Black Holes and Thermogeometric Optimization

TL;DR

The paper develops Thermogeometric Optimization (TGO), a finite-time, geodesic-based framework that uses Hessian thermodynamic metrics to study thermal fluctuations and optimal state-to-state protocols in black holes. A simple scale factor $\\epsilon$ is introduced to guarantee positive thermodynamic length in non-equilibrium settings, with its sign connected to thermodynamic curvature and Davies phase transitions in entropy representation. Applying TGO to Schwarzschild and Kerr BHs across entropy, energy, and Helmholtz representations reveals that optimal fluctuations can drive full evaporation in some representations and that Davies points manifest differently across representations, notably detected in entropy space but not in energy space. The framework yields evaporation- and accretion-driven pathways with distinct time scales and length measures, offering a geometric lens on black hole thermodynamics and finite-time processes and suggesting extensions to AdS, GTD, and holographic contexts. These results bridge information geometry and finite-time thermodynamics in gravitational systems, highlighting representation-dependent signatures of criticality and providing a platform for exploring black hole heat engines and cosmological fluctuations.

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.

Figures (6)

• Figure 1: The evaporation profiles of energy and entropy of a solar-mass Schwarzschild black hole are shown via two processes: (a) fluctuations (optimal processes) represented by solid curves, and (b) the Hawking evaporation process shown by dashed red curves. Both profiles are normalized using the energy and entropy of a solar-mass Schwarzschild black hole (see App. \ref{['appNDF1']}).
• Figure 2: Hawking evaporation profiles (\ref{['eqEJprofsKerH']}) and (\ref{['eqSHKerrProf']}) for a solar-mass Kerr black hole at different specific spins. The numeric data for these profiles is provided in App. \ref{['appfigKerr0208']}.
• Figure 3: The optimal fluctuation-driven evaporation profiles of energy (solid red), entropy (solid green), and angular momentum (solid blue) for a solar-mass Kerr black hole in entropy representation. The corresponding Hawking evaporation profiles are indicated by dashed curves, with the Hawking evaporation rates all negative and evaluated according to (\ref{['eqHERESJ']}). Their values for $a_*=0.2$, $a_*=0.4$, $a_*=0.8$, and $a_*=0.99$ are listed in Appendix \ref{['appfigKerr0208']}. The time scale is measured in $10^{58}$ billions of years. The information geometry is elliptic ($\epsilon > 0$) for initial spins below the Davies point ($\tilde{a}_* = 0.681$) and hyperbolic ($\epsilon < 0$) for initial spins above this threshold. The thin purple vertical line denotes the end time of the numerical solutions to the thermodynamic geodesic equations.
• Figure 4: The data for these profiles is: (a)$\dot E_0=-4.6 \,\dot E_{H,0}$, $\dot J_0=-8.3 \,\dot J_{H,0}$, $\dot S_0=-4.0 \,\dot S_{H,0}$; (b)$\dot E_0=-6.1 \,\dot E_{H,0}$, $\dot J_0=-4.3 \,\dot J_{H,0}$, $\dot S_0=-22.3 \,\dot S_{H,0}$. The Hawking evaporation rates are negative and evaluated according to (\ref{['eqHERESJ']}). Their values for $a_*=0.5$ and $a_*=0.995$ are listed in Appendix \ref{['appfigKerr0208']}.
• Figure 5: The optimal fluctuation profiles of energy (solid red curves), entropy (solid green curves), and angular momentum (solid blue curves) for a solar-mass Kerr black hole are displayed in energy representation. The corresponding Hawking evaporation profiles are indicated by dashed curves, with the Hawking evaporation rates all negative and evaluated according to (\ref{['eqHERESJ']}). Their values for $a_*=0.2$, $a_*=0.4$, $a_*=0.8$, and $a_*=0.99$ are listed in Appendix \ref{['appfigKerr0208']}. The time scale is measured in $10^{58}$ billions of years. The information geometry is elliptic ($\epsilon > 0$) for initial spins above the Davies point ($a_* = 0.681$) and hyperbolic ($\epsilon < 0$) for initial spins below this threshold. The thin purple vertical line denotes the end time of the numerical solutions to the thermodynamic geodesic equations.