3.3 Stability and Completeness of 3 Quasi-Normal Modes of Black 3.1 Schwarzschild 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 tex2html_wrap_inline3004 coordinates is

  eqnarray359

with

equation366

M is the mass and tex2html_wrap_inline3008 the rotational parameter of the Kerr metric. The zeros of tex2html_wrap_inline3010 are

equation368

and determine the horizons. For tex2html_wrap_inline3012 the spacetime admits locally a timelike Killing vector. In the ergosphere region

equation371

the Killing vector tex2html_wrap_inline3014 which is timelike at infinity, becomes spacelike. The scalar wave equation for the Kerr metric is

  eqnarray374

where tex2html_wrap_inline3016 for scalar, electromagnetic or gravitational perturbations, respectively. As the Kerr metric outside the horizon tex2html_wrap_inline3018 is globally hyperbolic, the Cauchy problem for the scalar wave equation (37Popup Equation) is well posed for data on any Cauchy surface. However, the coefficient of tex2html_wrap_inline3020 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 tex2html_wrap_inline3022, where tex2html_wrap_inline3024 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 tex2html_wrap_inline3026, a separation in angular and radial variables was found by Teukolsky [196]:

equation401

Note that in contrast to the case of spherical harmonics, the separation is tex2html_wrap_inline2624 -dependent. To be a solution of the wave equation (37Popup Equation), the functions R and S must satisfy

  equation406

  equation413

where tex2html_wrap_inline3034, tex2html_wrap_inline3036, and E is the separation constant.

For each complex tex2html_wrap_inline3040 and positive integer m, equation (39Popup Equation) together with the boundary conditions of regularity at the axis, determines a singular Sturm-Liouville eigenvalue problem. It has solutions for eigenvalues tex2html_wrap_inline3044, tex2html_wrap_inline3046 . The eigenfunctions are the spheroidal (oblate) harmonics tex2html_wrap_inline3048 . They exist for all complex tex2html_wrap_inline3050 . For real tex2html_wrap_inline3050 the spheroidal harmonics are complete in the sense that any function of tex2html_wrap_inline3054, absolutely integrable over the interval [-1, 1], can be expanded into spheroidal harmonics of fixed m  [181]. Furthermore, functions tex2html_wrap_inline3060 absolutely integrable over the sphere can be expanded into

equation424

For general complex tex2html_wrap_inline3050 such an expansion is not possible. There is a countable number of ``exceptional values'' tex2html_wrap_inline3050 where no such expansion exists [148Jump To The Next Citation Point In The Article].

Let us pick one such solution tex2html_wrap_inline3066 and consider some solutions tex2html_wrap_inline3068 of (40Popup Equation) with the corresponding tex2html_wrap_inline3070 . Then tex2html_wrap_inline3072 is a solution of (37Popup Equation). Is it possible to obtain ``all'' solutions by summing over tex2html_wrap_inline3074 and integrating over tex2html_wrap_inline2624 ? 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 tex2html_wrap_inline2624 is real. The coefficient tex2html_wrap_inline3080 will solve equation (40Popup 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 tex2html_wrap_inline3082 be the Laplace transform of a solution determined by data tex2html_wrap_inline3084 and tex2html_wrap_inline3086, while tex2html_wrap_inline2626 is analytic in s for real tex2html_wrap_inline3092, and has an analytic continuation onto the half-plane tex2html_wrap_inline3094 . For real s we can expand tex2html_wrap_inline2662 into a converging sum of spheroidal harmonics [148]

  equation444

tex2html_wrap_inline3100 satisfies the radial equation (9Popup Equation) with tex2html_wrap_inline3102 . This representation of tex2html_wrap_inline2662 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 (42Popup Equation) exists Popup Footnote .

To define quasi-normal modes we first have to define the correct Green function of (40Popup Equation) which determines tex2html_wrap_inline3100 from the data tex2html_wrap_inline3114 . As usual, this is done by prescribing decay for real s for two linearly independent solutions tex2html_wrap_inline3118 and analytic continuation. Out of the Green function for tex2html_wrap_inline3100 and tex2html_wrap_inline3108 for real s we can build the Green function tex2html_wrap_inline3126 by a series representation like (42Popup Equation). Analytic continuation defines G on the half plane tex2html_wrap_inline3130 . For non exceptional s we have a series representation. (We must define G by this complicated procedure because the partial differential operator corresponding to (39Popup Equation), (40Popup 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 tex2html_wrap_inline3100 .

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 Point In The Article, 185Jump To The Next Citation Point In The Article, 123Jump To The Next Citation Point In The Article, 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 tex2html_wrap_inline2880 can be written as a set of tex2html_wrap_inline3146 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 tex2html_wrap_inline3146 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 In The Article].



3.3 Stability and Completeness of 3 Quasi-Normal Modes of Black 3.1 Schwarzschild Black Holes

image Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt
http://www.livingreviews.org/lrr-1999-2
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de