The Real Scalar Field Equation for Nariai Black Hole in the 5D Schwarzschild-de Sitter Black String Space

TL;DR

This paper investigates the behavior of a massless scalar field near a 5D Schwarzschild-de Sitter black string, solving the field equation and analyzing reflection and transmission coefficients, comparing them with 4D results.

Contribution

It presents a method to solve the scalar field equation in 5D black string spacetime and compares 5D and 4D scattering coefficients.

Findings

Reflection and transmission coefficients are obtained for the 5D case.

Comparison shows differences between 5D and 4D scalar field scattering.

Method provides a boundary value approach for higher-dimensional black holes.

Abstract

The Nariai black hole, whose two horizons are lying close to each other, is an extreme and important case in the research of black hole. In this paper we study the evolution of a massless scalar field scattered around in 5D Schwarzschild-de Sitter black string space. Using the method shown by Brevik and Simonsen (2001) we solve the scalar field equation as a boundary value problem, where real boundary condition is employed. Then with convenient replacement of the 5D continuous potential by square barrier, the reflection and transmission coefficients ($R, T$) are obtained. At last, we also compare the coefficients with usual 4D counterpart.