Insights in Black Hole Thermodynamics using Von Neumann Algebras

Abstract

In recent years, algebraic quantum field theory (AQFT) and modular theory have provided significant insights into gravitational degrees of freedom. Notably, the works [1–3] demonstrate that the inclusion of gravitational degrees of freedom transforms the algebra of quantum fields on a curved spacetime from a Type III to a Type II crossed product algebra. Moreover, the authors in [2, 3] have shown that the generalized entropy of the black hole exterior, evaluated at the bifurcation surface, equals the algebraic entropy of the associated Type II crossed product algebra.

We have extended these results to arbitrary cuts on the horizon for black hole solutions in general relativity (GR) [4]. Specifically, we show that for QFT (including perturbative gravitons) in a static black hole spacetime, the generalized entropy equals the algebraic entropy at any cut on the horizon, with the construction relying on the Hartle–Hawking state. These results can also be extended to Kerr black holes under the assumption of a Hadamard stationary state. Furthermore, using the crossed product construction and modular theory, we present an algebraic proof of a local version of the generalized second law (GSL), where each step is manifestly finite—thanks to the Type II nature of the algebra. This finiteness provides a natural renormalization scheme, thereby addressing a key assumption in Wall’s proof of the GSL [5]. In addition, we have studied deformations of modular operators and derived the averaged null energy condition (ANEC) for a class of spacetimes.

We further generalize the crossed product construction and the relation between generalized entropy and algebraic entropy beyond GR to arbitrary diffeomorphism-invariant theories [6]. In particular, we prove that the equality between generalized entropy and the entropy of the Type II crossed product algebra holds at the bifurcation surface in any such theory. In this broader context, we also provide a weaker form of the GSL.

To study local algebras in higher curvature theories (HCT) and to better understand the nature of HCT itself, we investigate their causal structure. In particular, we analyze Generalized Quadratic Gravity (GQG) and Einsteinian Cubic Gravity (ECG). It is known that gravitons in higher curvature theories can propagate superluminally. This has important consequences for black holes: if the Killing horizon is not a characteristic surface for the fastest propagating mode, it cannot serve as a causal barrier.

We show that GQG, which possesses genuine fourth-order equations of motion, admits only null characteristic surfaces, thereby ensuring that the black hole horizon remains a causal barrier. We also perform a detailed characteristic analysis of ECG, finding that while all null surfaces are characteristic, not all characteristic surfaces are null. Despite the presence of non-null characteristic surfaces, we establish that the black hole horizon in ECG remains a characteristic surface