Elcio Abdalla

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Perturbations around Black Hole solutions

• Elcio Abdalla

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Classical (non-relativistic) Black Hole

• The escape velocity is equal to the velocity of
  light
• Therefore,

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The Schwarzschild Black Hole

• Birckhoff Theorem a static spherically
• Symmetric solution must be of the form
• Schwartzschild solution in Dd1 dimensions
  (dgt2)

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Properties of the BH solution

• Considering the solution for a large (not too
  heavy) cluster of matter (i.e. radius of
  distribution gt 2M, G1c). In this case one finds
  the Newtonian Potential
• For heavy matter (namely highly concentrated)
  with radius
• Rlt 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 R2M

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Properties of the BH solution

• Only the region rgt 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

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Reissner-Nordstrom solution

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

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Cosmological Constant

• Einstein Equations with a nonzero cosmological
  constant are
• ?gt0 corresponds to de Sitter space
• ?lt0 corresponds to Anti de Sitter space

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Lovelock Gravity
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Lovelock Gravity
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Black Holes with nontrivial topology
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Black Holes with nontrivial topology
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Black Holes with nontrivial topology
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21 dimensional BTZ Black Holes

• General Solution
• where J is the angular momentum

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21 dimensional BTZ Black Holes

• AdS space
• where -l2 corresponds to the inverse of the
  cosmological constant ?

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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?

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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.

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Wave dynamics in the asymptotically flat
space-time

• Schematic Picture of the wave evolution
• Shape of the wave front (Initial Pulse)
• Quasi-normal ringing
• Unique fingerprint to the BH existence
• Detection is expected through GW observation
• Relaxation
• K.D.Kokkotas and B.G.Schmidt, gr-qc/9909058

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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

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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

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Sensitivity of Detectors

• Amplitude of the gravitational wave
• for stellar BH
• for galactic BH
• Where 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)

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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

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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 CFT
• J.S.F.Chan and R.B.Mann, PRD55,7546(1997)PRD59,06
  4025(1999)
• G.T.Horowitz and V.E.Hubeny, PRD62,024027(2000)CQ
  G17,1107(2000)
• B.Wang et al, PLB481,79(2000)PRD63,084001(2001)P
  RD63,124004(2001) PRD65,084006(2002)

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Quasi normal modes in RN AdS

• We consider the metric

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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 Zerillis eq.

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Quasi normal modes in RN AdS

• We use the expansion

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Quasi normal modes in RN AdS
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Quasi normal modes in RN AdS

• Decay constant as a function of the Black Hole
  radius

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Quasi normal modes in RN AdS

• Dependence on the angular momentum (l)

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Quasi normal modes in RN AdS

• Solving the numerical equation

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Quasi normal modes in RN AdS

• Solving the numerical equation

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Quasi normal modes in RN AdS

• Result of numerical integration

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Quasi normal modes in RN AdS

• Approaching criticality

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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in AdS topological Black Holes
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Quasi normal modes in 21 dimensions

• For the AdS case

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Quasi normal modes in 21 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.

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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)

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Perturbations in dS spacetimes

• Small dependence on the charge of the BH
• Characteristic of space-time (cosmological
  constant)

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21-dimensional dS spacetime
The metric of 21-dimensional dS spacetime is
The horizon is obtained from
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Perturbations in the dS spacetimes

• Scalar perturbations is described by the wave
  equation
• Adopting the separation
• The radial wave equation reads

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Perturbations in the dS spacetimes

• Using the Ansatz
• The radial wave equation can be reduced to the
  hypergeometric equation

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Perturbations in the dS spacetimes

• For the dS case

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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

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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

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Perturbations in the dS spacetimes

• We obtain
• where
• The two point correlator can be got analogously
  to
• hep-th/0106113
• NPB625,295(2002)

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Perturbations in the dS spacetimes

• Using the separation
• The two-point function for QNM is

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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

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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