3.2 Kerr black holes

The Kerr metric represents an axisymmetric, black hole solution to the source free Einstein equations. The metric in (t,r,πœƒ,φ ) coordinates is
( ) 2 ds2 = − 1 − 2M--r dt2 − 4M-ar-sin-πœƒdtd φ + Σ-dr2 + Σd πœƒ2 ( Σ Σ) Δ 2M a2rsin2πœƒ + r2 + a2 + ------------- sin2πœƒd φ2, (33 ) Σ
with
2 2 2 2 2 Δ = r − 2M r + a , Σ = r + a cos πœƒ. (34 )
M is the mass and 0 ≤ a ≤ M the rotational parameter of the Kerr metric. The zeros of Δ are
2 2 1βˆ•2 r± = M ± (M − a ) , (35 )
and determine the horizons. For r+ ≤ r < ∞ the spacetime admits locally a timelike Killing vector. In the ergosphere region
r ≤ r < M + (m2 − a2 cos2πœƒ)1βˆ•2, (36 ) +
the Killing vector ∂βˆ•∂t which is timelike at infinity, becomes spacelike. The scalar wave equation for the Kerr metric is
[ ] [ ] (r2 +-a2-)2 2 2 ∂2χ- 4M--ar ∂2-χ- a2- --1--- ∂2-χ Δ − a sin πœƒ ∂t2 + Δ ∂t∂φ + Δ − sin2πœƒ ∂ φ2 ( ) ( ) [ ] − Δ −σ ∂- Δ σ+1∂-χ − -1---∂-- sinπœƒ ∂χ- − 2σ a(r-−-M-)-+ i-cosπœƒ ∂χ- ∂r ∂r sin πœƒ ∂πœƒ ∂πœƒ Δ sin2 πœƒ ∂φ [ 2 2 ] ( ) − 2σ M-(r-−--a-)− r − ia cosπœƒ ∂χ-+ σ2cot2 πœƒ − σ χ = 0, (37 ) Δ ∂t
where σ = 0,±1, ±2 for scalar, electromagnetic or gravitational perturbations, respectively. As the Kerr metric outside the horizon (r > r+) is globally hyperbolic, the Cauchy problem for the scalar wave equation (37View Equation) is well posed for data on any Cauchy surface. However, the coefficient of ∂2χβˆ•∂ φ2 becomes negative in the ergosphere. This implies that the time independent equation we obtain after the Fourier or Laplace transformation is not elliptic!

For linear hyperbolic equations with time independent coefficients, we know that solutions determined by data with compact support are bounded by ce γt, where γ is independent of the data. It is not known whether all such solutions are bounded in time, i.e. whether they are stable.

Assuming harmonic time behavior χ = eiωtˆχ(r,πœƒ,φ), a separation in angular and radial variables was found by Teukolsky [196]:

imφ ˆχ(r,πœƒ,φ ) = R(r,ω )S (πœƒ,ω)e . (38 )
Note that in contrast to the case of spherical harmonics, the separation is ω-dependent. To be a solution of the wave equation (37View Equation), the functions R and S must satisfy
[ ] [ 2 2 ] --1--d-- sin πœƒdS- + a2ω2 cos2πœƒ + 2a ωσ cosπœƒ − m--+-σ-+2m--σ-cosπœƒ+E S = 0, (39 ) sinπœƒ dπœƒ d πœƒ sin2 πœƒ
[ ] [ ] Δ −σ-d- Δ σ+1dR- + 1- K2 + 2iσ(r − 1)K − Δ(4iσr ω + λ) R = 0, (40 ) dr dr Δ
where 2 2 K = (r + a )ω + am, 2 2 λ = E + a ω + 2am ω − σ(σ + 1), and E is the separation constant.

For each complex a2 ω2 and positive integer m, Equation (39View Equation) together with the boundary conditions of regularity at the axis, determines a singular Sturm–Liouville eigenvalue problem. It has solutions for eigenvalues E(β„“,m2, a2ω2 ), |m | ≤ β„“. The eigenfunctions are the spheroidal (oblate) harmonics Sβ„“|m |(πœƒ). They exist for all complex 2 ω. For real 2 ω the spheroidal harmonics are complete in the sense that any function of z = cosπœƒ, absolutely integrable over the interval [− 1,1], can be expanded into spheroidal harmonics of fixed m [181]. Furthermore, functions A(πœƒ,φ ) absolutely integrable over the sphere can be expanded into

∑∞ +∑ β„“ A (πœƒ,φ) = A (β„“,m, ω)Sβ„“|m |(πœƒ)eimφ. (41 ) β„“=0m= −β„“
For general complex 2 ω such an expansion is not possible. There is a countable number of “exceptional values” 2 ω where no such expansion exists [148Jump To The Next Citation Point].

Let us pick one such solution Sβ„“|m |(πœƒ,ω ) and consider some solutions R(r,ω ) of (40View Equation) with the corresponding E(β„“,|m |,ω ). Then R β„“|m|(r,ω)Sβ„“|m|eim φe−iωt is a solution of (37View Equation). Is it possible to obtain “all” solutions by summing over β„“,m and integrating over ω? For a solution in spacetime for which a Fourier transform in time exists at any space point (square integrable in time), we can expand the Fourier transform in spheroidal harmonics because ω is real. The coefficient R (r,ω,m ) will solve Equation (40View Equation). Unfortunately, we only know that a solution determined by data of compact support is exponentially bounded. Hence we can only perform a Laplace transformation.

We proceed therefore as in Section 2. Let χˆ(s,r,πœƒ,φ) be the Laplace transform of a solution determined by data χ (t,r,πœƒ,φ ) = 0 and ∂ χ(t,r,πœƒ,φ ) = ρ t, while χ is analytic in s for real s > γ ≥ 0, and has an analytic continuation onto the half-plane Re (s) ≥ γ. For real s we can expand χˆ into a converging sum of spheroidal harmonics [148]

∞ +β„“ ˆχ(s,r,πœƒ,φ ) = ∑ ∑ R (β„“,m, s)S (− s2,πœƒ)eimφ. (42 ) β„“=0 m= −β„“ β„“|m|
R (β„“,m, s) satisfies the radial equation (9View Equation) with iω = s. This representation of ˆχ does, however, not hold for all complex values in the half-plane on which it is defined. Nevertheless is it true that for all values of s which are not exceptional an expansion of the form (42View Equation) exists2.

To define quasi-normal modes we first have to define the correct Green function of (40View Equation) which determines R(β„“,m, s) from the data ρ. As usual, this is done by prescribing decay for real s for two linearly independent solutions ±R (β„“,m, s) and analytic continuation. Out of the Green function for R (β„“,m, s) and Sβ„“|m |(− s2,πœƒ) for real s we can build the Green function G (s,r,r′,πœƒ,πœƒ′,Ο•,Ο• ′) by a series representation like (42View Equation). Analytic continuation defines G on the half plane Re (s) > γ. For non exceptional s we have a series representation. (We must define G by this complicated procedure because the partial differential operator corresponding to (39View Equation), (40View Equation) is not elliptic in the ergosphere.)

Normal and quasi-normal modes appear as poles of the analytic continuation of G. Normal modes are determined by poles with Re(s) > 0 and quasi-normal modes by Re (s) < 0. Suppose all such values are different from the exceptional values. Then we have always the series expansion of the Green function near the poles and we see that they appear as poles of the radial Green function of R (β„“,m, s).

To relate the modes to the asymptotic behavior in time we study the inverse Laplace transform and deform the integration path to include the contributions of the poles. The decay in time is dominated either by the normal mode with the largest (real) eigenfrequency or the quasi-normal mode with the largest negative real part.

It is apparent that the calculation of the QNM frequencies of the Kerr black hole is more involved than the Schwarzschild and Reissner–Nordström cases. This is the reason that there have only been a few attempts [135Jump To The Next Citation Point185Jump To The Next Citation Point123Jump To The Next Citation Point160] in this direction.

The quasi-normal mode frequencies of the Kerr–Newman black hole have not yet been calculated, although they are more general than all other types of perturbation. The reason is the complexity of the perturbation equations and, in particular, their non-separability. This can be understood through the following analysis of the perturbation procedure. The equations governing a perturbing massless field of spin σ can be written as a set of 2 σ + 1 wavelike equations in which the various different helicity components of the perturbing field are coupled not only with each other but also with the curvature of the background space, all with four independent variables as coordinates over the manifold. The standard problem is to decouple the 2σ + 1 equations or at least some physically important subset of them and then to separate the decoupled equations so as to obtain ordinary differential equations which can be handled by one of the previously stated methods. For a discussion and estimation of the QNM frequencies in a restrictive case refer to [124Jump To The Next Citation Point].


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