Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qfthep.sinp.msu.ru/talks/2010.09.13_Golitsyno_Rannu.pdf Дата изменения: Sun Sep 19 19:04:29 2010 Дата индексирования: Mon Oct 1 19:54:38 2012 Кодировка:

Internal structure of Maxwell-Gauss-Bonnet black hole

K.A. Rannu
SAI MSU

QFTHEP-2010 Golistyno 13.09.2010


Subject
The talk is based on the paper S.O. Alexeyev, A. Barrau, K.A. Rannu "Internal structure of a Maxwell-Gauss-Bonnet black hole" Phys. Rev. D 79 067503 (2009) S.O. Alexeyev, M.V. Pomazanov Phys. Rev. D 55 2110 (1997) S.O. Alexeyev, M.V. Sazhin Gen. Relativ. Grav 30 1187 (1998) S.O. Alexeyev, M.V. Sazhin, M.V. Pomazanov Int. J. Mod. Phys D 10 225 (2001)


Action and metric
Low-energy effective string action with second order curvature corrections: S= 1 16 d4 x -g -R + 2µ µ - e-2 F SGB = Rijkl R
ijkl µ 2

F

µ

+ e-2 S

GB

,

- 4Rij Rij + R

Spherically-symmetric metric in GHS coordinates: 1 ds2 = dt2 - dr2 - f 2 (d 2 + sin2 d 2 ) Anzats for Maxwell tensor: F = q sin d d Variable change: E = e-2 , E0 = e-20


Lagrange-Euler (Einstein) equations
0 f - 4 f E 4 E(f 2 - 1) = f - 4 f E 2(1 - 2 E ) 8 f E -2 (f 2 - 1) / r -4 f f / r = f 2 E / r E

f2E 2 - 4 E2 2 q f E 2 2E 3 - 2 f - + 4 2 f E 2 E2 f q2 E E2 2E 2 - f ( f + 2f ) + f 2 2 - 4 2 f 2 E E f E Constraint:

(1 + f ( ) - ff - (f ) ) + 4E (1 - 3(f )2 ) - Eq2 f

2

2

2

-2

=0


Asymptotics
Asymptotics near the horizon (r - rh ) 1:

= d1 (r - rh ) + d2 (r - rh )2 + O (r - rh )2 f = f0 + f1 (r - rh ) + f2 (r - rh )2 + O (r - rh )2 E = E0 + 1 (r - rh ) + 2 (r - rh )2 + O (r - rh )2 Asymptotics at the infinity -- GM-GHS solution: 2M 2M dt2 - 1 - ds = 1 - r r
2 -1

q2 E dr - r r - M
2

0

d

E = E0 -

q2 Mr


Curvature invariant in GHS coordinates

R

ijkl

R

ijkl

=

2

f2 f + 4 2 + 82 f f
2

2 2

ff 4 f2 f + 8 2 + 4 - 8 4 + 42 f f f f

4 4

Determining curvature curvature neglected

the singularity type: invariant diverges real scalar singularity (Clarke, 1993) invariant is finite coordinate singularity, that can be by appropriate coordinate transformation


Metric function (r)


Metric function f (r)


Dilatonic exponent exp (-2 (r))


Curvature invariant RijklRijkl(f )


Curvature invariant RijklRijkl(q, f )
critical charge qcr q>qcr; 10 10 10 10
8

singular horizon rx

singularity rs
6

4

2

q -2

10 10

-4

10

-6

q

Metric function f


Asymptotics near the particular points

Metric function f (r) should be regarded as a radial coordinate. f (r rs ) = fs + fs2 ( r - rs )2 + fs3 ( r - rs )3 + . . . f (r rx ) = fx + fx1 r - rx + fx2 ( r - rs )2 + . . . for for f fs f f
x

Rijkl Rijkl const1 в (f - fs )-1 Rijkl R
ijkl

const2 в (f - fx )-5


Results
When the black hole charge becomes larger than the critical value the singularity rs is replaced by a local minimum of the fuction (r) and the solution exists till the singular horizon rx . Function f (r) is the radius of S2 , so it plays the role of the radial coordinate. If q < qcr it decreases monotonously till r = rs like in GHS. When rs disappears the function f (r) reaches its zero in the new point rx . Curvature invariant increases much more rapidly (as (r - rx )-5 ) near the singular horizon rx than near the singularity rs (as (r - rs )-1 ), so the singularity in rx is much stronger than the one in rs . New kind of singularity inside black hole was found. Unfortunately Maxwell-Gauss-Bonnet black hole cannot help wormholes' or multiverse theories because this singularity is very strong.


Thank you for attention


Critical charge
Critical charge value -- a meaning at which the inner singularity disappears being replaced by a local minimum for (r)
35 30

25

Critical charge qcr

20

15

10

5

0 0 20 40 60 80 100

Black hole mass M