Focusing and the Holographic Hypothesis

TL;DR

The paper analyzes Susskind's screen map as a realization of the holographic hypothesis, examining how horizon images project onto a distant screen. It shows that while many images can be contracting due to focal points, at least one image is expanding, in line with the focusing equation $\rho' = \rho^2/2 + \sigma_{ab}\sigma^{ab} + R_{ab}k^a k^b$ under the null energy condition. The primary screen map, defined as the boundary of the past (or future) of the screen, obeys an area theorem and is expanding, tying holography to Hawking's area theorem. Through axisymmetric static black-hole spacetimes (e.g., Schwarzschild and Majumdar–Papapetrou-type configurations), the work clarifies when holographic mappings preserve area and how energy conditions govern the expansion, informing the scope and limitations of holographic surface descriptions.

Abstract

The ``screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is studied. For a single screen time, there are an infinite number of images of a black hole event horizon, almost all of which have smaller area on the screen than the horizon area. This is consistent with the focusing equation because of the existence of focal points. However, the {\it boundary} of the past (or future) of the screen obeys the area theorem, and so always gives an expanding map to the screen, as required by the holographic hypothesis. These considerations are illustrated with several axisymmetric static black hole spacetimes.

Figures (5)

• Figure 1: Past screen map in black hole spacetime, showing intersection with horizon. The multiple coverings of the horizon come from rays that orbit the hole before crossing the horizon. Only the first covering lies in the primary screen map.
• Figure 2: Three orbits from screen to a Schwarzschild black hole, drawn in $(r,\phi)$ plane.
• Figure 3: Screen image of Schwarzschild horizon. The inner disc is the primary cover, and all the rest of the covers are annuli. The higher order covers accumulate at an impact parameter of $3^{3/2} M$, the capture radius. Only the first and second covers are shown explicitly. The rest are too narrow to show to scale. Even the second cover has smaller area (0.42 $A_H$) than the horizon.
• Figure 4: Focal point for the second cover of the screen map to a Schwarzschild horizon.
• Figure 5: Extreme rays of the primary cover of the second black hole. None of the rays in this cover cross the axis.