The holographic principle

TL;DR

The paper argues that the information content of spacetime regions is fundamentally bounded by surface area, not volume, motivating the covariant entropy bound and the holographic principle. It develops light-sheets as the covariant construction relating a surface to adjacent spacetime and shows how the bound arises from black hole thermodynamics, unitarity, and gravitational focusing, with robust tests in cosmology and gravitational collapse. It surveys concrete realizations in string theory and AdS/CFT, and discusses holographic screens as a broader framework for holography in general spacetimes, including de Sitter space. The work highlights the tension between locality and holography, outlines two broad programmatic approaches to holographic theories, and outlines open questions about implementing holography beyond AdS and in cosmological settings.

Abstract

There is strong evidence that the area of any surface limits the information content of adjacent spacetime regions, at 10^(69) bits per square meter. We review the developments that have led to the recognition of this entropy bound, placing special emphasis on the quantum properties of black holes. The construction of light-sheets, which associate relevant spacetime regions to any given surface, is discussed in detail. We explain how the bound is tested and demonstrate its validity in a wide range of examples. A universal relation between geometry and information is thus uncovered. It has yet to be explained. The holographic principle asserts that its origin must lie in the number of fundamental degrees of freedom involved in a unified description of spacetime and matter. It must be manifest in an underlying quantum theory of gravity. We survey some successes and challenges in implementing the holographic principle.

Figures (10)

• Figure 1: A hypersurface of equal time. The spacelike entropy bound attempts to relate the entropy in a spatial region, $V$, to the area of its boundary, $B$. This is not successful.
• Figure 2: The worldvolume of a ball of gas, with one spatial dimension suppressed. (a) A time slice in the rest frame of the system is shown as a flat plane. It intersects the boundary of system on a spherical surface, whose area exceeds the system's entropy. (b) In a different coordinate system, however, a time slice intersects the boundary on Lorentz-contracted surfaces whose area can be made arbitrarily small. Thus the spacelike entropy bound is violated. (c) The light-sheet of a spherical surface is shown for later reference (Sec. \ref{['sec-els']}). Light-sheets of wiggly surfaces may not penetrate the entire system (Sec. \ref{['sec-nearnull']}).---The solid cylinder depicted here can also be used to illustrate the conformal shape of Anti-de Sitter space (Sec. \ref{['sec-ads']}).
• Figure 3: The four null hypersurfaces orthogonal to a spherical surface $B$. The two cones $F_1$, $F_3$ have negative expansion and hence correspond to light-sheets. The covariant entropy bound states that the entropy on each light-sheet will not exceed the area of $B$. The other two families of light rays, $F_2$ and $F_4$, generate the skirts drawn in thin outline. Their cross-sectional area is increasing, so they are not light-sheets. The entropy of the skirts is not related to the area of $B$.---Compare this figure to Fig. \ref{['fig-spheb']}.
• Figure 4: Local definition of "inside". (a) Ingoing rays perpendicular to a convex surface in a Euclidean geometry span decreasing area. This motivates the following local definition. (b) Inside is the direction in which the cross-sectional area decreases ($A'\leq A$). This criterion can be applied to light rays orthogonal to any surface. After light rays locally intersect, they begin to expand. Hence, light-sheets must be terminated at caustics.
• Figure 5: Penrose diagram for an expanding universe (a flat or open FRW universe, see Sec. \ref{['sec-cosmo']}). The thin curve is a slice of constant time. Each point in the interior of the diagram represents a sphere. The wedges indicate light-sheet directions. The apparent horizon (shown here for equation of state $p=\rho$) divides the normal spheres near the origin from the anti-trapped spheres near the big bang. The light-sheets of any sphere $B$ can be represented by inspecting the wedge that characterizes the local domain and drawing lines away from the point representing $B$, in the direction of the wedge's legs.