Black hole stability in EinsteinGauss-Bonnet gravity
In collaboration with Reinaldo Gleiser
Class & Quant Gravity 22 (2005) L1
and gr-qc/0503117
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
1
Einstein-Gauss-Bonnet gravity
G( 0 )
Tb
a
a
G(1)
b
 G b   g b  k R b 

a
a
a

a
b
1
2
a
Rg
b
 
G( 2 )
a
b
Special case of Lovelock gravity:
G( p)
a
b
 g
[a
[b
R
i1i 2
i1i 2
R
i3 i 4
i3 i 4
..... R
i 2 p 1ii
2p
]
i 2 p 1i 2 p ]
[( d  1 ) / 2 ]
Gb 
a

p0
c p G( p)
a
b
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
2
Some static, symmetric solutions
ds
2
  f ( r ) dt 
2
dr
2
 r hij dx dx


f (r )
2
i
j
horizonmanifold

 n , dim n , curv
f (r )    r  (r )
2
 ´
2
Cosmological solutions:

 k ´   ´ 
r
n 1
  0
Black hole solutions:
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
  0
3
Linear perturbations
g ab  g ab   hab
h ij ( t , r , x )  r  ( r , t ) h ij ( x )
Tensor type:
h ij g
ij
(h transverse traceless)
2
0,
D h ij  0 ,
i
D D k h ij   h ij
k
Vector type: h  i  F  V i , hij  HV ij
D Vi  0 ,
i
D D k Vi   Vi ,
k
V ( ij )  D ( i V j )
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
4
Tensor linear perturbations
 (r , t )  K (r )  (r ) e
 t
get a (stationary) Schrödinger-like equation for
d 
 (r )
2
dr
2
 V      E 
2
negative E value in spectrum signals instability
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
5
S-deformation technique
E

lower bound of
r2
 * ( "  V) dr  , H
r1
, H 

r2
r1
D 
d
dr
S
2
D
dr 

r2
~
2
V |  | dr
r1
dS
~
2
V V 
S
dr
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
6
Found S deformed potential that factors Laplace-Beltrami
~
eigenvalue out, V  ( 2   )U
, H 

r2
r1
2
D
r2
2
dr  (2   ) U |  | dr
r1
(2   )  0 and grows for higher harmonics , thus:
U (r )  0 for all r  (r1 , r2 )  stable spacetime
U (r )  0 for some r  (r1 , r2 )  unstable spacetime
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
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Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
8
What about vector and scalar
perturbations ?
Vector mode:
-Could work out the Schrödinger potential, up to spacetime
dimension 11
-Found S-deformed potential that factors out harmonic
eigenvalue in every case.
-Analysis not done yet.
Scalar mode:
-Preliminary computations suggest there is no
convenient S-deformation that gives a factored potential
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
9
Stability under tensor
perturbations
•Cosmological solutions found to be stable.
(positive string tension) BHs are stable if the spacetime
dimension is different from 6.
•D = 6 BH instability: BHs with positive curvature horizons
are unstable if their masses are smaller than a critical value.
•D = 6 BH instability: BHs with negative curvature horizons
are unstable if their masses are bigger than a critical value.
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
10
Conclusions
• Linearization of EGB gravity around symmetric static
solutions give eqns that can be cast in Schrödinger form.
• The Schrödinger potentials reproduce previously known
results in the Einstein gravity limit.
• For tensor and vector perturbations, S-deformed
potentials can be found such that definite conclusions
about stability can be arrived to in every case. This does
not happen for scalar perturbations.
• 6 dimensional BHs are special, there is a critical mass
below (above) which those with positively (negatively)
curved horizons become unstable.
Problemas Actuales en Física de
Gravitación - Córdoba - 28-05-2005
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Black hole stability in Einstein-Gauss