The Einstein's linear equation of a space-time with a homogeneous section of low dimension

TL;DR

This paper studies solutions to Einstein's linear equations in higher-dimensional space-times with a homogeneous section, revealing decay laws at infinity and implications for black hole stability.

Contribution

It provides a detailed analysis of perturbations in high-dimensional space-times, deriving decay behaviors and confirming the uniqueness of vacuum solutions.

Findings

Solutions decay at infinity according to a universal law.

No regular perturbations exist outside the event horizon that are well-behaved at infinity.

The analysis supports the stability of high-dimensional black holes.

Abstract

The Einstein's linear equation of a small perturbation in a space-time with a homogeneous section of low dimension, is studied. For every harmonic mode of the horizon, there are two solutions which behave differently at large distance $r$. In the basic mode, the behavior of one of the solutions is ${{(-{r^2}+{t^2})}^{\frac{1-n}{2}}}$ where $n$ is dimension of space. These solutions occur in an integral form. In addition, a main statement of the article is that a field in a black hole decays at infinity according to a universal law. An example of such a field is an eigentensor of the Einstein's linear operator that corresponds to an eigenvalue different from Zero. Possible applications to the stability of black holes of high dimension are discussed. The analysis we present is of a small perturbation of space-time. The perturbation analysis of higher order will appear in a sequel. We determine perturbations of space-time in dimension 1+$n\ge$ 4 where the system of equations is simplified to the Einstein's linear equation, a tensor differential equation. The solutions are some integral transformations which in some cases reduce to explicit functions. We perform some perturbation analysis and we show that there exists no perturbation regular everywhere outside the event horizon which is well behaved at the spatial infinity. This confirms the uniqueness of vacuum space-time within the perturbation theory framework. Our strategy for treating the stability problem is applicable to other space-times of high dimension with a cosmological constant different from Zero.