A Summary of the Black Hole Perturbation Theory Steven Hochman - - PowerPoint PPT Presentation
A Summary of the Black Hole Perturbation Theory Steven Hochman Introduction Many frameworks for doing perturbation theory The two most popular ones Direct examination of the Einstein equations -> Zerilli-Regge-Wheeler equations for
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.
In spherical polar coordinates the flat space Minkowski metric can be written as where
ds2 = −dt2 + dr2 + r2dΩ2 r2dΩ2 = r2dθ2 + r2 sin2 θdφ2
The Schwarzschild metric is a vacuum solution The coordinates above fail at R = 2M
ds2 = −
r
dr2
r
+ r2dΩ2
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.
Two Killing vectors: conservation of the direction
plane Energy conservation is shown in the timelike Killing vector Magnitude of the angular momentum conserved by the final spacelike Killing vector
Kµ = (∂t)µ = (1, 0, 0, 0) Rµ = (∂φ)µ = (0, 0, 0, 1)
The geodesic equation can be written after some simplification as The potential is
V (r) = 1 2ǫ − ǫGM r + L2 2r2 − GML3 r3 1 2 dr dλ 2 + V (r) = ε,
Null cones close up Replace t with coordinate that moves more slowly where
dt dr = ±
r −1 t = ±r∗ + constant r∗ = r + 2GM ln
2GM − 1
R = 2GM -> - infinity Transmission Reflection
ds2 =
r
Null cones Unlike the tortoise the event horizon is not infinitely far away, and is defined by Vishveshwara
ds2 = −32G3M 3 r e−r/2GM(−dT 2 + dR2) + r2dΩ2 T = ±R + constant T = ±R
where and Angular momentum
ds2 = −
ρ2
ρ2 (dtdφ + dφdt) + ρ2 ∆ dr2 +ρ2dθ2 + sin2 θ ρ2 [(r2 + a2)2 − a2∆ sin2 θ]dφ2
∆(r) = r2 − 2GMr + a2 ρ2(r, θ) = r2 + a2 cos2 θ
Can also be written as
Rµν − 1 2Rgµν = 8πGTµν Rµν = 8πG(Tµν − 1 2Tgµν)
For a perturbation Inserting this in But
Rµν = 0 g′
µν = gµν + hµν
Rµν + δRµν = 0 δRµν = 0
Regge and Wheeler - Spherical Harmonics Stability? Gauge invariance? Physical Continuity? “Ring down” Zerilli - Falling particle
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.
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.
The odd waves contain three unknown functions: The even waves contain seven unknown functions:
Odd/Magnetic/Axial parity = Y M
L ×
−h0(t, r)(
1 sin θ)( ∂ ∂φ)
h0(t, r)(sin θ)( ∂
∂θ)
−h1(t, r)(
1 sin θ)( ∂ ∂φ)
h1(t, r)(sin θ)( ∂
∂θ)
Sym Sym h2(t, r)[(
1 sin θ)( ∂2 ∂θ∂φ) 1 2h2(t, r)[( 1 sin θ)( ∂2 ∂φ∂φ) + (cos θ)( ∂ ∂θ)
−(cos θ)(
1 sin2 θ)( ∂ ∂φ)]
− sin θ ∂2
∂θ∂θ)]
Sym Sym Sym −h2(t, r)[(sin θ)( ∂2
∂θ∂φ) − (cos θ)( ∂ ∂φ)
Even/Electric/Polar Parity = Y M
L ×
(1 − 2M/r)H0(t, r) H1(t, r) h0(t, r)( ∂
∂θ)
h0(t, r)( ∂
∂φ)
Sym (1 − 2M/r)−1H2(t, r) h1(t, r)( ∂
∂θ)
h1(t, r)( ∂
∂φ
Sym Sym r[K(t, r) r2G(t, r)[(∂2/∂θ∂φ) +G(t, r)( ∂2
dθ2)]
−(cos θ)(
1 sin θ)( ∂ ∂φ)]
Sym Sym Sym r2[K(t, r) sin2 θ +G(t, r)[( ∂2
∂φ2)
+(sin θ)(cos θ)( ∂
∂θ)
: (h0, h1, h2). (H0, H1, H2, G, K, h0, h1).
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 = xα + ξα
g′
µν + h′ µν = gµν + ξµ;ν +ξν;µ +hµν
hnew
µν
= hold
µν + ξµ;ν +ξν.
The gauge vector that simplifies the general odd wave has the form The final canonical form for an odd wave L, M = 0 is The gauge vector that simplifies the general even wave has the form The final canonical form for an even wave L, M = 0 is
ξ0 = 0; ξ1 = 0; ξµ = Λ(T, r); ǫµν(∂/∂xν)Y M
L (θ, φ), (µ, ν = 2, 3)
hodd
µν = e(−ikT)(sin θ)(∂/∂θ)PL(cos θ) ×
h0(r) h1(r) Sym Sym
ξ0 = M0(T, r)Y M
L (θ, φ);
ξ1 = M1(T, r)Y M
L (θ, φ);
ξ2 = M(T, r)(∂/∂θ)Y M
L (θ, φ); ξ3 = M(T, r)(1/ sin2 θ)(∂/∂φ)Y M L (θ, φ).
heven
µν
= e(−ikT)PL(cos θ) ×
H0(1 − 2M/r) H1 Sym H2(1 − 2M/r)−1 r2K r2K sin2 θ
There are now only two unknown functions for the
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)
Even/Electric/Polar Odd/Magnetic/Axial L = 0,1,2.... Static k=0
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.
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 with L = 0 no perturbation L = 1 addition of angular momentum
d2Ψodd dr∗2 + k2(r)Ψodd = 0
d2Ψodd dr∗2 − d2Ψodd dt2 + V (r)Ψodd = 0,
V (r) = [−L(L + 1)/r2 + 6M/r3](1 − 2M/r).
For even waves there are seven non-trivial equations: One algebraic relation, three first-
Can be expressed as a wave equation known as the Zerilli Equation In time domain with L = 0 addition of mass L = 1 shift of the cm
d2Ψeven dr∗2 + k2(r)Ψeven = 0. d2Ψeven dr∗2 − d2Ψeven dt2 + V (r)Ψeven = 0,
V (r) =
r 1 λ2
r5
− 12M
r3 (L − 1)(L + 2)
r
r2
r .
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 To determine distribution in time use Fourier No static perturbations for L>=2
is (1/625)(m2
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.
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 The projections of the Weyl tensor (used heavily in NP formalism in place of G.. and R..) then become
equations that (− → l , − → n , − → m, − → m)
− → l · − → n = 1 = −− → m · − → m.
Ψ0 = −Cµνρσlµmνlρmσ Ψ1 = −Cµνρσlµnνlρmσ Ψ2 = −Cµνρσlµmνmρnσ Ψ3 = −Cµνρσlµnνmρnσ Ψ4 = −Cµνρσnµmνnρmσ.
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
we arrive at the Teukolsky equation. where While not possible to achieve angular separation in the time domain, in the frequency domain it is separable.
(r2+a2)2
∆
− a2 sin θ
∂t2 + 4Mar ∆ ∂2Ψ ∂t∂ψ +
∆ − 1 sin2 θ
∂φ2
−∆−s ∂
∂r
∂r
1 sin θ ∂ ∂θ
∂θ
a(r−M)
∆
+ i cos θ
sin2 θ
∂φ
−2s
M(r2−a2)
∆
− r − ia cos θ
∂t + (s2 cot θ − s)Ψ = 0,
∆(r) = r2 − 2GMr + a2
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.
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