A Covariant Entropy Conjecture

TL;DR

The paper proposes a universal, covariant entropy bound S ≤ A/4 that applies to any spacetime satisfying Einstein's equations with the dominant energy condition. By formulating the bound on light-sheets—null hypersurfaces generated by surface-orthogonal null geodesics with non-positive expansion—the author provides a time-reversal invariant, background-free perspective on holography. Extensive discussion establishes a practical recipe, explains how caustics protect the bound, and demonstrates recovery of Bekenstein's bound as a special case via the Spacelike Projection Theorem. Cosmological and gravitational-collapse tests show the bound can be saturated but not exceeded, supporting a fundamental limit on the number of degrees of freedom and offering a framework for broader holographic principles in physics.

Abstract

We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: Let A be the area of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion. Let S be the entropy on L. Then S does not exceed A/4. We present evidence that the bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes. For systems with limited self-gravity it reduces to Bekenstein's bound. Because the conjecture is manifestly time reversal invariant, its origin cannot be thermodynamic, but must be statistical. Thus it places a fundamental limit on the number of degrees of freedom in nature.

Figures (7)

• Figure 1: There are four families of light-rays projecting orthogonally away from a two-dimensional surface $B$, two future-directed families (one to each side of $B$) and two past-directed families. At least two of them will have non-positive expansion. The null hypersurfaces generated by non-expanding light-rays will be called "light-sheets." The covariant entropy conjecture states that the entropy on any light-sheet cannot exceed a quarter of the area of $B$.
• Figure 2: Light-rays from a spherical surface into a massive body. If the body is spherically symmetric internally (left), the light-rays all meet at a "radial" caustic in the center of the sphere. If the internal mass distribution is not spherically symmetric, most rays get deflected into angular directions and end on "angular caustics." The picture on the right shows an example with four overdense regions. The thick lines represent caustics.
• Figure 3: Two Penrose diagrams for collapsing spherical objects of identical mass. Each point represents a two-sphere. The thick lines are light-rays. In an object which is spherically symmetric internally (left), the future-directed ingoing light-rays originating from its surface $B$ (at a suitable time) just make it to the center of the object before reaching the singularity. In a highly disordered system, on the other hand, light-rays will be deflected into angular directions (right). We are taking the surface $B$ to have the same area as before. The light-ray path looks timelike because the angular directions are suppressed in the diagram. We see that the light-rays will not sweep the entire interior of the disordered system before meeting the singularity.
• Figure 4: Penrose diagram for a closed FRW universe dominated by dust. The horizontal lines correspond to $S^3$ spacelike sections. Every point represents a two-sphere. The apparent horizons divide the space-time into four regions. The directions of light-sheets in each region are indicated by small wedges.
• Figure 5: Penrose diagram for a flat universe with matter and a negative cosmological constant. Two apparent horizons divide the space-time into an anti-trapped, a trapped, and a normal region. The wedges show the light-sheet directions. The future-directed ingoing light-sheet $L_1$ of the surface $B_1$ is complete. By the spacelike projection theorem (Sec. \ref{['sec-spt']}), the entropy on the spatial region $V_1$ is bounded by the area of $B_1$. An apparent horizon surface $B_2$ at a later time, however, does not possess a complete future-directed ingoing light-sheet: $L_2$ is truncated by the future singularity. Therefore the area of $B_2$ bounds only the entropy on its three light-sheets, not on its spatial interior $V_2$.