Black hole thermodynamics at null infinity. Part 1: Dual Generalized Second Law

TL;DR

This work develops a dual generalized second law (GSL) for black holes from the perspective of asymptotic observers at future null infinity, using algebraic quantum field theory and modular theory to define a monotone thermodynamic potential built from Bondi data. Depending on the chosen vacuum state—Hartle–Hawking or its hard/soft regularizations—the monotone quantity is either a free energy $\mathcal G$ or a grand potential $\mathcal G$, with explicit expressions $\mathcal G = M - T_H S$ or $\mathcal G = M - \sum_l \mu_{\omega l} \Delta n_{\omega l m} - T_H S$, respectively. The paper provides a rigorous algebraic proof of the dual GSL by exploiting relative entropy monotonicity for nested algebras, linking modular Hamiltonians to the Bondi mass and horizon area through semiclassical Einstein equations. It shows that, for the hard regularization, the dual law reduces to a decrease of the free energy, while for soft regularization, a generalized grand potential governs the evolution, reflecting mode-dependent backreaction via chemical potentials. These results complement the standard horizon-based GSL and illuminate how asymptotic observers experience irreversible black hole dynamics through boundary observables, with potential implications for open quantum systems and energy-work exchanges in Hawking radiation.

Abstract

The generalized second law (GSL) of black hole thermodynamics asserts the monotonic increase of the generalized entropy combining the black hole area and the entropy of quantum fields outside the horizon. Modern proofs of the GSL rely on information theoretic methods and are typically formulated using algebras of observables defined on the event horizon together with a vacuum state invariant under horizon symmetries, inducing a geometric modular flow. In this work, we formulate a dual version of the generalized second law from the perspective of asymptotic observers at future null infinity, who do not have access to the black hole area. Our approach exploits the dependence of the second law on the choice of algebra of observables and of a reference state invariant under suitable symmetries, in close analogy with open quantum thermodynamics. Using algebraic quantum field theory and modular theory, we analyze several physically motivated vacuum states, including the Hartle Hawking state and two classes of regularized vacua. We show that, at null infinity, the monotonic quantity governing an irreversible evolution is no longer the generalized entropy, but rather a thermodynamic potential constructed from asymptotic observables. Depending on the chosen vacuum, this potential takes the form of the free energy or of a generalized grand potential built from the Bondi mass and additional (angular) mode dependent chemical potentials. The resulting inequalities define a dual generalized second law at future null infinity, which can be consistently combined with the standard GSL involving variations of the black hole area.

Figures (5)

• Figure 1: Penrose diagram of the maximal extension of the Schwarschild solution. The solid blue (reps. red) regions depict ${\cal I}_R$ (resp. ${\cal I}_L$), whereas the dashed blue (resp. red) regions depict $\mathcal{H}_L$ (resp. $\mathcal{H}_R$). Hence, the whole blue (resp. red) region represent $\Sigma_f$ (resp. $\Sigma_i$).
• Figure 2: Conformal extension of the black hole spacetime centered on spacelike infinity. Regions $III$ and $IV$ (respectively the black and white hole regions) are not depicted here, since they lie beyond the horizons. The region shaded in blue is called $\mathcal{R}_\infty$ (see Faulkner:2024gst) and does not exist in the physical spacetime, but can be constructed in the conformal spacetime, although it may contain singularities. It is the dual of the black hole regions in the conformal spacetime.
• Figure 3: The trajectory in red is generated by a vector $\bar{\chi}$ in the bulk whose vertical component is given by the asymptotic vector $\chi$ (generating the dashed red trajectory in the region $U \geq U_0$ of ${\cal I}_R$). It resembles a boost near the cut $U = U_0$ but it looks like a usual time translation at late times. The blue-shaded region represents the portion of spacetime where an observer traveling along the red curve can determine the physics using only the local measurements they can perform. It is delimited by the hypersurfaces $U = U_0$ and $U = + \infty$ (the event horizon).
• Figure 4: 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 5: The two spacelike hypersurfaces $\Sigma_1$ and $\Sigma_1$ we consider in this proof, as well as the regions on the horizon and null infinity on which the algebras are defined.