The Real Scalar Field in Schwarzschild-de Sitter Spacetime

TL;DR

This paper numerically solves the real scalar field equation in Schwarzschild-de Sitter spacetime using a novel polynomial approximation method, revealing detailed wave behavior near horizons and the impact of the cosmological constant.

Contribution

Introduces a polynomial approximation method for high-precision numerical solutions of scalar fields in Schwarzschild-de Sitter spacetime, improving accuracy over tangent approximation.

Findings

Wave function is harmonic near horizons

Wave function amplitude increases away from horizons

Potential peak height increases as cosmological constant decreases

Abstract

In this paper, the real scalar field equation in Schwarzschild-de Sitter spacetime is solved numerically with high precision. A method called polynomial approximation is introduced to derive the relation between the tortoise coordinate x and the radius r. This method is different from the tangent approximation [1] and leads to more accurate result. The Nariai black hole is then discussed in details. We find that the wave function is harmonic only near the horizons as I. Brevik and B. Simonsen [1] found. Howerver the wave function is not harmonic in the region of the potential peak, with amplitude increasing instead. Furthermore, we also find that, when cosmological constant decreases, the potential peak increases, and the maximum wave amplitude increases.