Optimal Control Theory of the (2+1)-Dimensional BTZ Black Hole

TL;DR

This work develops a finite-time geometric optimization framework for BTZ black holes by leveraging thermodynamic geometry with Weinhold (energy) and Ruppeiner (entropy) metrics. It derives geodesic equations in the macrostate spaces, obtaining minimally dissipative trajectories that describe optimal evaporation or accretion processes; static BTZ cases admit analytic evaporation timelines, while rotating BTZ cases reveal a rich set of end-states in the entropy representation. In the energy representation, geodesics invariably drive toward static, non-rotating end-states, and complete finite-time evaporation is not achieved, whereas the entropy representation permits near-extremal asymptotics and finite-time static finales. The results connect thermodynamic curvature and the thermodynamic length to process probability and duration, offering a robust geometric framework for exploring black-hole thermodynamics and extending to other (2+1)D spacetimes and holographic contexts.

Abstract

We apply a finite-time geometric optimization framework to investigate thermal fluctuations and (non)equilibrium optimal processes in the $(2+1)$-dimensional BTZ black hole. Employing Hessian thermodynamic information metrics, we construct geodesic trajectories that define optimal protocols connecting distinct thermodynamic configurations. Finite-time state transitions are described by paths that extremize entropy production or energy dissipation, depending on the chosen thermodynamic representation. This work presents the first formulation of a geometric optimal control theory for the BTZ black hole.

Figures (7)

• Figure 1: The Weinhold thermodynamic curvature $R$ is presented as a function of $S$ and $J$ in Planck units with $\ell=1$. An increase in the interaction strength among the horizon degrees of freedom is observed near the extremal BTZ curve (dashed red), where a phase transition takes place. In the asymptotic regime, far from extremality, the thermodynamic geometry of the BTZ black hole approaches flatness, indicating weak or negligible interactions. Since $\epsilon > 0$ ensures a positive-definite thermodynamic length, the curvature $R$ remains positive, implying that the information geometry is elliptic.
• Figure 2: Profiles for $\phi=0^\circ$. (a) Time evolution of $E$ (orange), $S$ (green), and $J$ (blue). (b) The dependence of $E$, $S$, and $J$ on the specific spin $a$. The final configuration corresponds to a static BTZ black hole with $E_\tau= 151$ and $S_\tau= 55$ in Planck units. (c) Time evolution of the specific spin $a(t)$. It peaks at $a_{\text{peak}} \approx 0.994$ indicating the state closest to extremality, and then gradually decreases to zero. (d) The geodesic trajectory of states (green curve) in $(S, J)$ space, with a starting point at $S_0=5$ and $J_0=1$ (the black dot), terminates when the angular momentum vanishes. The extremal states are depicted by the dashed red curve.
• Figure 3: Profiles for $\phi=240^\circ$. (a) Time evolution of $E$ (orange), $S$ (green), and $J$ (blue). (b) The dependence of $E$, $S$, and $J$ on the specific spin $a$. The final configuration corresponds to a static BTZ black hole with $E_\tau= 0.4$ and $S_\tau= 2.9$ in Planck units. (c) The evolution of $a(t)$ is strictly monotonic, tending towards zero. (d) The geodesic trajectory of states (green curve) in $(S, J)$ space, with a starting point $S_0=5$ and $J_0=1$ (black dot), terminates when the angular momentum vanishes. Extremality is depicted by the dashed red curve.
• Figure 4: Geodesics paths of states in $(S, J)$ space. (a) The thermodynamic curvature $R$ is represented as a colored background. The initial state $(S_0 = 5, J_0 = 1)$ lies in a relatively strong curved region (yellow) with high interactions, though it remains sufficiently distant from the extremal boundary (the dashed red curve). (b) A magnified view near the initial point. Polar circles help visualize the initial angles $\phi$, which determine the direction of each geodesic. All paths naturally avoid extremality and drive the system to a static BTZ black hole configuration.
• Figure 5: Profiles for $\phi=0^\circ$. (a) Time evolution of $E$ (orange), $S$ (green), and $J$ (blue). (b) The dependence of $E$, $S$, and $J$ on the specific spin $a$. The BTZ black hole does not settle to a final state. It fluctuates between near-extremal states forever asymptotically approaching extremality. (c) Time evolution of the specific spin $a(t)$. The spin asymptotically increases towards the extremal value. (d) The geodesic trajectory of states (orange curve) in the $(E, J)$ space starts at $E_0 = 5$ and $J_0 = 1$ (black dot), and asymptotically approaches the extremal curve (dashed red) over time.