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 {\em 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.
Highlights
The coalescence of two black holes (BHs) is a fascinating process from a theoretical perspective: two regions devoid of matter interact violently to produce enormous amounts of gravitational radiation, leaving behind a stationary, vacuum solution of the field equations
In general relativity (GR), this process is interesting since both the initial BH spacetimes and the final state are characterized by only two parameters: their mass and angular momentum [1,2]
Structure through gravitational waves, relaxing to the final stationary solution. This exponential approach to equilibrium is a property of many physical systems. It can be mathematically described as a small deviation away from the final state, or as the time window in which the spacetime response to generic perturbations can be decomposed into a set of quasinormal modes (QNMs) characterized by complex frequencies [3,4]
Summary
The coalescence of two black holes (BHs) is a fascinating process from a theoretical perspective: two regions devoid of matter interact violently to produce enormous amounts of gravitational radiation, leaving behind a stationary, vacuum solution of the field equations. Effective field theory (EFT) arguments can produce the general form of the corrections in the absence of extra fundamental fields in the problem [12] Such corrections can be used to study BH solutions and their dynamics, most notably their QNM spectra [13]. Once the linearized equations of motion around a BH spacetime are known, our results can be used to find the corresponding QNM frequencies In this first work, for simplicity, we assume that there are no couplings between the perturbation equations. Many known modified theories of gravity yield separable equations for nonspinning background spacetimes, but it is unclear whether this feature is generic. In Appendices A and B we prove that our assumed form for the perturbative potentials is general enough for our present purposes, and in Appendix C we study the large-l (eikonal) limit of our approximation
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