Proof of a New Area Law in General Relativity

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.