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Brane Localized Black Holes Classical BH evaporation conjecture . Prog . Theor . Phys. 121 1133 (2009) (arXiv:0709.3674) TT arXiv:0910.5376 KK, NT, AF, TT arXiv:0910.5303 NT, TT. Takahiro Tanaka (YITP, Kyoto university) in collaboration with

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brane localized black holes classical bh evaporation conjecture

BraneLocalized Black HolesClassical BH evaporation conjecture

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

arXiv:0910.5376 KK, NT, AF, TT

arXiv:0910.5303 NT, TT

Takahiro Tanaka (YITP, Kyoto university)

in collaboration with

N. Tanahashi, K. Kashiyama, A. Flachi

slide2

:

AdS curvature radius

l

6

Negative cosmological

constant

L

=

-

2

l

3

s

=

Brane tension

4pG

l

5

Infinite extra-dimension: Randall-Sundrum II model

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

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

L

??

z

Brane

AdS

Bulk

BUT

  • No Schwarzshild-like BH solution????
slide3

Black string solution

( Chamblin, Hawking, Reall (’00) )

Metric induced on the brane

is exactly Schwarzschild solution.

z

However, this solution is singular.

  • Cmnrs Cmnrs ∝ z 4

behavior of zero mode

Moreover, this solution is unstable.

  • Gregory Laflamme instability

“length ≳ width”

slide4

AdS/CFT correspondence

( Maldacena (’98) )

( Gubser (’01) )

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))

( Hawking, Hertog, Reall (’00) )

Boundary metric

Counter terms

z0⇔ cutoff scale parameter

Brane position

brane tension

4D Einstein-Hilbert action

slide5

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 )

classical black hole evaporation conjecture
Classical black hole evaporation conjecture

(T.T. (’02), Emparan et al (’02))

4D Einstein+CFT with the lowest order quantum correction

AdS/CFT

correspondence

Classical 5D dynamics in RS II model

equivalent

number of field of CFT

4D BH with CFT

5D BH on brane

equivalent

Classical evaporation of 5D BH

Hawking radiation in 4D Einstein+CFT picture

equivalent

Time scale of BH evaporation

10±5M◎BH+K-type star X-ray binaryA0620-00

ℓ < 0.132mm (10M◎BH is assumed)

(Johansen, Psaltis, McClintock

arXiv:0803.1835)

(Johansen arXiv:0812.0809)

slide7

most-probable shape

of a large BH

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

brane

R

Assume Gregory-Laflamme instability at the cap region

Black string

region

Droplet escaping to the bulk

Droplet formation

Local proper time scale: l

R on the brane due to redshift factor

Area of a droplet: l3

bulk

Rzb/z~ l

BH cap

Area of the black hole: A~lR2

numerical brane bh
Numerical brane BH

Kudoh, Nakamura & T.T. (‘03)

Kudoh (’04)

  • Static and spherical symmetric configuration

T, R andCare functions ofzandr.

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

5D Sch.

4D Sch.

k is surface gravity

  • It becomes more and more difficult to construct brane BH solutions numerically for larger BHs.
    • Small BH case (k –1 < ℓ ) is beyond the range of validity of the AdS/CFT correspondence.
slide9

Let’s assume that the followings are all true,

1) Classical BH evaporation conjecture is correct.

Namely, there is no static large localized BH solution.

2) Static small localized BH solutions exist.

3) A sequence of solutions does not disappear suddenly.

then, what kind of scenario is possible?

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

s = 0 & L = 0

Un-warped two-brane model

M

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

x

non-uniform

black string

localized BH

x

floating BH

uniform black string

deformation degree

(Kudoh & Wiseman (2005))

warped two brane model rs i

UV

IR

Warped two-brane model (RS-I)

L ≠ 0 & s is fine-tuned

In the warped case the stable position of a floating black hole shifts toward the UV (+vetention) brane.

Acceleration acting on a test particle in AdS bulk is

1/ℓ

UV (+ve tention)

IR (-ve tention)

Compensating forcetoward the UV brane is necessary.

Self-gravity due to the mirror images

on the other side of the branes

When RBH > ℓ,

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

mirror

image

mirror

image

Pair annihilation of two sequences of localized BH,

which is necessary to be consistent with AdS/CFT.

Large floating BHs become large localized BHs.

phase diagram for warped two brane model
Phase diagram for warped two-brane model

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

M

localized BH

as large as

brane separation

x

non-uniform

black string

x

uniform black string

floating BH

localized small BH

deformation degree

model with detuned brane tension
Model with detuned brane tension

Karch-Randall model JHEP0105.008(2001)

s< sRS

Effectively four-dimensional negative cosmological constant

Background configuration:

Braneplaced at a fixed y.

single brane

warp factor

0

y

RS limit

tension-less limit

y→∞

slide14

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

brane

Ueff=log(g00)

y

y

Very large BHs cannot float,

There are stable and unstable floating positions.

finite distance

size

large localized BH

Phase diagram for detuned tension model

necessarily touch the brane.

critical configuration

stable floating BH

floating

BH

distance from

the UV brane

small localized BH

slide15

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

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

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

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

4D AdS

curvature scale

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

mBH > mpl(ℓL)1/2.

2

(Hawking & Page ’83)

slide16

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

4-dimensional static asymptotically AdSstar made of thermal CFT

Floating BH in 5D

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

SL-3/2l-1/2

2

1

M/L

T(lL)1/2

L2rc

102

106

10-2

(central density)

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

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

r→∞

g00→ 0

slide17

Sequence of sols with a BH in 4D CFT picture

4-dimensional asymptotically AdS space with radiation fluid+BH

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

radiation fluid

with

BH

, does not

without back reaction

pure gravity

empty zone with thicknessDr~rh

log10T(Ll)1/2

radiation star

Temperature for the Killing vector

∂t normalized at infinity,

diverge even in the limit rh->0.

Stability changing critical points

BH+radiation

(plot for l=L/40)

log10M/L

slide18

Floating BHs in 5D AdS picture

brane

Numerical construction of static BH solutions is necessary.

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

We study time-symmetric initial data just solving

the Hamiltonian constraint,

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

slide19

Time-symmetric initial data for floating BHs

work in progress N. Tanahashi & T.T.

We use 5-dimensional Schwarzschild AdS space as a bulk solution.

Hamiltonian constraint is automatically satisfied in the bulk.

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

5D Schwarzschild AdS bulk:

brane

Bulk

Brane:=3 surface in 4-dimensional space.

t=constant slice

r0

Hamiltonian constraint

on the brane

Trace of extrinsic curvature

of this 3 surface

slide20

Asymptotically

static

Mass

maxmum

M/L

Abott-Desser mass

Mass

minimum

SL3/2l1/2

Critical value where mass minimum

(diss)appears is approximately read as

SL3/2l1/2

T(lL)1/2

Critical value is close to

expected from the 4dim calculation,

M/L

slide21

Comparison of the four-dim effective energy density for the mass-minimum initial data with

four-dim CFT star.

r L2

radius/L

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

(k –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 AdSbrane

numerical brane bh1
Numerical brane BH

Kudoh, Nakamura & T.T. (‘03)

Kudoh (’04)

  • Static and spherical symmetric configuration

T, R andCare functions ofzandr.

  • It becomes more and more difficult to construct brane BH solutions numerically for larger BHs.
    • Small BH case (k –1 < ℓ ) is beyond the range of validity of the AdS/CFT correspondence.

Numerical error?

or

Physical ?

Yoshino (’09)

model with detuned brane tension1
Model with detuned brane tension

Karch-Randall model JHEP0105.008(2001)

Background configuration:

warp factor

single brane

Braneplaced at a fixed y.

0

y

UV

y→-\

s → 6/ℓ (RS limit)

Warp factor increases for y>0

y→ 0

s → 0

Zero-mode graviton is absent since it is not normalizable

(Karch & Randall(2001), Porrati(2002))

slide25

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

brane

Ueff=log(g00)

y

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

y

Large BHs necessarily touch the brane.

There are stable and unstable floating positions.

1) When ds = s - 6/ℓis very small, stable floating BH is very far from the brane.

2) When s goes to zero, no floating BH exists.

slide26

Phase diagram for detuned tension model

size

This region is not clearly understood.

large localized BH

critical size

stable floating BH

floating

BH

small localized BH

distance from

the UV brane

From the continuity of sequence of solutions,

large localized BHs are expected to exist above the critical size.

Showing presence will be easier than showing absence.

slide27

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

Why does static BHs not exist in asymptotically flat spacetime?

Hartle-Hawking (finite temperature) state has regular Tmn on the BH horizon, but its fall-off at large distance is too slow.

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

4D AdS

curvature scale

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

2

(Hawking & Page ’83)

size

size

smaller ds

In the RS-limit, stable floating BH disappears

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

(ℓL)1/2

stable floating BH

floating

BH

tensionless limit (s →0)

distance from the UV brane

distance from the UV brane

slide28

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

although the BH size goes to zero.

-4

-2

2

-1

Smaller black hole

-2

metric perturbations induced on the brane
Metric perturbations induced on the brane

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

  • Not exactly Schwarzschild ⇒ ℓ << 1mm
  • 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

Gravity on the brane looks like 4D GR approximately,

BUT

  • No Schwarzshild-like BH solution????
slide30

Transition between floating and localized sequences

Configuration at the transition in un-warped two-brane model:

Configuration at the transition in de-tuned tension model:

predicted

known

“Event horizon” = “0-level contour of the lapse function, a (>0)”

3r +T→r +3P for perfect fluid in 4D

Possible contours of a :

3r +T=-s d (y) on the brane

max

3r +T=12/ℓ 2in the bulk

Maximum of a is possible only when

3r +T is negative, i.e. only on the brane.

max

X saddle

point

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

max

slide31

At the linear level, correspondence is perfect

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

CFT

CFT contribution

to the graviton self-energy

Graviton propagator

RS

slide32

Also in cosmology, correspondence is perfect

( Shiromizu & Ida (’01) )

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

CFT

(A)

trace anomary by CFT

RS

(B)

(A) and (B) give the same

modified Friedmann equation

brack hole solution in 3 1 braneworld
Brack Hole solution in 3+1 braneworld

( Emparan, Horowitz, Myers (’00) )

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

The above metric is a solution with this effective energy momentum tensor.

  • At the lowest order there is no black hole.

Hence, no Hawking radiation is consistent.

Emparan et al (’02)

slide34

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

boundary brane

our brane

If the radius at the bottle neck is > l,

there is no Gregory-Laflamme instability.

y+

y-

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

This configuration will not be directly related to our discussion?

slide35

Time scale of BH evaporation

Xray binary A0620-00: 10±5M BH+K-star

ℓ < 0.132mm(assuming 10M BH )

(Johansen, Psaltis, McClintock

arXiv:0803.1835)

J1118+480: 6.8±0.4M BH+K~M-star

ℓ < 0.97mm (assuming 6.8M BH )

(Johansen

arXiv:0812.0809)

Possible constraint from gravitational waves

Neutron star

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

BH

LISA-1.4M NS+3M BH: ℓ < 0.1mm?

numerical brane bh 2
Numerical brane BH (2)

Yoshino (’08)

  • The same strategy to construct static and spherical symmetric BH configuration localized on the brane.

Error measured by the variance in surface gravity

non-systematic growth of error

Absence of even small black holes?

location of the outer boundary

numerical brane bh 2 conti
Numerical brane BH (2) ~conti.

Non-systematic growth of error occurs for further outer boundary for better resolution.

Error measured by the variance in surface gravity

For further outer boundary, simply we need better resolution.

location of the outer boundary