Hyperboloidal slicing approach to quasinormal mode expansions: The Reissner-Nordström case

Abstract

We study quasi-normal modes of black holes, with a focus on resonant (or quasi-normal mode) expansions, in a geometric frame based on the use of conformal compactifications together with hyperboloidal foliations of spacetime. Specifically, this work extends the previous study of Schwarzschild in this geometric approach to spherically symmetric asymptotically flat black hole spacetimes, in particular Reissner-Nordström. The discussion involves, first, the non-trivial technical developments needed to address the choice of appropriate hyperboloidal slices in the extended setting as well as the generalization of the algorithm determining the coefficients in the expansion of the solution in terms of the quasi-normal modes. In a second stage, we discuss how the adopted framework provides a geometric insight into the origin of regularization factors needed in Leaver's Cauchy based foliations, as well as into the discussion of quasi-normal modes in the extremal black hole limit.

Quasinormal modes and dual resonant states on de Sitter space

“…We shall reconcile our late time asymptotics (21) (leading to the full set of QNMs ( 22)) with the wave evolution in [4] by analyzing the dependence of the coefficients c j in (20) on the initial data (9) of Φ, ultimately showing in §III C that the qualitative change observed in [4, §IV.D] is not present when the initial data are allowed to be nonzero near the dS horizon. In §III D, we shall also connect our analysis to the Green's function computations presented in [4,12,13].…”

“…5 We stress that one can choose the asymptotic data u (0) ± so that the innermost sum in ( 21) is a nonzero function for any desired value of n, i.e. for all of the QNMs in (22) a mode solution indeed exists, and thus all QNMs contribute to the long-time dynamics (20) of the field.…”

“…(This equation is equivalent to ( 8)-( 9) for the choice f = [2 − m 2 , Θ(t * )](Φ 0 + t * Φ 1 ).) Assuming that the j-th QNM ω j is simple, the map assigning to f the coefficient c j in the QNM expansion (20) is linear, hence given by the formula…”

“…This makes it an ideal starting point for the investigation of a large variety of physical phenomena in universes, such as our own [1,2], which undergo an accelerated expansion. The decay of classical fields propagating on dS spacetimes and its generalizations containing black holes-such as Schwarzschild-de Sitter (SdS), Kerr-de Sitter (KdS), Reissner-Nordströmde Sitter (RNdS), and Kerr-Newman-de Sitter (KNdS) spacetimes-has been studied in detail via numerical evolutions [3][4][5], the computation of quasinormal modes (QNMs) [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], and mathematical investigations [24][25][26][27][28][29][30][31][32][33][34][35][36].…”

“…We give a characterization of v j which is valid in all cases, and call such generalized purely incoming solutions dual resonant states. 1 This characterization requires working in a hyperboloidal slicing of dS [19,20,23]; the failure of time evolution to be unitary in this slicing is the reason for the asymmetry between the definitions of mode solutions u j and dual resonant states v j . The notion of dual resonant states thus allows one to discern what triggers the appearance of any particular QNM frequency in the expansion (1).…”