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JHEP 06 (2004) 053, hep-th/0406002
The Hebrew The Hebrew UniversityUniversity
July 27 2006Freie Universität Berlin
Class.Quant.Grav. 22 (2005) 3935-3960, hep-th/0505009
Talk at:
• The goal and related works• Description of the method matched asymptotic
expansion in– Example: monopole match
• Results and implications for the phase diagram• Summary
2,1 1.dR S
2,1 1d S - solution for a small BH
Analytical
• Harmark 03’, (Harmark and Obers 02’) • Karasik et al. 04’
• Chu, Goldberger and Rothstein 06’
5d
Interpolating coordinates
EFT formalism 22ndnd order order Only monopole
match
• Sorkin, Kol and Piran 03’ 5d
• Kudoh and Wiseman 03’, 04’ 5d, 6d
Numerical
0.1 0.2 0.3 0.4 0.6 0.8 1
0.5 1
1.5 2
2.5 3
M / Mcrit
n / nbs
Coordinates for small caged BH
r
z
cylindrical coordinates
spherical coordinates
2,1 1dR S
( , )
( , )r z
0L
+two dimensionful parameters: 0 , L
one dimensionless parameter: 0
L 1
• A small parameter
• Input:- An exact solution- Boundary conditions
0
L Perturbative expansion
of Einstein’s equations
Two zones:
Near horizon Asymptotic0
0 L2 2 0r zz z L
The exact
solutionSchwarzschild-Tangherlini Minkowsky + periodic
b.c.Fixed parameter
Small parameter
0 L1L
0
Large Overlap Region0 L
L0 The near zone
The asymptotic zone
0
L The overlap region
0 L
3ll
ll d
A B
• Asymptotic zone - post-Newtonian expansion• Near zone – Black hole static perturbations
(0) (1) 2 (2) ...g g g g
( ) ( 1) ( 2) (1)( ) ( , ,..., )n n nnL g F g g g
(4d - Regge Wheeler 57’)
The solution is determined up to solutions of the homogeneous equation
...L 0
Einstein’s equations
3l
ll
l d
A B
The leading terms in the radial part
Weak field
4,1 1R S
BH
4S
0 1 2 3 4 5 6 7
(6d)
monopolequadrupoleHexadecapole
Asymptotic zone
Near zone
0
L
order in
A dialogue of multipolesA dialogue of multipoles
Asymptotic Asymptotic zonezone
NearNear zonezone
OverlapOverlap RegionRegion
1tt Ng
2 2 2 2 22
113 dCds C f dt f d d
d
3
031
d
df
0
1ttg C
01
30
32 2 2( )
d
N dn r z nL
0C
2
2N
20
2
L
20
0 2C
L
30
0 32 ( 3)d
ddL
30
03
d
N d
angular terms
1
1( )n n
T dSdM dL
12
22:
12
d
d d
2dS The area of a unit
1
1( )n n
30( 2) ( 3)
3
dd dd L
Near zone Asymptotic zone
32 0( 2) 31
16 2( 2)
dd
d
d dMG d
3 O
2 20 2 14 2
dd
d
SG
2
0
3 14 2dT
2 32 0( 3)
32
dd
d
dG
• Eccentricity
• The “Archimedes” effect
24
10
2( 3) 2 ( 1)348( 2)3
...dd d
dLd
d
L
12 polesL
0
1132
1 12 3
...polesdL L
d
The BH “repels” the space of the
compact dimension
6dNumerical results: Kudoh and Wiseman
03’, 04’
0.1 0.2 0.3 0.4 0.6 0.8 1n nbs
0.5
1
1.5
2
2.5
3
M Mcrit
GL
BH
US
NUS
11stst order order
M / Mcrit
n / nbs0.1 0.2 0.3 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
US
11stst order order
GL
M / Mcrit
n / nbs
BH
NUS
0.1 0.2 0.3 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
GL
BH
US
NUS
11stst order order22ndnd order order
n / nbs
M / Mcrit
6dNumerical results: Kudoh and Wiseman
03’, 04’
0.1 0.2 0.3 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
GL
BH
US
NUS
11stst order order22ndnd order order
n / nbs
M / Mcrit
6dNumerical results: Kudoh and Wiseman
03’, 04’
6dNumerical results: Kudoh and Wiseman
03’, 04’
0.1 0.2 0.3 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
GL
BH
US
NUS
11stst order order22ndnd order order
n / nbs
M / Mcrit
Inflection point??
• The method yields approximations to the whole metric providing not only the thermodynamic quantities but also BH eccentricity and "the BH Archimedes effect"
• A comparison with the numerical simulation in 6d shows an excellent agreement in the first order approximation when the second order indicates that there should be an inflection point which is not seen in the simulations so far.
• Matched asymptotic expansion was used to obtain an approximate analytical solution for a small BH in . The method can be carried in principle to an arbitrarily high order in the small parameter.
2,1 1dR S