Black-hole normal modes: A WKB approach. II. Schwarzschild black holes

Abstract

We employ a semianalytic technique, based on a modified WKB approach, to determine the complex normal-mode frequencies of Schwarzschild black holes. It yields a simple analytic formula that gives the real and imaginary parts of the frequency in terms of the parameters of the black hole and of the field whose perturbation is under study, and in terms of the quantity (n ++), where n =0, k 1 , k2, . . . , and labels the fundamental mode, first overtone mode, and so on. In the case of the fundamental gravitational normal modes of the Schwarzschild black hole, the WKB estimates agree with numerical results to better than 0.13% in the real part of the frequency and 0.22% in the imaginary part. The agreement for both the real and imaginary parts of the low overtones is better than 0.5%. The relative agreement improves with increasing angular harmonic.

Quasinormal modes of charged, dilaton black holes

“…This approach has been applied to the Schwarzchild [17] and Reissner Nordström [18] cases, and for the fundamental quadrupole mode l = 2 agrees with other approaches [19] within 1% both for the real and the immaginary parts of the first 3-4 modes. The agreement improves with increasing angular harmonic and decreasing mode number.…”

Split fermion quasinormal modes

“…Note that this analytic method has been used extensively in various BH cases [16], where comparisons with other numerical results have been found to be accurate up to around 1% for both the real and the imaginary parts of the frequencies for low-lying modes with n < l (where n is the mode number and l is the spinor angular momentum quantum number). Furthermore, we have also included the n = l modes in our results, but the inclusion…”

“…It is worth mentioning that in the spin-1/2 case it does not seem possible to solve for r max for an arbitrary value of d, that is, to find an analytic solution for the roots (unlike the case for fields of other spins [16]). In the case of a bulk spin-0 field and the graviton tensor perturbations on a d-dimensional Schwarzschild background a similar analytic expression can be found for r max in d dimensions, see reference [18].…”

An extensive search for overtones in Schwarzschild black holes

“…We have compared our data to those generated via 6th-order WKB computations, from [5] and 3rd-order WKB from [6]. When it comes to notation, we have employed for the multipole index (as before), n for the overtone index (n = 0 for the fundamental, n = 1 for the first overtone, and so on).…”