Parametrized black hole quasinormal ringdown: Decoupled equations for nonrotating black holes

Abstract

Black hole solutions in general relativity are simple. The frequency spectrum of linear perturbations around these solutions (i.e., the quasinormal modes) is also simple, and therefore it is a prime target for fundamental tests of black hole spacetimes and of the underlying theory of gravity. The following technical calculations must be performed to understand the imprints of any modified gravity theory on the spectrum: 1. Identify a healthy theory; 2. Find black hole solutions within the theory; 3. Compute the equations governing linearized perturbations around the black hole spacetime; 4. Solve these equations to compute the characteristic quasinormal modes. In this work (the first of a series) we assume that the background spacetime has spherical symmetry, that the relevant physics is always close to general relativity, and that there is no coupling between the perturbation equations. Under these assumptions, we provide the general numerical solution to step 4. We provide publicly available data files such that the quasinormal modes of any spherically symmetric spacetime can be computed (in principle) to arbitrary precision once the linearized perturbation equations are known. We show that the isospectrality between the even-and odd-parity quasinormal mode spectra is fragile, and we identify the necessary conditions to preserve it. Finally, we point out that new modes can appear in the spectrum even in setups that are perturbatively close to general relativity.

Note on the parametrized black hole quasinormal ringdown formalism

“…· · · hold. 2 We report that e − j with j ≥ 3 evaluated from the numerical data of e − 0 , e − 2 , e − 7 in [3] agree with that in the previous work.…”

“…In the parametrized black hole quasinormal ringdown formalism [3], the deviation of QNM frequency from GR is given by j α ± j e ± j . While the parameters α ± j , which depend on physical situations, can be read from the master equation, we need to numerically calculate the model independent coefficients e ± j in advance [3]. Especially, it was technically not easy task to compute e ± j with high j in a high accuracy.…”

“…Repeating this process, we can write the effective potential δV in terms of (r H /r) j with j = 0, 1, 2, 7. 3 This implies that we only need to consider the case…”

“…δV denotes the small deviation of the effective potential from the GR case. In [3], the form of δV is assumed to be…”

“…In this sence, e ± j can be considered as the model independent coefficients. In the previous work [3], the coefficients e ± j were calculated numerically, but to obtain them in a high accuracy was technically not easy, especially for large j. In this paper, we derive recursion relations among e ± j with different j.…”