Brane

??

z

AdSBulk

L

dxdxdz

zds 2

2

22

z

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

l

l

l

5

2

3

6AdS curvature radius :

4pGs

-L 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. Crs Crs ∝ z 4

behavior of zero mode

Moreover, this solution is unstable. Gregory Laflamme instability

( Chamblin, Hawking, Reall (’00) )

“length width” ≳

brane tension

Z[q]=∫d[f] exp(-SCFT[f,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

-- 2

5525

122

1

RgxdSEH

-- KqxdSGH4

25

1

-- qxdS 425

13

RqxdS 4425

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 model2

4

2

number 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))

Time scale of BH evaporation

32

32

1species

ofNumber

MGMGMM

NN

year120mm1 23

SolarMM

10±5M◎BH+K-type star X-ray binary A0620-00　　　　　　　　　　　ℓ < 0.132mm (10M◎BH is assumed)

(Johansen, Psaltis, McClintock arXiv:0803.1835)(Johansen arXiv:0812.0809 )

Rzb/z ~ l

Black stringregion

BH cap

most-probable shape of a large BH

Droplet escaping to the bulk

Rl

dtdA 3

3

211Rl

dtdA

AdtdM

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

bulk

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)

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

p 4A

0 1 2 3 4 5 6Log @L kD

1

1.2

1.4

1.6

1.8

2

2.2

k!!!!!

!!!!!!

A 4p

100*10005D 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.

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 modelM

deformation degree uni

form

bla

ck s

tring

x localized BH

non-uniform

black string x floating BH

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

(Kudoh & Wiseman (2005))

s = 0 & L = 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.

L ≠ 0 & s 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

tring x

non-uniform

black string x

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)

222224

/cosh AdSdsydyds

Brane placed at a fixed y.

single brane

RS limit

y 0

warp factor

--L--

- xdgRg

xdS 44

5

5 22

s

Background configuration:

tension-less limit

　 s < sRSEffectively 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 r

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

　 g00 → 0　 r → ∞

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

L2rc

M/LT(lL)1/2

S L-3/2l-1/2

10-2 102 106

2

1 LrTT 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 r

BH

radiation fluid with

empty zone with thicknessD r~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+radiationradiation star

Temperature for the Killing vector 　∂ t normalized at infinity,

diverge even in the limit rh -> 0.L

rTT circlocr

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 work in progress 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

113Ll

-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

AreaS44

2

Abo

tt-D

esse

r mas

s

100/1/ lL

3.43 lLScrit

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

6.6/ 3 lLS

expected from the 4dim calculation,

M/L

Massminimum

Massmaxmum

Asymptoticallystatic

M/L

T(lL

)1/2

SL3/

2 l1/2

SL3/2l1/2

radius/L

r L2

Comparison 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

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)

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.

Numerical error?or

Physical ?

Yoshino (’09)

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

222224

/cosh AdSdsydyds

ytanh6

5 -s

Brane placed at a fixed y.

y→-\

Zero-mode graviton is absent since it is not normalizable(Karch & Randall(2001), Porrati(2002))

UV

Warp factor increases for y>0

single brane

y

s → 6/ℓ (RS limit) y→ 0 s → 0

0

warp factor

--L--

- xdgRg

xdS 44

5

5 22

s

Background configuration:

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

There are stable and unstable floating positions.

Ueff=log(g00)

y

1) When ds s - 6/ℓ is very small, stable floating BH is very far from the brane. 2) When s goes to zero, no floating BH exists.

brane

Large BHs necessarily touch the brane.

y

Since this distance is finite, very large BHs cannot float.

size

distance from the UV brane

large localized BH

stable floating BH floating

BH small localized BH

Phase diagram for detuned tension model

critical size

From the continuity of sequence of solutions, large localized BHs are expected to exist above the critical size.

This region is not clearly understood.

Showing presence will be easier than showing absence.

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

Why does static BHs not exist in asymptotically flat spacetime?

size

distance from the UV brane

stable floating BH

floatingBH

(ℓ L)1/2

In AdS, temperature drops at infinity by 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.

2100 /1/1/1 LrrgT - -

Quantum state consistent with static BHs will exist if the BH mass is as large as mpl(ℓL)1/2.

In the RS-limit, stable floating BH disappears

size

distance from the UV brane

5d-AdS-Schwarzschild with brane on the equatorial plane

tensionless limit (s →0)

smaller ds

(Hawking & Page ’83)

4D AdS curvature scale

2

At the transition point, the temperature is finite at Tcrit ,Ll1

BHrrBH rr

gerg

gT 11 0000

although the BH size goes to zero.

rrgg00log -

rlog

Smaller black hole

-4 -2 2

-2

-1

Static spherical symmetric case (Garriga & T.T. (’99))

• Not exactly Schwarzschild ⇒ ℓ << 1mm

Metric perturbations induced on the brane

ijij rrGMh d

2

2

312

2

2

00 3212rr

GMh

• For static and spherically sym. Configurations, second or higher order perturbation is well behaved. 　　

Correction to 4D GR=O (ℓ 2/R2star)

» Giannakis & Ren (’00) outside» Kudoh, T.T. (’01) interior» Wiseman (’01) numerical

No Schwarzshild-like BH solution????

Gravity on the brane looks like 4D GR approximately, BUT

Transition between floating and localized sequencesConfiguration at the transition in un-warped two-brane model:

Configuration at the transition in de-tuned tension model:

“Event horizon” = “0-level contour of the lapse function, a >0” ara TD 3

65

max Maximum of a is possible only when 3r T is negative, i.e. only on the brane.

max

max

3r T →r 3P for perfect fluid in 4D

3r T =-s d y on the brane3r T =12/ℓ 2 in the bulk

Possible contours of a :

X saddle point

This type of transition is not allowed for the model with tensionless branes.

knownpredicted

Integrate out CFT ( Duff & Liu (’00) )

At the linear level, correspondence is perfect

Graviton propagator

CFT contribution to the graviton self-energy

aaa

- 14

21 DxdS

CFT

DTD DTDDT -

--

-- pTpTpGpTpT

pGph pp

3116

2116~

2

- const.ln

4 2

22

pp

RS

-- pTpT

pph

212~

2

25

42

22

1

0 24

ln2

pOp

ppKpKpKK

-

-D

42

2

pOpp KK D

-D- pTpTpKK

312

5

　　　　 effective Einstein eqs. (Shiromizu-Maeda-Sasaki(99))

- GGGGTGG 4444

2

44

31

48

p

Also in cosmology, correspondence is perfect

CFT

RS ppp EGTGG - 254

4 88

( Shiromizu & Ida (’01) )

trace anomary by CFT

- 2

24

44

31

488 TTTGTGG

pp

CFTTTGG p 44 8

(A)

(B)

424

4 /8 LOTGG p (A) and (B) give the same modified Friedmann equation

p 2T rs

sr CnnE 5

srrp2

82

42 GH

2221

22 2121 drdrr

dtr

ds

-

--

-

Brack Hole solution in 3+1 braneworld

Metric induced on the brane looks like Schwarzschild solution,But

This static solution is not a counter example of the conjecture. Casimir energy of CFT fields on with is given by

At the lowest order there is no black hole. Hence, no Hawking radiation is consistent.

( Emparan, Horowitz, Myers (’00) )

p

p 2

234

3/1 <<D

-

21

1

8 33rG

T CFT

p

The above metric is a solution

with this effective energy momentum tensor.

3G 22222 drdrdtds - 3/123/4 p D

Emparan et al (’02)

our brane boundary brane

What is the meaning of the presence of stable black string configuration?

If the radius at the bottle neck is > l, there is no Gregory-Laflamme instability.

L- 54

35 2

2RgxddyMS

444235 RgxdyadyMS

y

y -

-

Action and hence total energy are dominated by the boundary brane side, and diverge in the limit y+ →∞.

y- y+

This configuration will not be directly related to our discussion?

42

Time scale of BH evaporation

32

32

1species

ofNumber

MGMGMM

NN

year100mm1 23

SolarMM

LISA- 1.4M NS+3M BH: ℓ < 0.1mm?

BH

Neutron star

Roughly speaking, if DM/M ～ 1/N, where N is the number of observed cycles, DM will be detectable.

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

(Johansen, Psaltis, McClintock arXiv:0803.1835)

Possible constraint from gravitational waves

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

(Johansen arXiv: 0812.0809 )

Numerical brane BH (2)• The same strategy to construct static and spherical

symmetric BH configuration localized on the brane.

Yoshino (’08)

hout rrlog

Error measured by the variance

in surface gravity

location of the outer boundary

hr

hr

hr

non-systematic growth of error

Absence of even small black holes?

Numerical brane BH (2) ~conti.

hout rrlog

Error measured by the variance

in surface gravity

location of the outer boundary

Non-systematic growth of error

occurs for further outer boundary for

better resolution.

For further outer boundary, simply we need better

resolution.

hr