VOLUME 52, NUMBER 16 PHYSICAL REVIEW LETTERS Oscillations of a Black Hole 16 APRIL 1984 Valeria Ferrari ' and Bahram Mashhoon 1nstitut fiir Theoretische Physik, Universita tzu'Koln, D 500-0 Cologne 41, Federal Republic of Germany (Received 24 August 1983) A new analytic method for the determination of quasinormal oscillations of a black hole is presented. It is based on a connection between the quasinormal modes and the bound states of the inverted black-hole potentials. With use of this method, the quasinormal frequencies of a Schwarzschild, a Reissner-Nordstrom, and a slowly rotating Kerr black hole are deter- mined in the eikonal approximation, Moreover, we provide evidence for the stability of an extreme Kerr black hole. PACS numbers: 04.30. + x, 03.80, + r, 04.20.Jb, 97.60.Lf Massive rotating black holes may provide the en- ergy source for the activity observed in galactic nu- clei and quasistellar sources. ' Rotating black holes are also expected to be strong sources of gravita- tional radiation and the detection of this radiation in the laboratory may provide direct evidence for their existence. Over the past decade, numerical investi- gations of the response of a black hole to external perturbations have revealed a common feature: At late times the black-hole response is generally dominat- ed by characteristic damped oscillations. Incident matter or radiation excites the quasinormal modes (QNM's) which are then manifested in the black- hole oscillations at late times. These vibrations therefore carry the imprint of a black hole on its response to external perturbations. For the dom- inant excitations of a black hole, the oscillations are strongly damped. It has been suggested, however, that extreme Kerr black holes are, in some sense, marginally unstable since for rapidly rotating black holes (near the extreme Kerr limit) some of the quasinormal oscillations are rather weakly damped, and the damping factor approaches zero in the ex- treme Kerr limit. In this Letter we present a gen- eral method for determining the QNM's of a black hole and show, in particular, that for a rotating black hole the contribution of the weakly damped modes to the black-hole response is small and it, in fact, vanishes in the extreme Kerr limit, so that these modes cannot render the extreme Kerr black hole marginally unstable; moreover, they are not expected to be significant for gravitational wave ex- periments. A general gravitational (or electromagnetic) per- turbation of a Kerr black hole may be written as a sum of modes each of the form4 ei(rat — ms)+, (g)~, (t ) where ao, j, and m are the frequency and the angular momentum parameters of the perturbation, and s indicates the spin of the perturbing field. The radi- U(x;p) = U( — ix;p'). (3) al part R„J, (r) = Q(x) satisfies the equation d'y/dx'+ [to' — V(x;to) ]y =~ (x;co), where x is the generalized Regge-Wheeler coordi- nate and V(x;to) is a real effective potential. 6 The QNM's are defined to be the solutions of the homogeneous form of Eq. (2) with the boundary conditions corresponding to outgoing waves at spa- tial infinity and incoming waves at the horizon. Though QNM's are of the form (1), they do not in general represent a perturbation of the black hole since Q may not be finite everywhere. Consider ra- diation of frequency cu incident on the black hole from infinity and let R (co) and T(to) be the reflec- tion and transmission amplitudes, respectively. The QNM's correspond to the singularities of R (z), the extension of R (to) to the complex frequency plane, such that Re(z) A0 and T(z)/R (z) is regular. Kramers and Heisenberg pointed out long ago' that the singularities of the scattering amplitude are re- lated to the bound states of the potential. A black hole has in general an effective barrier potential; nevertheless, a method has been developed that connects the QNM's to the bound states of the in verted black-hole potentials. ' To illustrate this method, consider first a spheri- cal black hole where the effective potential U(x) is nonnegative and falls off exponentially for x — ~ and as x 2 for x +~. It can be shown that the quasinormal frequencies must all be complex and distributed symmetrically with respect to the imaginary axis in the upper half of the com- plex plane. Let p be a set of parameters which may belong to the potential; or they may be introduced as scaling factors. The potential and the modes are then functions of these parameters as well, U= U(x;p), Q=Q(x;p), and co=co(p). Consider the formal transformations x — ix, p p' = n (p) such that the potential is left invariant: 1984 The American Physical Society 1361