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
6
AdS 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. C mnrs C mnrs ∝ 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
255
25
12
2
1
RgxdSEH
KqxdSGH4
25
1
qxdS 425
1
3
RqxdS 44
25
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
1
species
ofNumber
MGMGM
M
NN
year120
mm123
SolarM
M
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
R
l
dt
dA 3
3
211
R
l
dt
dA
Adt
dM
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.
4A
0 1 2 3 4 5 6Log @L kD
1
1.2
1.4
1.6
1.8
2
2.2
k
!!!!!!
!!!!!
A4
p
100*1000
5D Sch.
4D Sch.
log
4D Sch.
5D 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.
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 uni
form
bla
ck s
trin
g x
localized BH
non-uniform
black stringx
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
trin
g x
non-uniform
black stringx
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)
22222
4/cosh AdSdsydyds
Brane placed at a fixed y.
single brane
RS limit
y 0
warp factor
xdgRg
xdS 44
5
5 22
Background configuration:
tension-less limit
s < sRS
Effectively 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 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.
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)” T 3
65
max Maximum of a is possible only when 3 r +T is negative, i.e. only on the brane.
max
max
3 r +T → r +3P for perfect fluid in 4D
3 r +T =- s d (y) on the brane
3 r +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.
known
predicted
Integrate out CFT ( Duff & Liu (’00) )
At the linear level, correspondence is perfect
Graviton propagator
CFT contribution to the graviton self-energy
14
2
1DxdS
CFT
DTD DTDDT
pTpTpGpTpT
p
Gph
3
116
2
116~2
const.ln
4 2
22
p
p
RS
pTpT
pph
2
12~2
25
42
22
1
0 24
ln2
pOp
ppK
pKpKK
42
2
pOpp KK
pTpTpKK
3
125
effective Einstein eqs. (Shiromizu-Maeda-Sasaki(99))
GGGGTGG 4444
2
44
3
1
48
Also in cosmology, correspondence is perfect
CFT
RS EGTGG 254
4 88
( Shiromizu & Ida (’01) )
trace anomary by CFT
2
24
44
3
1
4
88 TTT
GTGG
CFTTTGG 44 8
(A)
(B)
424
4 /8 LOTGG (A) and (B) give the same modified Friedmann equation
2T
CnnE 5
2
82
42 GH
2221
22 21
21
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) )
2
23
43/1
2
1
1
8 33rG
T CFT
The above metric is a solution
with this effective energy momentum tensor.
3G 22222 drdrdtds 3/123/4
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.
54
35 2
2Rgxddy
MS
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
MGMGM
M
NN
year100
mm123
SolarM
M
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 log
Error measured by the variance
in surface gravity
location of the outer boundary
h
h
h
non-systematic growth of error
Absence of even small black holes?
Numerical brane BH (2) ~conti.
hout log
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.
h