Refinement of the Bousso-Engelhardt Area Law

TL;DR

This work extends the Bousso-Engelhardt area law from leaves of holographic screens to arbitrary subregions, showing that area monotonicity persists under translation along the screen's leaf-orthogonal fibration. The authors define a fibration $h$ with $h=\alpha l+\beta k$ and leverage a zig-zag construction using null surfaces to relate subregion evolution to leaf evolution; they derive the local derivative $\frac{d}{dr}\lVert A_r\rVert = \int_{A_r} \sqrt{g^{\sigma_r}}\,\alpha\,\theta^l>0$, ensuring strict monotonic growth of subregion area. This subregion area law strengthens the link between holographic screens and entanglement-like quantities by aligning with the screen entanglement conjecture, including a Page bound $S(A_r)\le \min(\lVert A_r\rVert,\lVert\sigma_r\setminus A_r\rVert)$. The result supports a local second-law-like behavior in general spacetimes and clarifies how degrees of freedom encoded by holographic screens distribute locally along the fibration.

Abstract

Past holographic screens are codimension-one surfaces of indefinite signature that are foliated by marginally anti-trapped surfaces called leaves. Future holographic screens are defined similarly except with marginally trapped leaves. Bousso and Engelhardt recently showed that the leaves of past and future holographic screens have monotonic area. We prove a stronger area law that shows that subregions of leaves also have monotonic area. For every past and future holographic screen, there exists a family of leaf-orthogonal curves called the fibration of the screen. Any region in a leaf can be translated along the fibration to a leaf of larger area. Our result states that the area of the subregion grows as it is translated.

Figures (1)

• Figure 1: We show that $A_r$ has monotonic area by comparing $A_r$ with the region $f_r(A_0)$. As depicted here, $A_r$ and $f_r(A_0)$ are identical at linear order in $r$.

Theorems & Definitions (4)

• Lemma 1
• proof
• Theorem 1
• proof