Late time dynamics of scalar perturbations outside black holes: I. A shell toy-model

TL;DR

This paper introduces a new analytic method to study the late time decay of scalar fields outside black holes, successfully applied to a shell model and promising for Kerr black holes, advancing understanding of black hole perturbations.

Contribution

The paper develops a novel analytic technique for analyzing late time scalar field decay on black hole backgrounds, applicable to Kerr black holes where previous methods were lacking.

Findings

Reproduces known power-law decay tails for scalar fields

Provides explicit relation between initial data and late time behavior

Applicable to static shell and future Kerr black hole geometries

Abstract

We present a new analytic approach for the study of late time evolution of linear test-fields, propagating on the exterior of black holes. This method provides a calculation scheme applicable to Kerr black holes (for which case no analytic calculation of the late time tails has been presented so far). In this paper we develop the new technique and apply it to the case of massless scalar waves evolving on the background geometry of a static spherically-symmetric thin shell with a Schwarzschild exterior. The late time behavior of the scalar field at null infinity is calculated, and is explicitly related to the form of (quite arbitrary) initial data. This reproduces the well-known late time power-law decaying tails. In an accompanying paper we apply our approach to the complete Schwarzschild black hole geometry, where we obtain the familiar inverse-power late time tails at null infinity, as well as at time-like infinity and along the event horizon. A calculation of the late time power-law tails in the Kerr geometry, based on the same approach, will be presented in a forthcoming paper.