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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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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
AdS curvature radius
l
6
Negative cosmological
constant
L
=

2
l
3
s
=
Brane tension
4pG
l
5
Infinite extradimension: RandallSundrum II model
Volume of the bulk is finite due to warped geometry although its extension is infinite.
L
??
z
Brane
AdS
Bulk
BUT
( Chamblin, Hawking, Reall (’00) )
Metric induced on the brane
is exactly Schwarzschild solution.
z
However, this solution is singular.
behavior of zero mode
Moreover, this solution is unstable.
“length ≳ width”
( 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 Smatter2(WCFT+ S3))
( Hawking, Hertog, Reall (’00) )
Boundary metric
Counter terms
z0⇔ cutoff scale parameter
Brane position
brane tension
4D EinsteinHilbert action
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 )
(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+Ktype star Xray binaryA062000
ℓ < 0.132mm (10M◎BH is assumed)
(Johansen, Psaltis, McClintock
arXiv:0803.1835)
(Johansen arXiv:0812.0809)
of a large BH
Structure near the cap region will be almost independent of the size of the black hole. ~discrete selfsimilarity
brane
R
Assume GregoryLaflamme 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
Kudoh, Nakamura & T.T. (‘03)
Kudoh (’04)
T, R andCare functions ofzandr.
Comparison of 4D areas with 4D and 5D Schwarzschild sols.
5D Sch.
4D Sch.
k is surface gravity
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.
RSI (two branes)
KarchRandall (AdSbrane)
M
We do not consider the sequences which produce BH localized on the IR(right) brane.
x
nonuniform
black string
localized BH
x
floating BH
uniform black string
deformation degree
(Kudoh & Wiseman (2005))
IR
Warped twobrane model (RSI)L ≠ 0 & s is finetuned
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.
Selfgravity due to the mirror images
on the other side of the branes
When RBH > ℓ,
selfgravity (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.
Deformation of nonuniform BS occurs mainly near the IR brane. (Gregory(2000))
M
localized BH
as large as
brane separation
x
nonuniform
black string
x
uniform black string
floating BH
localized small BH
deformation degree
KarchRandall model JHEP0105.008(2001)
s< sRS
Effectively fourdimensional negative cosmological constant
Background configuration:
Braneplaced at a fixed y.
single brane
warp factor
0
y
RS limit
tensionless limit
y→∞
Effective potential for a test particle (=no selfgravity).
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
Large localized BHs above the critical size are consistent with AdS/CFT?
Why doesn’t static BHs exist in asymptotically flat spacetime?
HartleHawking (finite temperature) state has regular Tmn on the BH horizon, but its falloff at large distance is too slow to be compatible with asymptotic flatness.
In AdS, temperature drops at infinity owing to the redshift 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)
CFT star in 4D GR as counter part of floating BH
4dimensional static asymptotically AdSstar made of thermal CFT
Floating BH in 5D
The case for radiation fluid has been studied by Page & Phillips (1985)
SL3/2l1/2
2
1
M/L
T(lL)1/2
L2rc
102
106
102
(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
Sequence of sols with a BH in 4D CFT picture
4dimensional 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
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 timesymmetric initial data just solving
the Hamiltonian constraint,
extrinsic curvature of tconst. surface Kmn=0.
Timesymmetric initial data for floating BHs
work in progress N. Tanahashi & T.T.
We use 5dimensional 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 4dimensional space.
t=constant slice
r0
Hamiltonian constraint
on the brane
Trace of extrinsic curvature
of this 3 surface
static
Mass
maxmum
M/L
AbottDesser 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
Comparison of the fourdim effective energy density for the massminimum initial data with
fourdim CFT star.
r L2
radius/L
(k –1≫ℓ )brane BH solution in RSII brane world.
which is consistent with AdS/CFT correspondence.
As a result, we predicted new sequences of black objects.
1) floating stable and unstable BHs
2) large BHs localized on AdSbrane
Kudoh, Nakamura & T.T. (‘03)
Kudoh (’04)
T, R andCare functions ofzandr.
Numerical error?
or
Physical ?
Yoshino (’09)
KarchRandall 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
Zeromode graviton is absent since it is not normalizable
(Karch & Randall(2001), Porrati(2002))
Effective potential for a test particle (=no selfgravity).
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.
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.
Large localized BHs above the critical size are consistent with AdS/CFT?
Why does static BHs not exist in asymptotically flat spacetime?
HartleHawking (finite temperature) state has regular Tmn on the BH horizon, but its falloff at large distance is too slow.
In AdS, temperature drops at infinity by the redshift 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 RSlimit, stable floating BH disappears
5dAdSSchwarzschild 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
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
Static spherical symmetric case (Garriga & T.T. (’99))
Gravity on the brane looks like 4D GR approximately,
BUT
Transition between floating and localized sequences
Configuration at the transition in unwarped twobrane model:
Configuration at the transition in detuned tension model:
predicted
known
“Event horizon” = “0level 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
At the linear level, correspondence is perfect
Integrate out CFT ( Duff & Liu (’00) )
CFT
CFT contribution
to the graviton selfenergy
Graviton propagator
RS
Also in cosmology, correspondence is perfect
( Shiromizu & Ida (’01) )
effective Einstein eqs. (ShiromizuMaedaSasaki(99))
CFT
(A)
trace anomary by CFT
RS
(B)
(A) and (B) give the same
modified Friedmann equation
( 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.
Hence, no Hawking radiation is consistent.
Emparan et al (’02)
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 GregoryLaflamme 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?
Xray binary A062000: 10±5M BH+Kstar
ℓ < 0.132mm(assuming 10M BH )
(Johansen, Psaltis, McClintock
arXiv:0803.1835)
J1118+480: 6.8±0.4M BH+K~Mstar
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
LISA1.4M NS+3M BH: ℓ < 0.1mm?
Yoshino (’08)
Error measured by the variance in surface gravity
nonsystematic growth of error
Absence of even small black holes?
location of the outer boundary
Nonsystematic 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