The perturbation equation of a static symmetrical homogeneous space-time

TL;DR

This paper develops methods to analyze perturbations in static symmetrical homogeneous space-times, deriving decay rates, energy-momentum relations, Green's functions, and solving linear perturbation equations explicitly.

Contribution

It introduces a quasi-transformation approach to solve perturbation equations and provides explicit solutions and decay behaviors in higher-dimensional black hole space-times.

Findings

Decay of massless fields far from black hole horizons

Relation between energy and momentum in hyper black holes

Explicit solutions for linear perturbation equations

Abstract

In absence of explicit solutions of the perturbation equation of a static symmetrical homogeneous space-time, the best we can do is to construct a {\it quasi-}transformation. In this framework, we solve the perturbation equation with initial data and a number of results are derived. Far from the horizon of a black hole of even space dimension $N$, a mass-less field decays as ${r^l} {{(-{r^2}+{t^2})}^{\frac{1-N}{2}-l}}$ in space-time, where $l$ is a harmonic number of the sphere. A relation of energy and momentum of a particle with mass in a hyper black hole is discovered and a solution to the equation of Klein-Gordon in the metric of Schwarzschild-Tangherlini with initial data on the hypersphere is proposed. Also, the Green's function of the Klein-Gordon equation in Schwarzschild coordinates is calculated. This function is a sum on the harmonic modes of the sphere. The first term is a double integration on the spectrum of energy and the momentum of the particle. Far from the horizon, the double integration is approximated by an integration on a line defined by the relation of energy and momentum of a free particle. From here, the potential of Yukawa is derived. Finally, the linear perturbation equations are derived and solved exactly.