Transcript Localized brane black hole part II

Brane Localized Black Holes
Classical BH evaporation conjecture
Takahiro Tanaka
(YITP, Kyoto university)
Prog. Theor. Phys. 121 1133 (2009)
(arXiv:0709.3674) TT
arXiv:0910.5376 KK, NT, AF, TT
arXiv:0910.5303 NT, TT
in collaboration with
N. Tanahashi, K. Kashiyama, A. Flachi
Infinite extra-dimension:
Randall-Sundrum II model
Volume of the bulk is finite due to warped geometry although its
extension is infinite.

2

ds2  2 dz2    dx dx
z
??
x

L
z
Brane
AdS
Bulk
 : AdS curvature radius
6
L  - 2 Negative cosmological

constant
s  3 Brane tension
4pG5 
• Extension is infinite,
but 4-D GR seems
to be recovered!
z
BUT
 No Schwarzshild-like BH solution????
Black string solution
2


ds  2 dz2  g Sch dx dx
z
2

Metric induced on the brane g  x 
is exactly Schwarzschild solution.
However, this solution is singular.

Crs Crs ∝ z 4
behavior of zero mode
Moreover, this solution is unstable.

Gregory Laflamme instability
“length ≳ width”
( Chamblin, Hawking, Reall (’00) )
z
AdS/CFT correspondence
( Maldacena (’98) )
( Gubser (’01) )
( Hawking, Hertog, Reall (’00) )
Z[q]=∫d[f] exp(-SCFT[f,q])
Boundary=∫d[gbulk] exp(- SHE- SGH+S1+S2+S3)≡ exp(-WCFT[q])
metric
S EH
SGH
12 
 5 
 - 2 d x - g  R 2 
2 5
 

1
 - 2  d 4x - q K
1
5
5
Counter terms
3
S1  - 2  d 4 x - q
5 

S2  - 2  d 4 x - q
4 5
S3  
z0→ 0 limit is well defined with the counter terms.
z0 ⇔ cutoff scale parameter
Brane position
4 
brane tension
∫d[g] exp(- SRS) = ∫d[g] exp(- 2(SEH+ SGH) + 2S1- Smatter )
= exp(- 2S2 -Smatter- 2(WCFT+ S3))
4D Einstein-Hilbert action
R
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 )
Classical black hole evaporation conjecture
(T.T. (’02), Emparan et al (’02))
4D Einstein+CFT with
the lowest order
quantum correction
2

number of
 2
field of CFT

AdS/CFT
correspondence
equivalent
Classical 5D
dynamics in
RS II model
equivalent
5D BH on brane
equivalent
Classical
evaporation
of 5D BH
4
4D BH with CFT
Hawking radiation in 4D
Einstein+CFT picture
Time scale of BH evaporation
 M
  
 M Solar



3
2
 1mm 

 120year
  
M  Number of 
1
2
 
  2 3 
M  species  GN M
GN M


3
10±5M◎BH+K-type star X-ray binary A0620-00
(Johansen, Psaltis, McClintock
arXiv:0803.1835)
ℓ < 0.132mm (10M◎BH is assumed)
(Johansen arXiv:0812.0809 )
most-probable shape
of a large BH
brane
R
Assume Gregory-Laflamme
instability at the cap region
Droplet escaping to the bulk
Black string
region
bulk
Rzb/z ~ l
BH cap
Structure near the cap region will be
almost independent of the size of the
black hole. ~discrete self-similarity
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
dA l 3

dt R
1 dM 1 dA l 2

 3
M dt
A dt R
Numerical brane BH
Kudoh, Nakamura & T.T. (‘03)
Kudoh (’04)
• Static and spherical symmetric configuration
T, R and C are functions of z and r.
Comparison of 4D areas with
4D and 5D Schwarzschild sols.
 A4 p
100 * 1000
2. 2
5D Sch .
2
!!!!!!!!!!!
k A4 p
5D Sch.
4D Sch.
 is surface gravity
1. 8
1. 6
1. 4
1. 2
4D
.
It becomes more and more difficult
toSch
construct
brane BH
1
solutions numerically for larger BHs.
0
Small BH case ( < ℓ ) is beyond the
AdS/CFT correspondence.
–1
1
2
3
4
rangeLog
of validity
@L k D of
log 
5
the
6
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)
s=0 & L=0
Un-warped two-brane model
uniform black string
M
We do not consider the sequences which
produce BH localized on the IR(right) brane.
x
x
deformation degree
(Kudoh & Wiseman (2005))
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 (+ve tention) brane.

Acceleration acting on a test particle
in AdS bulk is
1/ℓ
UV (+ve tention)
ds2  dy2  e-2 y /  - dt2  dx2
IR (-ve tention)
a
log g tt , y
g yy
-
1

Compensating force toward the UV brane is necessary.
Self-gravity due to the mirror images
on the other side of the branes
mirror
image
mirror
image
UV
IR
Large floating BHs become
large localized BHs.
When RBH > ℓ,
self-gravity (of O(1/RBH) at most),
cannot be as large as 1/ℓ.
Pair annihilation of two
sequences of localized BH,
which is necessary to be
consistent with AdS/CFT.

Phase diagram for warped two-brane model
Deformation of non-uniform BS occurs
mainly near the IR brane. (Gregory(2000))
uniform black string
M
x
x
deformation degree
Model with detuned brane tension
Karch-Randall model
JHEP0105.008(2001)
-g
R - 2L - s  - g 4 d 4 x
S  - d x
2 5
Background configuration:
s < sRS
5
Effectively fourdimensional negative
cosmological constant
2
ds2  dy2  2 cosh2  y / dsAdS
4
Brane placed at a fixed y.
single brane
warp factor
0
RS limit
y→∞
tension-less
limit
y
Effective potential for a test
particle (=no self-gravity).
brane
Ueff=log(g00)
y
y
There are stable and unstable
floating positions.
Phase diagram for
detuned tension model
size
finite distance
Very large BHs cannot float,
necessarily touch the brane.
large
localized
BH
critical
configuration
small localized BH
distance from
the UV brane
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 T 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.
T  1 / g 00  1 / 1 - r -1  r / L 
2
4D AdS
curvature scale
Quantum state consistent with static BHs will exist
if the BH mass is large enough:
mBH > m2pl(ℓL)1/2.
(Hawking & Page ’83)
CFT star in 4D GR as counter part of floating BH
4-dimensional static asymptotically
AdS star made of thermal CFT
Floating BH in 5D
The case for radiation fluid has been studied by Page & Phillips (1985)
r  3P  aTloc4  1/ g002
S L-3/2l-1/2
2
T  lim Tloc
r 
1
M/L
10-2
102
T(lL)1/2
106 L2rc
(central density)
Sequence of static solutions does not
disappear until the central density diverges.
r→∞
g00 → 0
rcirc
L
In 5D picture, BH
horizon will be going
to touch the brane
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:
r  aTloc4  1/ g002
Temperature for the Killing vector
∂t normalized at infinity,
radiation fluid
rcirc
with Tloc  TBH / - g00
T  lim T
, does not
BH
r 
loc
L
diverge even in the limit rh -> 0.
without back reaction
empty zone with
thicknessD r~rh
pure gravity
-0.4
-0.5
-0.6
log10T(Ll)1/2
Stability changing
critical points
-0.7
-0.8
(plot for l=L/40)
-0.9
-1
-0.5
0.5
1
log10M/L
brane
Floating BHs in 5D AdS picture
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
extrinsic curvature of t-const. surface K=0.
the Hamiltonian constraint,
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:
dr2
ds  -U r dt 
 r 2 d2
U r 
2
2
Bulk
Brane:=3 surface in 4-dimensional space.
t=constant slice
r0
Hamiltonian constraint
on the brane
1 1
Trace of extrinsic curvature
3 2 - 2
of this 3 surface
l
L
Mass
maxmum
L / l  1 / 100
SL3/2l1/2
Critical value where mass minimum
(diss)appears is approximately read as
S / L3l  6.6
Critical value is close to
T(lL)1/2
Mass
minimum
SL3/2l1/2
M/L
Abott-Desser mass
2  Area
S
4G4l
Asymptotically
static
S crit L3l  4.3
expected from the 4dim calculation,
M/L
Comparison of the four-dim effective energy
density for the mass-minimum initial data with
four-dim CFT star.
S / L l  4.3
r L2
3
S / L l  0.89
3
radius/L
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.
As a result, we predicted new sequences of black objects.
1) floating stable and unstable BHs
2) large BHs localized on AdS brane
• Partial support for this scenario was obtained by comparing the
4dim asymptotic AdS isothermal star and the 5dim timesymmetric initial data for floating black holes.
Numerical brane BH
Kudoh, Nakamura & T.T. (‘03)
Kudoh (’04)
• Static and spherical symmetric configuration
T, R and C are functions of z and r.
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
JHEP0105.008(2001)
-g
R - 2L - s  - g 4 d 4 x
S  - d x
2 5
5
Background configuration:
2
ds2  dy2  2 cosh2  y / dsAdS
4
Brane placed at a fixed y.
6

 5s  - tanh
y→-\
y→ 0
single brane
y

s → 6/ℓ (RS limit)
s→0
0
warp factor
y
UV
Warp factor increases for y>0
Zero-mode graviton is absent
since it is not normalizable
(Karch & Randall(2001), Porrati(2002))
Effective potential for a test
particle (=no self-gravity).
brane
Ueff=log(g00)
y
y
There are stable and unstable
floating positions.
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.
Since this distance is
finite, very large BHs
cannot float.
Large BHs
necessarily
touch the brane.
Phase diagram for detuned tension model
size
This region is not
clearly understood.
large localized BH
critical size
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?
Hartle-Hawking (finite
temperature) state has regular T
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.
T  1 / g 00  1 / 1 - r -1  r / L 
2
4D AdS
curvature scale
Quantum state consistent with static BHs will exist
if the BH mass is as large as m2pl(ℓL)1/2. (Hawking & Page ’83)
size
(ℓ L)1/2
size
smaller ds
In the RS-limit, stable
floating BH disappears
distance from the UV brane
5d-AdS-Schwarzschild
with brane on the
equatorial plane
tensionless
limit (s →0)
distance from the UV brane
At the transition point, the temperature is finite at Tcrit
1 lL ,
although the BH size goes to zero.
TBH 
g
1  g 00
1
 e 00 
r
rBH
g rr r
  log- g00 grr 
-4
-2
2
log r
-1
Smaller black hole
-2
Metric perturbations induced on the brane
Static spherical symmetric case (Garriga & T.T. (’99))
2GM
h00 
r
 2 2 
1  2 
 3r 
2GM 
2 
1  2  d ij
hij 
r  3r 
• 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????
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, a >0”
Da 
5
3r  T a
6
Possible contours of a :
max
max
max
X saddle
point
3r T →r 3P for perfect fluid in 4D
3r T =-s d y on the brane
3r T =12/ℓ 2 in the bulk
Maximum of a is possible only when
3r T is negative, i.e. only on the brane.
This type of transition is not allowed
for the model with tensionless branes.
At the linear level, correspondence is perfect
CFT
Integrate out CFT


( Duff & Liu (’00) )
1 4
-1
S   d x   Da
  a  a CFT contribution
2
to the graviton self-energy
Graviton propagator
  D  DT
  DT - D DT
~
16p G 
1
1



h  p   - 2 T  p  -  T  p  - 16p G p T  p  -  T  p 
p 
2
3




2  p2
  p   -  ln 2  const.
4 

RS
~
2 52
h  p   - 2
p
1




T
p
 T  p 
 
2


1


-  52 D KK  p T  p  -   T  p 
3




  p   D KK  p   O p 2  4
2

- K 0  p 
  p 2 2
D KK  p  
 -  ln
 2   O p 2 4
pK1  p 
2
4




Also in cosmology, correspondence is perfect
4 
CFT

G  8pG4 T  TCFT
( Shiromizu & Ida (’01) )

2

1 4  4  
 4 
4  
4 
G  8pG4T   G G G G
4
3

(A)
trace anomary by CFT
RS 4  G

 8pG4T  8pG5  p  - E
effective Einstein eqs.
(Shiromizu-Maeda-Sasaki(99))
p   T 2  E  nr ns 5Crs
2
2


8
p
G


4 
4
G  8pG T 
T
4
4 
4

G  8pG4T  O 2 / L4




T

1 2
- T 
3 
(B)
(A) and (B) give the same
modified Friedmann equation

r2 
2

H  8p G4  r 
2s 

Brack Hole solution in 3+1 braneworld
-1
( Emparan, Horowitz, Myers (’00) )
 2  2  2 
2
2
2
ds2  -1 dt  1  dr  r d
r 
r 


Metric induced on the brane looks like Schwarzschild solution,
But
4p
D 
32 
1/ 3
<< 2p
This static solution is not a counter example of the conjecture.
Casimir energy of  G3 CFT fields on ds2  -dt2  dr2  r 2d 2
with D  4p / 32 1/ 3 is given by
CFT
T

1


 

1

3 
8pG3 r 
- 2 

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)
What is the meaning of the presence of stable black string configuration?
our brane
boundary brane
If the radius at the bottle neck is > l,
there is no Gregory-Laflamme instability.
y-
y+
M 53
4
S
dy
d
x - g R  2L 5 


2
S M
3
5

y-
y
dy a 2  y  d 4 x - g 4  R4 
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?
Time scale of BH evaporation
 M
  
 M Solar
3
  1mm 
 
 100year
  
2


2
M
1

of 
 Number

2
3
species
M
GN M
GN M


3
X ray binary A0620-00: 10±5M BH+K-star (Johansen, Psaltis, McClintock
ℓ < 0.132mm (assuming 10M BH ) arXiv:0803.1835)
J1118+480: 6.8±0.4M BH+K~M-star (Johansen
ℓ < 0.97mm (assuming 6.8M BH )arXiv: 0812.0809 )
Possible constraint from
gravitational waves
Neutron star
BH
Roughly speaking, if DM/M ～1/N,
where N is the number of observed
cycles, DM will be detectable.
42
LISA- 1.4M NS+3M BH: ℓ < 0.1mm?
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
rh 
rh 
rh 
Absence of even
small black holes?
 logrout rh 
location of the outer boundary
Numerical brane BH (2)
~conti.
rh 
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.
 logrout rh 
location of the outer boundary