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Brane Localized Black Holes Classical BH evaporation conjecture

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

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  1. 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

  2. : 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????

  3. 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”

  4. 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

  5. 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 )

  6. 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)

  7. 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

  8. 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.

  9. 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)

  10. 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))

  11. 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.

  12. 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

  13. 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→∞

  14. 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

  15. 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)

  16. 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

  17. 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

  18. 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.

  19. 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

  20. 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

  21. Comparison of the four-dim effective energy density for the mass-minimum initial data with four-dim CFT star. r L2 radius/L

  22. 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

  23. 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)

  24. 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))

  25. 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.

  26. 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.

  27. 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

  28. 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

  29. 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????

  30. 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

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

  32. 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

  33. 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)

  34. 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?

  35. 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?

  36. 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

  37. 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

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