Analytical quasinormal modes of spherically symmetric black holes in the eikonal regime

M S Churilova

doi:10.1140/epjc/s10052-019-7146-0

M S Churilova

Open Access

https://doi.org/10.1140/epjc/s10052-019-7146-0

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Abstract

Quasinormal modes in the high frequency (eikonal) regime can be obtained analytically as the Mashhoon–Will–Schiutz WKB formula is exact in this case. This regime is interesting because of the correspondence between eikonal quasinormal modes and null geodesics, as well as due to existence of potential eikonal instabilities in some theories of gravity. At the same time in a number of studies devoted to quasinormal modes of spherically symmetric black holes this opportunity was omitted. Here we find analytical quasinormal modes of black holes in various alternative and extended theories of gravity in the form of the Schwarzschld eikonal quasinormal modes and added corrections due to deviations from Einstein theory. We also deduce a generic formula for analytical calculations of the eikonal quasinormal modes for the class of asymptotically flat metrics in terms of small deviations from the Schwarzschild geometry.

Highlights

There exist a number of alternative gravitational theories representing attempts to solve fundamental problems, such as the nature of the gravitational singularity, dark matter and dark energy problems, quantum gravitational theory, hierarchy problem
In [31] it was shown that the correspondence is guaranteed for any stationary, spherically symmetric, asymptotically flat black hole provided that (a) effective potential has the form of the potential barrier with a single extremum, implying two turning points and decaying at the event horizon and infinity and (b) one is limited by perturbations of the test fields only, and not of the gravitational field itself or other fields, which are non-minimally coupled to gravity
We filled the gap in the current literature devoted to analytical calculations of quasinormal modes in the high frequency regime

Summary

Introduction

There exist a number of alternative gravitational theories representing attempts to solve fundamental problems, such as the nature of the gravitational singularity, dark matter and dark energy problems, quantum gravitational theory, hierarchy problem. In [31] it was shown that the correspondence is guaranteed for any stationary, spherically symmetric, asymptotically flat black hole provided that (a) effective potential has the form of the potential barrier with a single extremum, implying two turning points and decaying at the event horizon and infinity and (b) one is limited by perturbations of the test fields only, and not of the gravitational field itself or other fields, which are non-minimally coupled to gravity Another reason to study quasinormal modes in the eikonal regime is possibility of the so-called eikonal instability which may happen either for black branes [32] and holes [33] or even for wormholes [34].

Eikonal expansion

General approach for asymptotically flat metrics

Conclusions