Black Holes and Thermogeometric Optimization

Abstract

We suggest a finite-time geometric optimization framework to analyze thermal fluctuations and optimal processes in black holes. Our approach implement geodesics in thermodynamic space to define optimal pathways between equilibrium and non-equilibrium states. Since thermodynamic metrics need not be positive-definite, the method ensures a positive thermodynamic length by incorporating simple scale factor into the metric. We show that the scale factor is sensitive to phase transitions in entropy representation, addressing a key gap in Hessian thermodynamic geometry. Additionally, we link the scale factor to the sign of thermodynamic curvature, connecting it to the information geometry governing optimal processes. Our results indicate that optimal fluctuations can drive the evaporation of Schwarzschild and Kerr black holes, which may significantly differ from Hawking radiation. We also explore optimal accretion-driven processes supported by an external inflow of energy.

Minor update: corrected typo in Eq. (4.4) for the angular velocity: Ω=\frac{c r_+}{a (r_+ +r_-)} becomes Ω=\frac{c r_-}{a (r_+ +r_-)}

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