Black hole thermodynamics at null infinity. Part 2: Open systems, Markovian dynamics and work extraction from non-rotating black holes

TL;DR

The paper develops a unified open-system thermodynamics perspective for black hole dynamics at null infinity by constructing a dictionary that maps late-time boundary conditions to thermal reservoirs and vacuum representations to fixed points of Markovian evolution. It extends prior dual generalized second-law results to maximally extended null infinity and then to the Unruh vacuum in Schwarzschild and Kerr spacetimes, incorporating greybody factors and angular-momentum flux into grand-potential type laws. A key advance is showing that non-thermal vacua, such as the $\kappa_l$-vacua, enable autonomous work extraction from Hawking radiation via engine-inspired protocols, while the Unruh-Kerr generalization reveals additional work terms from angular momentum flux and superradiance. Overall, the work situates black hole radiation and horizon dynamics within the framework of open quantum thermodynamics, offering new tools for proving generalized second laws and for understanding energy-entropy exchanges in dynamical spacetimes.

Abstract

Black hole thermodynamics provides a unique setting in which general relativity, quantum field theory, and statistical mechanics converge. In semiclassical gravity, this interplay culminates in the generalized second law (GSL), whose modern proofs rely on information theoretic techniques applied to algebras of observables defined on null hypersurfaces. These proofs exhibit close structural parallels with the thermodynamics of open quantum systems governed by Markovian dynamics. In this work, we draw parallels between the dynamics of quantum fields in regions bounded by non expanding causal horizons and the thermodynamics of quantum systems weakly coupled to equilibrium reservoirs. We introduce a dictionary relating late time boundary conditions to the choice of reservoir, vacuum states to fixed points of the dynamics, and modular Hamiltonians to thermodynamic potentials. Building on results from a companion paper on dual generalized second laws at future null infinity, we show that additional terms appearing in the associated thermodynamic potentials admit a natural interpretation as work contributions. We demonstrate that certain non thermal vacuum states at null infinity allow for the operation of autonomous thermal engines and enable work extraction from the radiation. Extending the analysis to the Unruh vacuum in Schwarzschild and Kerr backgrounds, we obtain generalized grand potential type laws incorporating grey body effects and angular momentum fluxes. Altogether, our results clarify the thermodynamic description of black hole dynamics and place it within the broader framework of open quantum thermodynamics.

Figures (5)

• Figure 1: Conformal extension of the black hole spacetime centered on spacelike infinity. Figure taken from the first part of this work RBVilatte251.
• Figure 2: Setup in which the hypersurfaces $\Sigma_1$ and $\Sigma_2$ both starts at the horizon bifurcation surface $\mathcal{B}$ and end at different cuts $U = U_1$ and $U = U_2$ respectively at ${\cal I}_R^+$. Are also depicted the regions $\mathcal{D}^{\mathcal{H}}$ and $\mathcal{D}_i^{{\cal I}}$. Figure taken from RBVilatte251.
• Figure 3: A picture depicting the BLPS engine. Because of the population inversion, we are able to extract some energy (heat) $Q_2$ from the hot temperature bath so that we can perform some work $W \leq 0$ to lift a load on an energy ladder. Figure taken from Rignon-Bret:2020nmt, written by one of us.
• Figure 4: Here we represented the eigenbasis of the tensor space spanned by our two qubits. We depicted the jumps induced by the reservoirs (in blue and red) on the composite qubit and the transition to a higher energy eigenstate of the ladder. Figure taken from Rignon-Bret:2020nmt, written by one of us.
• Figure 5: One region of the Penrose diagram of the maximal extension of the Kerr solution. The (anti)-trapped regions (black and white holes) are in shaded blue. In the eternal Kerr solution, the singularities are timelike (vertical dashed black lines). Therefore, contrary to the Schwarzschild solution, one can pile up, on top of each other, an infinite amount of regions like the one depicted in this figure, so that only a fraction of the maximal extension of the Kerr solution is represented here. Notice that as long as one focuses on the exterior regions, the diagram is very similar to the one of an eternal Schwarzschild black hole.