Focusing and the Holographic Hypothesis

TL;DR

This paper examines the holographic hypothesis through the lens of screen mapping, focusing on how black hole horizons are represented and how area theorems relate to the holographic principle.

Contribution

It analyzes the properties of screen mappings in black hole spacetimes, clarifying how area theorems and focusing equations support the holographic hypothesis.

Findings

Multiple images of black hole horizons have smaller areas, consistent with focusing equations.

The boundary of the past or future obeys the area theorem, ensuring an expanding map.

Illustrations include axisymmetric static black hole spacetimes.

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.