The Holographic Principle for General Backgrounds

TL;DR

This paper generalizes the holographic principle to arbitrary space-times via a covariant entropy bound: the entropy on any light-sheet $L(A)$ generated from a boundary surface $A$ obeys $S \leq A/4$, and the number of fundamental degrees of freedom on the light-sheet satisfies $N_{\rm dof} \leq A/4$. It introduces light-sheets as covariant, locally defined null hypersurfaces along which the cross-sectional area does not increase, enabling a universal bound that applies even in strongly gravitating or non-static settings, including closed FRW universes and black hole interiors. The work shows how the Bekenstein bound emerges as a special case, and presents new bounds for trapped surfaces, providing substantial evidence for the conjecture while noting that a complete proof remains open. It also demonstrates how to construct holographic screens in general space-times, arguing that bulk physics can be encoded on lower-dimensional surfaces, which supports pursuing a background-independent, holographic formulation of quantum gravity beyond AdS/CFT рода.

Abstract

We aim to establish the holographic principle as a universal law, rather than a property only of static systems and special space-times. Our covariant formalism yields an upper bound on entropy which applies to both open and closed surfaces, independently of shape or location. It reduces to the Bekenstein bound whenever the latter is expected to hold, but complements it with novel bounds when gravity dominates. In particular, it remains valid in closed FRW cosmologies and in the interior of black holes. We give an explicit construction for obtaining holographic screens in arbitrary space-times (which need not have a boundary). This may aid the search for non-perturbative definitions of quantum gravity in space-times other than AdS.

Figures (4)

• Figure 1: [In (a,b) we have suppressed one spatial dimension (surface $\rightarrow$ line). In (c,d) we have suppressed two (surface $\rightarrow$ point). A fixed light-like angle separates spacelike and timelike directions.] The spatial volume $V$ enclosed by a surface $A$ depends on time slicing (a). Thus the original formulation of the holographic principle was not covariant. However, $A$ is the 2D boundary of four 2+1D light-like hypersurfaces (b). They are covariantly generated by the past- and future-directed light-rays going to either side of $A$. E.g., for a normal spherical surface they are given by two cones and two "skirts" (b). In a Penrose diagram, where spheres are represented by points, the associated null hypersurfaces show up as the 4 legs of an X (c). Null hypersurfaces with decreasing cross-sectional area, such as the two cones in (b), are called light-sheets. The entropy passing through them cannot exceed $A/4$ (covariant entropy bound). The light-sheets for normal (d1), trapped (d2), and anti-trapped (d3) spherical surfaces are shown. If gravity is weak, as in (b), the light-sheet directions agree with our intuitive notion of "inside" (d1). For surfaces in a black hole interior, both of the future-directed hypersurfaces collapse (d2). Near the big bang, the cosmological expansion means that the area decreases on both past-directed hypersurfaces (d3).
• Figure 2: [Time and one spatial dimension are suppressed.] We define the "inside" of a 2D surface $A$ to be a light-like direction along which the cross-sectional area decreases (a): $A' \leq A$, or equivalently, $\theta \leq 0$. Such light-rays generate 2+1D light-sheets, the entropy on which is bounded by $A/4$. This definition can be applied to open surfaces as well (b). Light-sheets end on caustics, as $\theta$ becomes positive there. If one stops earlier, the bound can be strengthened to $(A-A')/4$FMW.
• Figure 3: The covariant bound implies the Bekenstein bound. Consider a closed surface $A$. We expect the Bekenstein bound to hold only if $A$ encloses a spatial region $V$ of limited self-gravity. This allows us to make two assumptions. 1. $A$ possesses a future-directed light-sheet $L$ going to the side defined by $V$ (otherwise, $V$ would contain trapped or anti-trapped surfaces, which implies strong gravity). 2. The light-sheet $L$ has no boundary other than $A$, i.e., it closes off in the center of the enclosed region (otherwise, the space-time would end within a light-crossing time, which implies strong gravity). By causality and the second law, all entropy on the spatial region $V$ (or more) must pass through the light-sheet; by the second assumption there are no holes through which it can escape. Therefore, $S(V) \leq S(L)$. By the covariant bound, $S(L) \leq A/4$. Thus we obtain the Bekenstein bound, $S(V) \leq A/4$.
• Figure 4: The closed FRW universe. A small two-sphere divides the $S^3$ spacelike sections into two parts (a). The covariant bound will select the small part, as indicated by the normal wedges (see Fig. \ref{['fig-hypersurfaces']}d) near the poles in the Penrose diagram (b). After slicing the space-time into a stack of light-cones, shown as thin lines (c), all information can be holographically projected towards the tips of wedges, onto an embedded screen hypersurface (bold line).