Download - 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