A Summary of the Black Hole Perturbation Theory. Steven Hochman

Transcription

1 A Summary of the Black Hole Perturbation Theory Steven Hochman

2 Introduction Many frameworks for doing perturbation theory The two most popular ones Direct examination of the Einstein equations -> Zerilli-Regge-Wheeler equations for Schwarzschild. Newman-Penrose formalism -> Bardeen-Press equation for the Schwarzschild type, and the Teukolsky equation for Kerr type black holes.

3 The Metric In spherical polar coordinates the flat space Minkowski metric can be written as where ds 2 = dt 2 + dr 2 + r 2 dω 2 r 2 dω 2 = r 2 dθ 2 + r 2 sin 2 θdφ 2

4 Schwarzschild The Schwarzschild metric is a vacuum solution ds 2 = ( 1 2M r ) dt 2 + dr 2 ( 1 2M r ) + r 2 dω 2 The coordinates above fail at R = 2M

5 Killing Vectors Killing vectors tell us something about the physical nature of the spacetime. Invariance under time translations leads to conservation of energy Invariance under rotations leads to conservation of the three components of angular momentum. Angular momentum as a three-vector: one component the magnitude and two components the direction.

6 Killing Vectors of Schwarzchild Two Killing vectors: conservation of the direction of angular momentum -> we can choose pi = 2 for plane Energy conservation is shown in the timelike Killing vector K µ =( t ) µ = (1, 0, 0, 0) Magnitude of the angular momentum conserved by the final spacelike Killing vector R µ =( φ ) µ = (0, 0, 0, 1)

7 Geodesics in Schwarzschild The geodesic equation can be written after some simplification as 1 2 ( dr dλ ) 2 + V (r) =ε, The potential is V (r) = 1 2 ɛ ɛgm r + L2 2r 2 GML3 r 3

8 The Event Horizon and the Tortoise Null cones close up ( dt dr = ± 1 2GM r ) 1 Replace t with coordinate that moves more slowly where t = ±r + constant r = r +2GM ln ( r 2GM 1 )

9 More Tortoise ds 2 = ( 1 2GM r ) ( dt 2 + dr 2 )+r 2 dω 2 R = 2GM -> - infinity Transmission Reflection

10 Kruskal Coordinates ds 2 = 32G3 M 3 Null cones r e r/2gm ( dt 2 + dr 2 )+r 2 dω 2 Unlike the tortoise the event horizon is not infinitely far away, and is defined by Vishveshwara T = ±R + constant T = ±R

11 Kerr ds 2 = ( 1 2GMr ) ρ 2 dt 2 2GMar sin2 θ ρ 2 (dtdφ + dφdt)+ ρ2 dr2 +ρ 2 dθ 2 + sin2 θ ρ 2 [(r 2 + a 2 ) 2 a 2 sin 2 θ]dφ 2 where and Angular momentum (r) =r 2 2GMr + a 2 ρ 2 (r, θ) =r 2 + a 2 cos 2 θ

12 Einstein Field Equation R µν 1 2 Rg µν =8πGT µν Can also be written as R µν =8πG(T µν 1 2 Tg µν)

13 Perturbations For a perturbation Inserting this in But R µν =0 g µν = g µν + h µν R µν + δr µν =0 δr µν =0

14 Schwarzschild Perturbations Regge and Wheeler - Spherical Harmonics Stability? Gauge invariance? Physical Continuity? Ring down Zerilli - Falling particle

15 Tensor Harmonics Separate the solution into a product of four factors, each a function of a single coordinate. This separation is best achieved by generalizing the method of spherical harmonics already established for vectors, scalars, and spinors.

16 Parity Scalar functions have even parity. Two kinds of vectors, each of different parity: One the gradient of a the spherical harmonic and has even parity. The pseudogradient of the spherical harmonic, and has odd parity. There are three kinds of tensors. One is given by the double gradient of the spherical harmonic and has even parity. Another is a constant times the metric of the sphere, also with even parity. The last is obtained by taking the double pseudogradient; it has odd parity.

17 Even and Odd Odd/Magnetic/Axial parity = Y M L 0 0 h 0 (t, r)( 1 sin θ φ h 0(t, r)(sin θ)( ) θ 0 0 h 1 (t, r)( 1 sin θ φ h 1(t, r)(sin θ)( ) θ Sym Sym h 2 (t, r)[( 1 sin θ )( 2 θ φ ) 1 2 h 2(t, r)[( 1 sin θ 1 (cos θ)( sin 2 θ φ Sym Sym Sym h 2 (t, r)[(sin θ)( 2 θ φ 2 )( ) + (cos θ)( ) φ φ θ 2 sin θ θ θ )] ) (cos θ)( The odd waves contain three unknown functions: Even/Electric/Polar Parity = Y M L (h 0,h 1,h 2 ) The even waves contain seven unknown functions: (1 2M/r)H 0 (t, r) H 1 (t, r) h 0 (t, r)( ) θ h 0(t, r)( φ Sym (1 2M/r) 1 H 2 (t, r) h 1 (t, r)( ) θ h 1(t, r)( φ Sym Sym r[k(t, r) r 2 G(t, r)[( 2 / θ φ) +G(t, r)( 2 )] dθ 2 (cos θ)( 1 sin θ φ Sym Sym Sym r 2 [K(t, r) sin 2 θ +G(t, r)[( 2 ) φ 2 +(sin θ)(cos θ)( ) θ (H 0,H 1,H 2, G, K, h 0,h 1 ) φ )

18 Gauge Transformations The Regge-Wheeler gauge a is unique fixed gauge The quantities are gauge invariant Any result can be expressed in a gauge invariant manner by substituting the Regge-Wheeler gauge quantities in terms of a general gauge Consider x α = x α + ξ α g µν + h µν = g µν + ξ µ ; ν +ξ ν ; µ +h µν h new µν = h old µν + ξ µ ; ν +ξ ν.

19 Regge-Wheeler Gauge The gauge vector that simplifies the general odd wave has the form ξ 0 = 0; ξ 1 = 0; ξ µ = Λ(T, r); ɛ µν ( / x ν )Y M L (θ, φ), (µ, ν =2, 3) The final canonical form for an odd wave L, M = 0 is h odd µν = e ( ikt ) (sin θ)( / θ)p L (cos θ) The gauge vector that simplifies the general even wave has the form The final canonical form for an even wave L, M = 0 is h 0 (r) h 1 (r) Sym Sym 0 0 ξ 0 = M 0 (T, r)y M L (θ, φ); ξ 1 = M 1 (T, r)y M L (θ, φ); ξ 2 = M(T, r)( / θ)y M L (θ, φ); ξ 3 = M(T, r)(1/ sin 2 θ)( / φ)y M L (θ, φ). h even µν = e ( ikt ) P L (cos θ) H 0 (1 2M/r) H Sym H 2 (1 2M/r) r 2 K r 2 K sin 2 θ

20 The Choice of Gauge There are now only two unknown functions for the odd case and four for the even case This helps tremendously with the differential equations But even perturbations increase with distance and remain in unchanging magnitude for odd 1/r? We can choose another gauge (Radiation)

21 Solutions Even/Electric/Polar Odd/Magnetic/Axial L = 0,1,2... Static k=0

22 Solutions for L values There is no L = 0 odd/magnetic perturbation L = 0, L = 1 even and L = 1 odd: the changes from perturbations in mass, velocity, and angular momentum, have exact solutions. L >=2 describe the radiation, no exact solutions.

23 Odd/Magnetic Solutions For odd waves there are three non-trivial equations Can be expressed as a wave equation known as the Regge-Wheeler Equation In time domain d 2 Ψ odd dr 2 + k 2 (r)ψ odd =0 d 2 Ψ odd dr 2 d2 Ψ odd dt 2 + V (r)ψ odd =0, with V (r) =[ L(L + 1)/r 2 +6M/r 3 ](1 2M/r) L = 0 no perturbation L = 1 addition of angular momentum

24 Even/Electric Solutions For even waves there are seven non-trivial equations: One algebraic relation, three firstorder equations, and three second-order equations. Can be expressed as a wave equation known as the Zerilli Equation In time domain d 2 Ψ even dr 2 + k 2 (r)ψ even =0. with V (r) = ( 1 2M r d 2 Ψ even dr 2 )[ 1 λ 2 [ 72M 3 d2 Ψ even dt 2 + V (r)ψ even =0, )[ [ r 5 12M r 3 ( (L 1)(L + 2) ( 1 3M r )] ] + (L 1)L(L+1)(L+2) r 2 λ = L(L + 1) 2+ 6M r L = 0 addition of mass L = 1 shift of the cm

25 Solutions for L>=2 Radiation Can not solve the equations explicitly Asymptotically at large r the perturbation is the sum or two traces tensor harmonics. Using a Green's function formed from high frequency-limit solutions, we obtain amplitudes for the ingoing r=2m and outgoing r=infinity radiation for a particle falling radially into the black hole. The amplitude peaks at approximately 3/16piM Integrating this, the estimated total energy radiated is (1/625)(m 2 o/m ) To determine distribution in time use Fourier No static perturbations for L>=2

26 Stability The Schwarzschild metric background gives an equilibrium state. If the metric is perturbed, however, will it remain stable? The collapsed Schwarzschild metric must be proven to be stable against small perturbations. A problem with coordinates chosen by Regge-Wheeler prevented from judging whether any divergence shown by the perturbations at the surface was real or due to the coordinate singularity at r=2m. Using new Kruskal coordinates, Vishveshwara was able to determine background metric finite at the surface and the divergence of the perturbations with imaginary frequency time dependence violate the small perturbation assumption. Thus perturbations with imaginary frequencies are physically unacceptable and the metric is indeed stable.

27 Newman-Penrose Formalism The second popular method for solving perturbation equations is the Newman-Penrose (NP) formalism. The NP formalism is a notation for writing various quantities and equations that appear in relativity. It starts by considering a complex null tetrad such that equations that ( l, n, m, m) l n = 1 = m m The projections of the Weyl tensor (used heavily in NP formalism in place of G.. and R..) then become Ψ 0 = C µνρσ l µ m ν l ρ m σ Ψ 1 = C µνρσ l µ n ν l ρ m σ Ψ 2 = C µνρσ l µ m ν m ρ n σ Ψ 3 = C µνρσ l µ n ν m ρ n σ Ψ 4 = C µνρσ n µ m ν n ρ m σ.

28 Kerr Perturbations Due to the complexity of the Kerr metric, it becomes difficult to use the Einstein equations directly to get a solvable perturbation equation. To obtain the perturbation equation for rotating black holes, Teukolsky used the Newman-Penrose formalism. Skipping over much laborious calculation we arrive at the Teukolsky equation. [ (r 2 +a 2 ) 2 s r a 2 sin θ ] 2 Ψ ( s+1 Ψ r 2s [ M(r 2 a 2 ) where [ + 4Mar 2 Ψ ] ) t 2 ( ) 1 [ sin θ θ ( sin θ Ψ θ t ψ + [ a 2 ] 1 2 Ψ sin 2 θ φ 2 ) 2s [ a(r M) r ia cos θ ] Ψ t +(s2 cot θ s)ψ =0, (r) =r 2 2GMr + a 2 While not possible to achieve angular separation in the time domain, in the frequency domain it is separable. + i cos θ sin 2 θ ] Ψ φ

29 Connections When a=0 in the Teukolsky equation you are then left with the Bardeen-Press equation for Schwarzchild black holes. The Bardeen-Press equation contains in its real and imaginary parts the Zerilli and the Regge-Wheeler equations respectively.

30 References [1] B.F. Whiting, Class Notes from General Relativity I (Transcribed by various students - usually Shawn Mitryk, Gainesville, 2008). [2] S.M. Carroll, Spacetime and Geometry (Addison-Wesley, San Fransisco, 2004). [3] J.B. Hartle, Gravity (Addison-Wesley, San Fransisco, 2003). [4] B.F Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 1985). [5] B.F Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, 1980). [6] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992). [7] S. Chandrasekhar, Selected Papers, Volume 6: The Mathematical Theory of Black Holes and of Colliding Plane Waves (University of Chicago Press, Chicago, 1991). [8] D.J. Griffiths, Introduction to Electrodynamics, Third Edition (Prentice-Hall, Upper Saddle River, 1999). [9] D.J. Griffiths, Introduction to Quantum Mechanics, Second Edition (Prentice-Hall, Upper Saddle River, 2005). [10] T. Regge and J.A. Wheeler, Stability of a Schwarzschild Singularity Phys. Rev. 108, 1063 (1957). [11] F.J. Zerilli, Gravitational Field of a Particle Fal ling in a Schwarzschild Geometry Analyzed in Tensor Harmonics Phys. Rev. D 2, 2141 (1970). [12] F.J. Zerilli, Effective Potential for Even-Parity Regge-Wheeler Gravitational Perturbation Equations Phys. Rev. Lett 24, 737 (1970). [13] C.V. Vishveshwara, Stability of the Schwarzschild Metric Phys. Rev. D 1, 2870 (1970). [14] S. Chandrasekhar, On the Equations Governing the Perturbations of the Schwarzschild Black Hole Proc. R. Soc. 343, 289 (1975). [15] S. Chandrasekhar, and S. Detweiler, The Quasi-Normal Modes of the Schwarzschild Black Hole Proc R. Soc. 344, 441 (1975). [16] S. Chandrasekhar, On One-Dimensional Potential Barriers Having Equal Reflection and Transmission Coefficients Proc. R. Soc. 369, 425 (1980). [17] S. Chandrasekhar, and S. Detweiler, On the Equations Governing the Axisymmetric Perturbation of the Kerr Black Hole Proc R. Soc. 345, 145 (1975). [18] J.M. Bardeen, and W.H. Press, Radiation Fields in the Schwarzschild Background J. Math. Phys. 14, 7 (1972). [19] J.M. Stewart M. Walker, Perturbations of Space-Times in General Relativity Proc. R. Soc. 341, 49 (1974). [20] S.A. Teukolsky, Perturbations of a Rotating Black Hole Astrophys J. 185, 635 (1973).