Holography in General Space-times

TL;DR

The paper develops a background-independent holographic framework by formulating a covariant entropy bound on light-sheets and introducing screens that store bulk information at a density of no more than one bit per Planck area. It provides a constructive recipe to generate screen-hypersurfaces and applies it to AdS, Minkowski, de Sitter, FRW, and Einstein static universes, revealing spacetime-specific screen types (null, timelike, spacelike) and their information-encoding capabilities. It discusses both dual boundary theories (as in AdS/CFT) and more general holographic theories with a varying number of degrees of freedom, arguing that geometry may be derived from entropy rather than assumed a priori. The work lays groundwork for emergent geometry from information and outlines how holographic principles constrain bulk descriptions and potential reconstructions of spacetime in quantum gravity regimes, with implications for information transfer in black hole contexts.

Abstract

We provide a background-independent formulation of the holographic principle. It permits the construction of embedded hypersurfaces (screens) on which the entire bulk information can be stored at a density of no more than one bit per Planck area. Screens are constructed explicitly for AdS, Minkowski, and de Sitter spaces with and without black holes, and for cosmological solutions. The properties of screens provide clues about the character of a manifestly holographic theory.

Figures (8)

• Figure 1: Conventions and methods used in all diagrams are spelled out in Sec. \ref{['sec-recipe']}. Anti-de Sitter space contains no apparent horizons; all spheres are normal. Spacelike projection is allowed. All null and spacelike projections are directed away from the center at $r=0$. Interior information can thus be projected onto a screen-hypersurfaces $H$ of constant area; $H$ encodes no exterior information. The screen at spatial infinity, $H_\infty$, is optimal and encodes all bulk information. --- The upper part of the figure shows a diagram for Schwarzschild-AdS. Since the future light-sheets of the screen surfaces are not complete (see dotted line), the black hole interior cannot be projected onto $H$ along space-like directions, but only along past light-cones.
• Figure 2: Like AdS, Minkowski space contains no trapped or anti-trapped spheres (a). Unlike AdS, the two allowed null projections lead to different preferred screens, $I^-$ (b) and $I^+$ (c). Either screen is sufficient to encode the entire spacetime. This can be viewed as an expression of the unitarity of the S-matrix.
• Figure 3: A classical black hole forms in a scattering process. The spheres within the apparent horizon are trapped (a). All information can be projected along past light-cones onto $I^-$ (b). But $I^+$ only encodes the information outside the black hole; this reflects the information loss in the classical black hole. The black hole interior can be projected onto the apparent horizon (c).
• Figure 4: A quantum black hole forms in a scattering process. Because negative energy particles cross the apparent horizon during the evaporating phase ($H$), its size decreases. The expansion of future light-cones immediately inside the apparent horizon changes from negative to positive in this process. Therefore the maximal area of the apparent horizon marginally exceeds the maximal area of the event horizon. The diagram shows the projection of this space-time along future light-cones onto screens formed by the apparent horizon and by $I^+$ (thick lines). The past light-cones would lead to the usual projection onto $I^-$.
• Figure 5: Penrose diagram for de Sitter space. The $\mathbf{S}^{D-1}$ spacelike slices would correspond to horizontal lines through the square. The diagonals are apparent horizons dividing the space-time into four regions (a). The $(D-2)$-spheres near past (future) infinity are trapped (anti-trapped); the spheres near the poles are normal. Null projection must be directed towards the tips of the wedges (see Sec. \ref{['sec-recipe']}). It follows that de Sitter space can be projected onto past and future infinity (b), which are spacelike, optimal screen-hypersurfaces of (exponentially) infinite size. A more interesting screen is obtained by applying the spacelike projection theorem to spheres near the event horizon $E$ of an observer at $\chi=0$ (a). By taking a limit, one can show that all information in the observable region of de Sitter space can be projected onto the preferred screen $E$, which is a null hypersurface of constant spatial area $4\pi H^{-2}$ (c).