Proof of a New Area Law in General Relativity

TL;DR

This paper proves a new area law for future holographic screens in classical General Relativity, showing that their area increases monotonically under certain conditions, with implications for gravitational collapse and thermodynamics.

Contribution

It establishes the first rigorous proof of a monotonic area law for holographic screens, extending the classical understanding of horizon dynamics beyond event horizons.

Findings

Future holographic screens have strictly increasing area.

The area law applies to both future and past holographic screens.

A thermodynamic interpretation as a Second Law is proposed.

Abstract

A future holographic screen is a hypersurface of indefinite signature, foliated by marginally trapped surfaces with area $A(r)$. We prove that $A(r)$ grows strictly monotonically. Future holographic screens arise in gravitational collapse. Past holographic screens exist in our own universe; they obey an analogous area law. Both exist more broadly than event horizons or dynamical horizons. Working within classical General Relativity, we assume the null curvature condition and certain generiticity conditions. We establish several nontrivial intermediate results. If a surface $\sigma$ divides a Cauchy surface into two disjoint regions, then a null hypersurface $N$ that contains $\sigma$ splits the entire spacetime into two disjoint portions: the future-and-interior, $K^+$; and the past-and-exterior, $K^-$. If a family of surfaces $\sigma(r)$ foliate a hypersurface, while flowing everywhere to the past or exterior, then the future-and-interior $K^+(r)$ grows monotonically under inclusion. If the surfaces $\sigma(r)$ are marginally trapped, we prove that the evolution must be everywhere to the past or exterior, and the area theorem follows. A thermodynamic interpretation as a Second Law is suggested by the Bousso bound, which relates $A(r)$ to the entropy on the null slices $N(r)$ foliating the spacetime. In a companion letter, we summarize the proof and discuss further implications.