Second law from the Noether current on null hypersurfaces

TL;DR

This work casts entropy in classical gravity as a local Noether-charge balance on null hypersurfaces within a covariant phase space framework that accounts for background fields. It introduces two dynamical entropy constructions, Dirichlet $S^D$ and York $S^Y$, each with its associated flux and charge, and analyzes their behavior under null energy conditions. The paper shows that York entropy increases along future-complete horizons and vanishes on Minkowski light cones, while Dirichlet entropy generically encodes a local thermodynamic balance near non-expanding horizons; it also reveals a phase-transition-like enlargement of the symmetry group by supertranslations during black-hole formation. Together, these results link horizon thermodynamics, boundary symmetries (BMSW), and positivity theorems to provide a nuanced, local second-law description of dynamical spacetimes with null boundaries.

Abstract

I study the balance law equation of surface charges in the presence of background fields. The construction allows a unified description of Noether's theorem for both global and local symmetries. From the balance law associated with some of these symmetries, I will discuss generalizations of Wald's Noether entropy formula and general entropy balance laws on null hypersurfaces based on the null energy conditions, interpreted as an entropy creation term. The entropy is generally the so-called improved Noether charge, a quantity that has recently been investigated by many authors, associated to null future-pointing diffeomorphisms. These local and dynamical definitions of entropy on the black hole horizon differ from the Bekenstein-Hawking entropy through terms proportional to the first derivative of the area along the null geodesics. Two different definitions of the dynamical entropy are identified, deduced from gravity symplectic potentials providing a suitable notion of gravitational flux which vanish on non-expanding horizons. The first one is proposed as a definition of the entropy for dynamical black holes by Wald and Zhang, and it satisfies the physical process first law locally. The second one vanishes on any cross section of Minkowski's light cone. I study general properties of its balance law. In particular, I look at first order perturbations around a non expanding horizon. Furthermore, I show that the dynamical entropy increases on the event horizon formed by a spherical symmetric collapse between the two stationary states of vanishing flux, i.e the initial flat light cone and the final stationary black hole. I compare this process to a phase transition, in which the symmetry group of the stationary black hole phase is enlarged by the supertranslations.

Figures (1)

• Figure 1: Spherical symmetric collapse up to the formation of a black hole. In this case, the event horizon is a light cone "bent" by spacetime curvature once some matter entered it (blue arrows on the picture). This event horizon possesses only one caustic, at point $O$. The entropy $S^Y$ evolves from $0$ on the Minkowski light cone to $\frac{A}{4}$ once the black hole has reached its stationary state.