The Numerical Solution of Scalar Field for Nariai Case in 5D Ricci-flat SdS Black String Space with Polynomial Approximation

TL;DR

This paper presents a numerical method using polynomial approximation to solve the scalar field equation around a 5D Ricci-flat Nariai black hole, improving accuracy near horizons compared to tangent approximation.

Contribution

It introduces a polynomial approximation technique for scalar field solutions in 5D Ricci-flat black string space, enhancing previous methods near horizons.

Findings

Wave solutions are nearly harmonic with the tortoise coordinate.

Waves accumulate similarly near both horizons.

Polynomial approximation improves solution accuracy near horizons.

Abstract

As one exact candidate of the higher dimensional black hole, the 5D Ricci-flat Schwarzschild-de Sitter black string space presents something interesting. In this paper, we give a numerical solution to the real scalar field around the Nariai black hole by the polynomial approximation. Unlike the previous tangent approximation, this fitting function makes a perfect match in the leading intermediate region and gives a good description near both the event and the cosmological horizons. We can read from our results that the wave is close to a harmonic one with the tortoise coordinate. Furthermore, with the actual radial coordinate the waves pile up almost equally near the both horizons.