Quasi-normal modes for de Sitter–Reissner–Nordström black holes

Abstract

The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by black holes. Here we consider scattering of massless uncharged Dirac fields propagating in the outer region of de Sitter-Reissner-Nordström black hole, which is spherically symmetric charged exact solution of the Einstein-Maxwell equations. Using the spherical symmetry of the equation and restricting to a fixed harmonic the problem is reduced to a scattering problem for the 1D massless Dirac operator on the line. The resonances for the problem are related to the resonances for a certain semiclassical Schrödinger operator with exponentially decreasing positive potential. We give exact relation between the sets of Dirac and Schrödinger resonances. The asymptotic distribution of the resonances is close to the lattice of pseudopoles associated to the non-degenerate maxima of the potentials.Using the techniques of quantum Birkhoff normal form we give the complete asymptotic formulas for the resonances. In particular, we calculate the first three leading terms in the expansion. Moreover, similar results are obtained for the de Sitter-Schwarzschild quasi-normal modes, thus improving the result of Sá Barreto and Zworski in [2].

Quasi-normal modes for Dirac fields in the Kerr–Newman–de Sitter black holes

“…In [26], using a special super-symmetric form of the radial Dirac equation, we show that resonances for dS-RN black holes can be obtained as solutions of one-dimensional Schrödinger equations and using the method of semiclassical Birkhoff normal form (as in [29], [30]) we obtain complete asymptotic expansions. Moreover, we get similar results for the wave equation in de Sitter-Schwarzschild metric thus improving the results in [4].…”

“…In [26], [27] we consider scattering of massless uncharged Dirac fields propagating in the outer region of de Sitter-Reissner-Nordström (dS-RN) black hole, which is spherically symmetric charged exact solution of the Einstein-Maxwell equations and is special case of non-rotating (a = 0) KN-dS black hole.…”

“…Note that the physicist treated the Dirac resonances exactly as solutions of the Schrödinger equation (see also [9], [8]). In [26] Theorem 1 shows a different point of view and gives exact relation between Schrödinger and Dirac resonances. Indeed, due to the symmetry of the equation, the set of non-zero Schrödinger resonances consists of two sets interposed: the set of Dirac resonances and its mirror image with respect to the imaginary axis.…”

“…In [27] we give an expansion of the massless Dirac fields in the outer region of dS-RN black hole in terms of resonances and describe the decay of local energy for compactly supported data. The methods extend to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.…”

“…Note that in the non-rotating case a = 0 (de Sitter-Reissner-Nordström black hole, the massless and chargeless case was studied in [26]), ∆ r = r 2 F (r), where F (r) = 1 − 2M r + Q 2 r 2 − Λ 3 r 2 and a 2 (x) = F (r)/r 2 = ∆ r /r 4 . Let Γ j , j = 0, 1, 2, 3, be any 4 by 4 Dirac matrices satisfying Γ i Γ j + Γ j Γ i = 2δ ij I 4 , and…”