Evolving test-fields in a black-hole geometry

TL;DR

This paper analyzes the late-time behavior of a scalar field in Schwarzschild black-hole spacetime using complex-frequency Green's functions, approximating quasinormal modes, power-law tails, and discussing generalizations to Kerr black holes.

Contribution

It provides a detailed complex-frequency approach to the scalar field evolution, including new approximations for quasinormal modes and late-time tails, and discusses early-time issues and generalizations.

Findings

Mode-sum converges after a well-defined time

Power-law tail with higher-order corrections derived

Complete late-time evolution description combining approximations

Abstract

We consider the initial value problem for a massless scalar field in the Schwarzschild geometry. When constructed using a complex-frequency approach the necessary Green's function splits into three components. We discuss all of these in some detail: 1) The contribution from the singularities (the quasinormal modes of the black hole) is approximated and the mode-sum is demonstrated to converge after a certain well defined time in the evolution. A dynamic description of the mode-excitation is introduced and tested. 2) It is shown how a straightforward low-frequency approximation to the integral along the branch cut in the black-hole Green's function leads to the anticipated power-law fall off at very late times. We also calculate higher order corrections to this tail and show that they provide an important complement to the leading order. 3) The high-frequency problem is also considered. We demonstrate that the combination of the obtained approximations for the quasinormal modes and the power-law tail provide a complete description of the evolution at late times. Problems that arise (in the complex-frequency picture) for early times are also discussed, as is the fact that many of the presented results generalize to, for example, Kerr black holes.