Brane Localized Black Holes Classical BH evaporation conjecture Takahiro Tanaka (YITP, Kyoto university) in collaboration with N. Tanahashi, K. Kashiyama, A. Flachi Prog. Theor. Phys (2009) (arXiv: ) TT arXiv: KK, NT, AF, TT arXiv: NT, TT

Extension is infinite, but 4-D GR seems to be recovered! Brane ?? z AdS Bulk  Volume of the bulk is finite due to warped geometry although its extension is infinite AdS curvature radius : 4G4G   Negative cosmological constant Brane tension Infinite extra-dimension: Randall-Sundrum II model  No Schwarzshild-like BH solution ???? BUT

Linear perturbation of R-S II model Brane ?? z AdS Bulk  Discrete spectrum –m=0: zero mode h  ∝ z – 2 Continuous spectrum –0< m< ℓ -1 : potential barrier ⇒ suppressed wave fn. on the brane –m> ℓ -1 : too heavy to excite V(z)V(z) z ～ z –2 ℓ -2  -fn

z Black string solution Metric induced on the brane is exactly Schwarzschild solution. However, this solution is singular. C  C  ∝ 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[  ] exp(-S CFT [ ,q]) =∫d[g bulk ] exp(- S HE - S GH +S 1 +S 2 +S 3 )≡ exp(-W CFT [q]) z 0 → 0 limit is well defined with the counter terms. ∫d[g] exp(- S RS ) = ∫d[g] exp(- 2(S EH + S GH ) + 2S 1 - S matter ) = exp(- 2S 2 -S matter - 2(W CFT + S 3 )) AdS/CFT correspondence Boundary metric Counter terms Brane position z 0 ⇔ 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 brane 4D BH with CFT equivalent Classical 5D dynamics in RS II model 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 10±5M ◎ BH+K-type star X-ray binary A ℓ < 0.132mm (10M ◎ BH is assumed) (Johansen, Psaltis, McClintock arXiv: ) (Johansen arXiv: )

Rz b /z ~ l Black string region BH cap most-probable shape of a large BH Droplet escaping to the bulk Droplet formation Local proper time scale: l R on the brane due to redshift factor Area of a droplet: l 3 Area of the black hole: A ～ lR 2 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. 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 model M deformation degree uniform black string 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))  = 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. 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 UVIR When R BH > ℓ, self-gravity (of O (1/ R BH ) 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 UV (+ve tention)IR (-ve tention) 1/ ℓ mirror image mirror image

Phase diagram for warped two-brane model M deformation degree uniform black string 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 BH as large as brane separation

Model with detuned brane tension Karch-Randall model JHEP (2001) Brane placed at a fixed y. single brane RS limit y 0 warp factor 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. U eff =log(g 00 ) 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 floating BH 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. Quantum state consistent with static BHs will exist if the BH mass is large enough: m BH > m pl (ℓL ) 1/2. (Hawking & Page ’83) 4D AdS curvature scale 2

CFT star in 4D GR as counter part of floating BH Floating 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 Sequence of static solutions does not disappear until the central density diverges. g 00 → 0 → ∞ → ∞ In 5D picture, BH horizon will be going to touch the brane L2cL2c M /L T (lL) 1/2 S L -3/2 l -1/ (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 BH radiation fluid with with empty zone with thickness   r ~ r h log 10 M/L log 10 T(Ll) 1/2 pure gravity without back reaction ( plot fo r l=L/40) BH+radiation radiation star Temperature for the Killing vector ∂ t normalized at infinity, diverge even in the limit r h  0., 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. brane 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 Brane:=3 surface in 4-dimensional space. t =constant slice Trace of extrinsic curvature of this 3 surface Hamiltonian constraint on the brane r0r0 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 Abott-Desser mass Critical value where mass minimum (diss)appears is approximately read as expected from the 4dim calculation, M/L Mass minimum Mass maxmum Asymptotically static M/L T(lL) 1/2 SL 3/2 l 1/2

radius /L  L 2 Comparison of the four-dim effective energy density for the mass-minimum initial data with four-dim CFT star.

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 tension Karch-Randall model JHEP (2001) 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  →  ℓ (RS limit) y →   →  0 warp factor Background configuration:

Effective potential for a test particle (=no self-gravity). There are stable and unstable floating positions. U eff =log(g 00 ) y 1) When  ℓ is very small, stable floating BH is very far from the brane. 2) When  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 floating BH ( 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. Quantum state consistent with static BHs will exist if the BH mass is as large as m pl (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 (  →0) smaller  (Hawking & Page ’83) 4D AdS curvature scale 2

At the transition point, the temperature is finite at T crit, although the BH size goes to zero. Smaller black hole

Static spherical symmetric case (Garriga & T.T. ( ’ 99)) Not exactly Schwarzschild ⇒ ℓ << 1mm Metric perturbations induced on the brane For static and spherically sym. Configurations, second or higher order perturbation is well behaved. Correction to 4D GR=O  (ℓ 2 /R 2 star ) »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 sequences Configuration 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,  ” max Maximum of  is possible only when 3  T is negative, i.e. only on the brane. max 3  T →  3P for perfect fluid in 4D 3  T =-  y  on the brane 3  T =  ℓ 2 in the bulk Possible contours of  : X saddle point This type of transition is not allowed for the model with tensionless branes. known predicted

Integrate out CFT ( Duff & Liu (’00) ) At the linear level, correspondence is perfect Graviton propagator CFT contribution to the graviton self-energy CFT RS

effective Einstein eqs. (Shiromizu-Maeda-Sasaki(99)) Also in cosmology, correspondence is perfect CFT RS ( Shiromizu & Ida (’01) ) trace anomary by CFT (A) (B) (A) and (B) give the same modified Friedmann equation

In both & pictures, we obtain the same equation for tensor-type perturbations. Cosmological perturbations, too RS ( Tanaka ) CFT non-local term Perturbations are linear but background is non-linear!

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) ) The above metric is a solution with this effective energy momentum tensor. Emparan et al (’02)

FRW-paper N=4 SU ( N ) super YM (strong coupling) There is a large mass gap on a sphere with radius R. # of massless degrees of freedom is ~ O(1). In 4D CFT picture, Hawking radiation is not efficient. In 5D picture, therefore there is no corresponding energy flux in the classical limit. Existence of Black String solution is a counter example against the BH evaporation conjecture. It is an unstable but stationary solution. Their claim is basically that presence of LBBHs is not inconsistent with AdS/CFT. (Fitzpatrick, Randall and Wiseman hep-th/ ) But if relevant modes are all massive, why long-ranged correlation is not exponentially suppressed???

BS is an unstable but stationary object with horizon. With fine tuning, it can persist for an arbitrary long period, avoiding Hawking evaporation. Answer) But BS is a singular object with curvature singularity deep inside the bulk. Therefore it will not have a counter part on CFT side. Black string is a counter example? What kind of CFT quantum state corresponding to black string can you imagine? In quantum mechanics unstable state is not stationary, is it?

IR cutoff branestableunstable In quantum mechanics unstable state is not stationary, is it? IR modification of CFT at the energy scale higher than T BH ~ R -1. R > ℓ < ℓ R  Wave function: Wave function is unnormalizable! This situation looks like…. effective potential

AdS/CFT in Karch-Randall brane Global embedding of K-R brane static AdS 5 metric “∂ t ” is Killing y:= fixed R →∞   brane t = const. surface Brane ≠ complete boundary of bulk

AdS/CFT in Karch-Randall brane communicate s through  CFT 1 t = const. surface CFT living on the other side is necessary CFT 2 CFT 1 CFT 2

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. Action and hence total energy are dominated by the boundary brane side, and diverge in the limit y + → ∞. y y y+ y+ This configuration will not be directly related to our discussion?

42 Time scale of BH evaporation LISA  1.4M NS+3M BH: ℓ < 0.1mm? BH Neutron star Roughly speaking, if  M/M ～ 1/N, where N is the number of observed cycles,  M will be detectable. X ray binary A : 10±5M BH+K-star ℓ < 0.132mm (assuming 10M BH ) (Johansen, Psaltis, McClintock arXiv: ) Possible constraint from gravitational waves J : 6.8±0.4M BH+K~M-star ℓ < 0.97mm (assuming 6.8M BH ) (Johansen arXiv: )

Numerical brane BH (2) The same strategy to construct static and spherical symmetric BH configuration localized on the brane. Yoshino (’08) Error measured by the variance in surface gravity location of the outer boundary non-systematic growth of error Absence of even small black holes?

Numerical brane BH (2) ~conti. 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.