Remark on the effective potential of the gravitational perturbation in the black hole background projected on the brane

TL;DR

This paper investigates the effective potential of gravitational perturbations in a black hole background projected onto a 4D brane, considering the induced metric from higher-dimensional Schwarzschild spacetime and analyzing the energy-momentum tensor components.

Contribution

It derives the effective potential for gravitational perturbations in a brane-world black hole, accounting for non-vacuum background effects and fixing most energy-momentum tensor components.

Findings

Effective potential with correct 4D limit derived

Analysis of energy-momentum tensor components under perturbation

Insights into gravitational perturbations in brane-world scenarios

Abstract

The polar perturbation is examined when the spacetime is expressed by a 4d metric induced from higher-dimensional Schwarzschild geometry. Since the spacetime background is not a vacuum solution of 4d Einstein equation, the various general principles are used to understand the behavior of the energy-momentum tensor under the perturbation. It is found that although the general principles fix many components, they cannot fix two components of the energy-momentum tensor. Choosing two components suitably, we derive the effective potential which has a correct 4d limit.