For linear hyperbolic equations with time independent coefficients, we know that solutions determined by data with compact support are bounded by , where 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 , a separation in angular and radial variables was found by Teukolsky :
For each complex and positive integer , Equation (39) together with the boundary conditions of regularity at the axis, determines a singular Sturm–Liouville eigenvalue problem. It has solutions for eigenvalues , . The eigenfunctions are the spheroidal (oblate) harmonics . They exist for all complex . For real the spheroidal harmonics are complete in the sense that any function of , absolutely integrable over the interval , can be expanded into spheroidal harmonics of fixed . Furthermore, functions absolutely integrable over the sphere can be expanded into.
Let us pick one such solution and consider some solutions of (40) with the corresponding . Then is a solution of (37). Is it possible to obtain “all” solutions by summing over and integrating over ? 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 is real. The coefficient 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 be the Laplace transform of a solution determined by data and , while is analytic in for real , and has an analytic continuation onto the half-plane . For real we can expand into a converging sum of spheroidal harmonics 2.
To define quasi-normal modes we first have to define the correct Green function of (40) which determines from the data . As usual, this is done by prescribing decay for real for two linearly independent solutions and analytic continuation. Out of the Green function for and for real we can build the Green function by a series representation like (42). Analytic continuation defines on the half plane . For non exceptional we have a series representation. (We must define 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 . Normal modes are determined by poles with and quasi-normal modes by . 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 .
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 can be written as a set of 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 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 .
© Max Planck Society and the author(s)