Brane Black Holes
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
Bran
e
??
z
AdSBulk
dxdxdz
zds 2
2
22
z
Volume of the bulk is finite due to warped geometry although its extension is infinite.
5
2
3
6
AdS curvature radius :
4G
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. CC∝ z 4
behavior of zero mode
Moreover, this solution is unstable.
Gregory Laflamme instability
( Chamblin, Hawking, Reall (’00) )
“length width” ≳
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) Yoshino(‘08)
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.
is surface gravity
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.
brane tension
Z[q]=∫d[] exp(-SCFT[,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 model
22 plMnumber 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))
8
9
Time scale of BH evaporation
32
32
1species
ofNumber
MGMGM
M
NN
year120
mm123
SolarM
M
X ray binary: orbital period derivative A0620-00: 10±5M BH+K-star ℓ < 0.132mm (assuming 10M BH )
(Johannsen, Psaltis, McClintock (2009))
J1118+480: 6.8±0.4M BH+K~M-star ℓ < 0.97mm (assuming 6.8M BH )
(Johannsen (2009))
In globular cluster RZ2109 located in an elliptical galaxy NGC4472 in the Virgo cluster
Compact object with rapid variability and broad emission line ~2000km/s ~10M BH
Assuming that the association with the globular cluster means old age of BH.
~10±3Gyr.
(Gnedin, Maccarone, Psaltis, Zepf (2009))
ℓ < 0.003mm
(Zepf, Stern, Maccarone, Kundu, Kamionkowski, Rhode, Salzer, Ciardullo & Gronwall (2008))
BH accreting at around Eddington limit From the luminosity
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
bulkbulk
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))
= 0 & = 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.
≠ 0 & 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
RS
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 T 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 → ∞
In 5D picture, BH horizon will be going to touch the brane
L2c
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 radiation fluid with with
empty zone with thicknessr~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 K=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 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
3.4/ 3 lLScrit
Critical value where mass minimum (diss)appears is approximately read as
6.6/ 3 lLS
expected from the 4dim calculation,
M/L
Massminimum
Massminimum
Massmaxmum
Massmaxmum
Asymptoticallystatic
Asymptoticallystatic
M/L
T(l
L)1/
2
SL-3
/2l-1
/2
SL-3/2l-1/2
radius/L
L
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
(–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