Quasinormal modes of extremal black holes

Abstract

The continued fraction method (also known as Leaver's method) is one of the most effective techniques used to determine the quasinormal modes of a black hole. For extremal black holes, however, the method does not work (since, in such a case, the event horizon is an irregular singular point of the associated wave equation). Fortunately, there exists a modified version of the method, devised by Onozawa et al. [Phys. Rev. D 53, 7033 (1996)], which works for neutral massless fields around an extremal Reissner-Nordström black hole. In this paper, we generalize the ideas of Onozawa et al. to charged massless perturbations around an extremal Reissner-Nordström black hole and to neutral massless perturbations around an extremal Kerr black hole. In particular, the existence of damped modes is analysed in detail. Similarities and differences between the results of the original continued fraction method for near extremal black holes and the results of the new continued fraction method for extremal black holes are discussed. Mode stability of extremal black holes is also investigated.

Spectroscopy of extremal and near-extremal Kerr black holes

“…We accompany this investigation with a similar one in NEK, which helps understand better the results in extremal Kerr. In particular, we numerically look both for poles and branch points of the retarded Green function in the upper frequency plane in extremal Kerr and report that we do not find any (in the case of poles, this is in agreement with [41,42]). We also give a simple analytical argument against the existence of poles in the upper plane.…”

“…In other words, there is no yet proof that in extremal Kerr there exist no unstable modes with m = 0, whether as poles or as branch points of the Green function in the upper frequency plane. We note, however, that there exists numerical support for the non-existence of such modes: in [40] by numerically solving the equation obeyed by azimuthal-m mode field perturbations, and in [41,42] by numerically looking for poles of the Green function in the upper complex-frequency plane and not finding any.…”

“…With regards to the lower frequency plane, we calculate and tabulate QNM frequencies for both NEK and extremal Kerr black holes. To the best of our knowledge, Richartz's [42] is the only work in the literature where QNMs of extremal Kerr black holes have been calculated. We reproduce Richartz's QNM values, extend the precision to 16 digits and obtain new frequencies for higher overtones (i.e., for larger magnitude of the imaginary part of the frequency).…”

“…We reproduce Richartz's QNM values, extend the precision to 16 digits and obtain new frequencies for higher overtones (i.e., for larger magnitude of the imaginary part of the frequency). Furthermore, we provide values for QNMs in extremal Kerr which were missed by [42] in extremal Kerr as well as by analyses [45,47] in NEK -notably, for the important gravitational modes with = m = 2 and 3 (the first one was observed -although not tabulated -in NEK in [43], the latter was not). Similarly, in NEK, our QNM values extend the precision to 16 digits of those tabulated in [44] and we include values for extra modes (particularly for the socalled zero-damping modes [45]).…”

An approach to stability analyses in general relativity via symplectic geometry

“…Note that the event (t, r) = (t s (0), 0) is now at (R, u) = (0, 0). The above differential equation (35) has a singular point at (X, u) = (0, 0) and if there exist null geodesics that meet this singularity, we can write along them,…”