Mathematical Modeling of Black Hole Thermodynamics

Abstract

Mathematical modeling of black hole thermodynamics explores the deep interconnection between gravitational dynamics and thermodynamic laws. This paper presents a mathematical approach that leverages methods such as geometric thermodynamics, information geometry, and extended phase space modeling to analyze key thermodynamic properties—entropy, temperature, heat capacity, phase transitions, and Smarr relations. We begin with the geometrothermodynamic method (GT-method), a differential-geometric framework parallel to general relativity, applicable to Schwarzschild, Kerr, Kerr–Newman black holes, and de Sitter space . Next, we explore thermodynamic modeling of black holes coupled with nonlinear electrodynamics (NED) in an anti–de Sitter (AdS) background, deriving first laws, generalized Smarr relations, Gibbs free energy profiles, and phase transition behaviors +1. We also examine the information geometry approach, where thermodynamic curvature (Ruppeiner curvature) is used to propose scaled equations of state parameterized by exponents, offering a thermodynamic procedure to constrain black hole models without explicit microphysics . Finally, we integrate phase transition analysis and higher-order corrections in rotating charged spacetimes (e.g., Kerr–Newman–Gödel black holes), incorporating log-area corrections to entropy and their impact on stability Overall, this mathematical modeling framework unifies geometric, thermodynamic, and statistical tools to provide deeper insight into black hole thermodynamics. The results reveal consistent determination of entropyarea relations, robust descriptions of phase behavior, and flexible modeling without requiring specific quantum gravity microstructure. This approach paves the way for future computational modeling, connecting astrophysical data (e.g., gravitational waves) to thermodynamic modeling parameters. Keywords: black hole thermodynamics; geometrothermodynamics; information geometry; nonlinear electrodynamics; Smarr relation; phase transitions; thermodynamic curvature.