The stable problem of the black-hole connected region in the Schwarzschild black hole

TL;DR

This paper investigates the stability of the Schwarzschild black hole's connected region using Painlevé coordinates, concluding it is stable from a physicist's perspective but unresolved mathematically due to real frequency perturbations.

Contribution

It provides a detailed analysis of the stability conditions of the black-hole-connected region, highlighting differences between physical and mathematical viewpoints.

Findings

Region is stable for perturbations with certain complex frequencies

Real frequency perturbations pose unresolved stability issues

Physicist and mathematician perspectives on stability differ

Abstract

The stability of the Schwarzschild black hole is studied. Using the Painlev\'{e} coordinate, our region can be defined as the black-hole-connected region(r>2m, see text) of the Schwarzschild black hole or the white-hole-connected region(r>2m, see text) of the Schwarzschild black hole. We study the stable problems of the black-hole-connected region. The conclusions are: (1) in the black-hole-connected region, the initially regular perturbation fields must have real frequency or complex frequency whose imaginary must not be greater than -1/4m, so the black-hole-connected regionis stable in physicist' viewpoint; (2) On the contrary, in the mathematicians' viewpoint, the existence of the real frequencies means that the stable problem is unsolved by the linear perturbation method in the black-hole-connected region.