Stability Conditions In Black Hole Problems

Abstract

This paper presents a systematic theoretical review of stability conditions in black hole spacetimes. Four interconnected aspects are examined: linear perturbation theory and its effective potential structures, quasinormal mode analysis including spectral instability, nonlinear stability and its rigorous mathematical proofs, and thermodynamic stability covering heat capacity and phase transitions in anti-de Sitter spacetime. The observational connection to gravitational-wave ringdown signals is discussed as empirical validation of the perturbative stability framework. Key open problems are identified alongside prospects offered by next-generation gravitational-wave detectors.