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. 121 1133 (2009) (arXiv:0709.3674) TT arXiv:0910.5376 KK, NT, AF, TT arXiv:0910.5303 NT, TT
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Brane Localized Black Holes Classical BH evaporation conjecture Takahiro Tanaka (YITP, Kyoto university) in collaboration with N. Tanahashi, K. Kashiyama,
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Brane Localized Black HolesClassical BH evaporation conjecture
Takahiro Tanaka (YITP, Kyoto university)
in collaboration with N. Tanahashi, K. Kashiyama, A. Flachi
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 Tmn 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 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 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 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+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 Kmn=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
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
3.43 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(l
L)1/
2
SL3/
2 l1/
2
SL3/2l1/2
radius/L
rL
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
( 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 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 ( k –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)
22222
4/cosh AdSdsydyds
y
tanh6
5
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
xdgRg
xdS 44
5
5 22
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 Tmn 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
BHrr
BH rr
ge
r
g
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 rr
GMh
2
2
31
2
2
2
00 3
21
2
rr
GMh
• For static and spherically sym. Configurations, second or higher order perturbation is well behaved.