perturbations around black hole solutions

Elcio Abdalla

Perturbations around Black Hole solutions

Classical (non-relativistic) Black Hole

• The escape velocity is equal to the velocity of light

• Therefore,

The Schwarzschild Black Hole• Birckhoff Theorem: a static spherically

Symmetric solution must be of the form

• Schwartzschild solution in D=d+1 dimensions (d>2):

Properties of the BH solution• Considering the solution for a large (not too heavy) cluster

of matter (i.e. radius of distribution > 2M, G=1=c). In this case one finds the Newtonian Potential

• For heavy matter (namely highly concentrated) with radius

R< 2M there is a so called event horizon where the g00 vanishes. To an outsider observer, the subject falling into the Black Hole takes infinite time to arrive at R=2M

Properties of the BH solution• Only the region r> 2M is relevant to external observers.

• Law of Black Hole dynamics: BH area always grows

• Quantum gravity: BH entropy equals 1/4 of BH area

• No hair theorem: BH can only display its mass (attraction), charge (Gauss law) and angular momentum (precession of gyroscope) to external observers

Reissner-Nordstrom solution

• For a Black Hole with mass M and charge q, in 4 dimensions, we have the solution

Cosmological Constant

• Einstein Equations with a nonzero cosmological constant are

• Λ>0 corresponds to de Sitter space

• Λ<0 corresponds to Anti de Sitter space

Lovelock Gravity

Lovelock Gravity

Black Holes with nontrivial topology

Black Holes with nontrivial topology

Black Holes with nontrivial topology

2+1 dimensional BTZ Black Holes

• General Solution

where J is the angular momentum

2+1 dimensional BTZ Black Holes

• AdS space

• where -l2 corresponds to the inverse of the cosmological constant Λ

Quasi Normal Modes

• First discovered by Gamow in the context of alpha decay

• Bell ringing near a Black Hole

• Can one listen to the form of the Black Hole?

• Can we listen to the form of a star?

Quasi normal modes expansion

QNMs were first pointed out in calculations of the scattering of gravitational waves by Schwarzschild black holes.

Due to emission of gravitational waves the oscillation mode frequencies become complex,

the real part representing the oscillation

the imaginary part representing the damping.

Wave dynamics in the asymptotically flat space-time

Schematic Picture of the wave evolution:

• Shape of the wave front (Initial Pulse)

• Quasi-normal ringingUnique fingerprint to the BH existence

Detection is expected through GW observation

• RelaxationK.D.Kokkotas and B.G.Schmidt, gr-qc/9909058

Excitation of the black hole oscillation

• Collapse is the most frequent source for the excitation of BH oscillation.

Many stars end their lives with a supernova explosion. This will leave behind a compact object which will oscillate violently in the first few seconds. Huge amounts of gravitational radiation will be emitted.

• Merging two BHs• Small bodies falling into the BH.• Phase-transition could lead to a sudden

contraction

Detection of QNM Ringing

• GW will carry away information about the BH• The collapse releases an enormous amount of

energy. • Most energy carried away by neutrinos. This is supported by the neutrino observations at the time of SN1987A.

• Only 1% of the energy released in neutrinos is radiated in GW

• Energy emitted as GW is of order 274 1010 cM

Sensitivity of Detectors

• Amplitude of the gravitational wave

for stellar BH

for galactic BHWhere E is the available energy, f the frequency and the r is the distance of the

detector from the source.

Anderson and Kokkotas, PRL77,4134(1996)

)15

()1

()10

(105 2/12/123

22

r

Mpc

f

kHz

cM

Eh

gw

)3

()1

()10

(103 2/12/123

18

r

Mpc

f

mHz

cM

Eh

gw

Sensitivity of Detectors

• An important factor for the detection of gravitational wave consists in the pulsation mode frequencies.

The spherical and bar detectors 0.6-3kHz

The interferometers are sensitive within 10-2000kHz

For the BH the frequency will depend on the mass and rotation:

10 solar mass BH 1kHz

100 solar mass BH 100Hz

Galactic BH 1mHz

Quasi-normal modes in AdS space-time

AdS/CFT correspondence:The BH corresponds to an approximately thermal state in the

field theory, and the decay of the test field corresponds to the decay of the perturbation of the state.The quasinormal frequencies of AdS BH have

direct interpretation in terms of the dual CFTJ.S.F.Chan and R.B.Mann, PRD55,7546(1997);PRD59,064025(1999)G.T.Horowitz and V.E.Hubeny, PRD62,024027(2000);CQG17,1107(2000)B.Wang et al, PLB481,79(2000);PRD63,084001(2001);PRD63,124004(2001);

PRD65,084006(2002)

Quasi normal modes in RN AdS

• We consider the metric

Quasi Normal Modes

• We can consider several types of perturbations

• A scalar field in a BH background obeys a curved Klein-Gordon equation

• An EM field obeys a Maxwell eq in a curved background

• A metric perturbation obeys Zerilli’s eq.

Quasi normal modes in RN AdS

• We use the expansion

Quasi normal modes in RN AdS

Quasi normal modes in RN AdS

• Decay constant as a function of the Black Hole radius

Quasi normal modes in RN AdS

• Dependence on the angular momentum (l)

Quasi normal modes in RN AdS

• Solving the numerical equation

Quasi normal modes in RN AdS

• Solving the numerical equation

Quasi normal modes in RN AdS

• Result of numerical integration

Quasi normal modes in RN AdS

• Approaching criticality

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in AdS topological Black Holes

Quasi normal modes in 2+1 dimensions

• For the AdS case

Quasi normal modes in 2+1 dimensional AdS BH

Exact agreement:

QNM frequencies & location of the poles of the retarded correlation function of the corresponding perturbations in the dual CFT.

A Quantitative test of the AdS/CFT correspondence.

Perturbations in the dS spacetimes

We live in a flat world with possibly a positive cosmological constant

Supernova observation, COBE satellite

Holographic duality: dS/CFT conjecture A.Strominger, hep-th/0106113

Motivation: Quantitative test of the dS/CFT conjecture E.Abdalla, B.Wang et al, PLB 538,435(2002)

Perturbations in dS spacetimes

• Small dependence on the charge of the BH

• Characteristic of space-time (cosmological constant)

2+1-dimensional dS spacetime

22

2212

2

2

22

2

2

2

22 )

2()

4()

4( dt

r

Jdrdr

r

J

l

rMdt

r

J

l

rMds

The metric of 2+1-dimensional dS spacetime is:

The horizon is obtained from

04 2

2

2

2

r

J

l

rM

Perturbations in the dS spacetimes

Scalar perturbations is described by the wave equation

Adopting the separation

The radial wave equation reads

0)(1 2

ggg

imti eerRrt )(),,(

Rg

Rmr

J

l

rMm

rrdr

dR

g

r

rdr

d

g rrrrrr

222

22

22

2 1])(

1[)(

1

Perturbations in the dS spacetimes

Using the Ansatz

The radial wave equation can be reduced to the hypergeometric equation

)()1()( zFzzzR

0])1([)1(2

2

abFdz

dFzbac

dz

Fdzz

Perturbations in the dS spacetimes

• For the dS case

Perturbations in the dS spacetimes

Investigate the quasinormal modes from the CFT side:

For a thermodynamical system the relaxation process of a small perturbation is determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation

Perturbations in the dS spacetimes

Define an invariant P(X,X’)associated to two points X and X’ in dS space

The Hadamard two-point function is defined as

Which obeys

BAAB XXXXP ')',(

0|)'(),(|0)',( XXconstXXG

0)',()( 22 XXGX

Perturbations in the dS spacetimes

We obtain

where

The two point correlator can be got analogously to

hep-th/0106113;

NPB625,295(2002)

)2/)1(,2/3,,(Re)( PhhFPG

2211 lh

**2

2)'(''lim rr

rG

l

rrddtdtd

Perturbations in the dS spacetimes

Using the separation:

The two-point function for QNM is

imti eerRrt )(),,(

)2

2/2/2/()

2

2/2/2/(

)2

2/2/2/()

2

2/2/2/()'(

]2

))((sinh

2

))((sinh2[

)''''exp(''

'

22

T

limh

T

limh

T

limh

T

limh

l

tilrir

l

tilrirtiimtiim

ddtdtd

mm

h

Perturbations in the dS spacetimes

The poles of such a correlator corresponds exactly to the QNM obtained from the wave equation in the bulk.

This work has been recently extended to four-dimensional dS spacetimes hep-th/0208065

These results provide a quantitative support of the dS/CFT correspondence

Conclusions and Outlook• Importance of the study in order to foresee

gravitational waves

• Comprehension of Black Holes and its cosmological consequences

• Relation between AdS space and Conformal Field Theory

• Relation between dS space and Conformal Field Theory

• Sounds from gravity at extreme conditions