Kerr black holes

3.2 Kerr black holes

The Kerr metric represents an axisymmetric, black hole solution to the source free Einstein equations. The metric in $(t,r,𝜃,φ )$ coordinates is

( ) 2 ds2 = − 1 − 2M--r dt2 − 4M-ar-sin-𝜃dtd φ + Σ-dr2 + Σd 𝜃2 ( Σ Σ) Δ 2M a2rsin2𝜃 + r2 + a2 + ------------- sin2𝜃d φ2, (33 ) Σ

with

$M$ is the mass and $0 ≤ a ≤ M$ the rotational parameter of the Kerr metric. The zeros of $Δ$ are

and determine the horizons. For $r+ ≤ r < ∞$ the spacetime admits locally a timelike Killing vector. In the ergosphere region

the Killing vector $∂∕∂t$ 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 𝜃 ∂t2 + Δ ∂t∂φ + Δ − sin2𝜃 ∂ φ2 ( ) ( ) [ ] − Δ −σ ∂- Δ σ+1∂-χ − -1---∂-- sin𝜃 ∂χ- − 2σ a(r-−-M-)-+ i-cos𝜃 ∂χ- ∂r ∂r sin 𝜃 ∂𝜃 ∂𝜃 Δ sin2 𝜃 ∂φ [ 2 2 ] ( ) − 2σ M-(r-−--a-)− r − ia cos𝜃 ∂χ-+ σ2cot2 𝜃 − 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 (37

) is well posed for data on any Cauchy surface. However, the coefficient of $∂2χ∕∂ φ2$ 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 $χ = eiωtˆχ(r,𝜃,φ)$ , a separation in angular and radial variables was found by Teukolsky [196 ]:

Note that in contrast to the case of spherical harmonics, the separation is $ω$ -dependent. To be a solution of the wave equation (37

), the functions $R$ and $S$ must satisfy

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 (39 ) together with the boundary conditions of regularity at the axis, determines a singular Sturm–Liouville eigenvalue problem. It has solutions for eigenvalues $E(ℓ,m2, a2ω2 )$ , $|m | ≤ ℓ$ . The eigenfunctions are the spheroidal (oblate) harmonics $Sℓ|m |(𝜃)$ . They exist for all complex $2 ω$ . For real $2 ω$ the spheroidal harmonics are complete in the sense that any function of $z = cos𝜃$ , absolutely integrable over the interval $[− 1,1]$ , can be expanded into spheroidal harmonics of fixed $m$ [181 ]. Furthermore, functions $A(𝜃,φ )$ absolutely integrable over the sphere can be expanded into

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 [148

Let us pick one such solution $Sℓ|m |(𝜃,ω )$ and consider some solutions $R(r,ω )$ of (40 ) with the corresponding $E(ℓ,|m |,ω )$ . Then $R ℓ|m|(r,ω)Sℓ|m|eim φe−iωt$ is a solution of (37 ). Is it possible to obtain “all” solutions by summing over $ℓ,m$ 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 (40 ). 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 $χˆ(s,r,𝜃,φ)$ be the Laplace transform of a solution determined by data $χ (t,r,𝜃,φ ) = 0$ and $∂ χ(t,r,𝜃,φ ) = ρ t$ , 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 ]

$R (ℓ,m, s)$ satisfies the radial equation (9

) 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 (42

) exists² .

To define quasi-normal modes we first have to define the correct Green function of (40 ) which determines $R(ℓ,m, s)$ from the data $ρ$ . As usual, this is done by prescribing decay for real $s$ for two linearly independent solutions $±R (ℓ,m, s)$ and analytic continuation. Out of the Green function for $R (ℓ,m, s)$ and $Sℓ|m |(− s2,𝜃)$ for real $s$ we can build the Green function $G (s,r,r′,𝜃,𝜃′,ϕ,ϕ ′)$ by a series representation like (42 ). 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 (39 ), (40 ) 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 $R (ℓ,m, s)$ .

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 [135 , 185 , 123 , 160 ] 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 [124 ].