The spectrum of perturbations inside the Schwarzschild black hole

TL;DR

This paper analyzes the spectrum of bound state perturbations inside Schwarzschild black holes across scalar, electromagnetic, and gravitational fields, revealing a close match with exterior quasi-normal modes and proposing a numerical calculation scheme.

Contribution

It provides an analytic expression for the interior perturbation spectrum and introduces a simple numerical method for accurate calculations.

Findings

Analytic spectrum matches high overtone quasi-normal modes.

Regularity at the black hole center determines the spectrum.

Numerical scheme achieves good accuracy in spectrum calculation.

Abstract

We study the spectrum of the bound state perturbations in the interior of the Schwarzschild black hole for the scalar, electromagnetic and gravitational perturbations. Demanding that the perturbations to be regular at the center of the black hole determines the spectrum of the bound state solutions. We show that our analytic expression for the spectrum is in very good agreement with the imaginary parts of the high overtone quasi normal mode excitations obtained for the exterior region. We also present a simple scheme to calculate the spectrum numerically to good accuracies.