Real Scalar Field Scattering with Polynomial Approximation around Schwarzschild-de Sitter Black-hole

TL;DR

This paper introduces a polynomial approximation method to analyze scalar field scattering around Schwarzschild-de Sitter black holes, improving the description near horizons by solving a Sturm-Liouville problem numerically.

Contribution

It applies polynomial approximation to simplify the complex coordinate relationship and solves the scattering problem numerically for different horizon configurations.

Findings

Enhanced description of scalar fields near horizons.

Numerical solutions for different black hole horizon configurations.

Improved accuracy over previous methods.

Abstract

As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild-de Sitter black-hole. The complex relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schr$\ddot{o}$dinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm-Liouville type problem. Then this boundary value problem can be solved numerically according to two limiting cases: the first one is the Nariai black-hole whose horizons are close to each other, the second one is when the horizons are widely separated. Compared with previous results (Brevik and Tian), the field near the event horizon and cosmological horizon can have a better description.