Proof of a New Area Law in General Relativity

TL;DR

This work defines future and past holographic screens as locally defined quasi-local hypersurfaces foliated by marginally trapped or anti-trapped leaves and proves a classical area law under the null curvature condition. The authors construct null sheets $N(\sigma)$ to partition spacetime into regions $K^+(\sigma)$ and $K^-(\sigma)$, then relate leaf foliations to a monotone spacetime splitting $K^+(r)$ that underpins a monotone growth of leaf area $A(r)$. The core contributions include a proof that the evolution vector $h^a$ has definite sign on holographic screens (i.e., $\alpha$ cannot change sign), a monotonicity theorem for the spacetime splitting, and a direct area-law result with an explicit derivative formula $\frac{dA}{dr}=\int_{\sigma(r)} \sqrt{h^{\sigma(r)}}\,\alpha\,\theta_l^{(\sigma(r))}$. By linking the area increase to the Bousso bound, the paper extends a Second-Law-like behavior to broad quasi-local horizons beyond event horizons, with implications for holography and gravitational thermodynamics.

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 $σ$ divides a Cauchy surface into two disjoint regions, then a null hypersurface $N$ that contains $σ$ splits the entire spacetime into two disjoint portions: the future-and-interior, $K^+$; and the past-and-exterior, $K^-$. If a family of surfaces $σ(r)$ 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 $σ(r)$ 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.

Figures (10)

• Figure 1: Penrose diagrams showing examples of holographic screens. The green diagonal lines show a null slicing of the spacetime; green dots mark the maximal area sphere on each slice. These surfaces combine to form a holographic screen (blue lines); we prove that their area increases monotonically in a uniform direction on the screen (blue triangles). (a) A black hole is formed by collapse of a star (inner shaded region); later another massive shell collapses onto the black hole (outer shaded region). At all other times an arbitrarily small amount of matter accretes (white regions); this suffices to satisfy our generic conditions. The black hole interior contains a future holographic screen that begins at the singularity and asymptotes to the event horizon. It is timelike in the dense regions and spacelike in the dilute regions. (b) In a closed universe filled with dust, marginally antitrapped spheres form a past holographic screen in the expanding region; its area increases towards the future. Marginally trapped spheres form a future holographic screen in the collapsing region; its area increases towards the past. The equator of the three-sphere at the turnaround time (black circle) belongs to neither the past nor the future screen; it is extremal in the sense of Ref. HubRan07.
• Figure 2: The null vectors $l^{a}$ and $k^{a}$ orthogonal to a leaf $\sigma$ of the foliation of $H$ at some point. The evolution of $H$ is characterized by vector $h^a$ normal to the leaves and tangent to $H$. Depending on the quadrant $h^a$ points to, $H$ evolves locally to the future, exterior, past, or interior (clockwise from top).
• Figure 3: (a) Each leaf $\sigma$ splits a Cauchy surface. This defines a partition of the entire spacetime into four regions, given by the past or future domains of dependence and the chronological future or past of the two partial Cauchy surfaces. (b) The pairwise unions $K^\pm$ depend only on $\sigma$, not on the choice of Cauchy surface. They can be thought as past and future in a null foliation defined by the lightsheets $N$.
• Figure 4: Proof that $K^+(r)$ grows monotonically under inclusion, for any foliation $\sigma(r)$ of a hypersurface $\cal H$ with $\alpha<0$. See the main text for details and definitions.
• Figure 5: An example illustrating Lemma \ref{['monotonicity']}: in Minkowski space, the spatial sphere $\chi$ is tangent to the null plane $N$ at $p$ and lies outside the past of $N$ near $p$. It is easy to see that this implies that $\chi$ is a cross-section of a future light-cone that shares one null generator with $N$. In this example it is obvious that $\chi$ expands faster than $N$ at $p$, as claimed in Lemma \ref{['monotonicity']}.

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Remark 2.4
• Remark 2.6
• Remark 2.7
• Definition 2.8
• Remark 2.9
• Definition 3.1
• Lemma 3.2
• Corollary 3.3