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