A Toy Penrose Inequality and its Proof

TL;DR

This paper introduces a simplified, toy model version of the Penrose inequality in a (2+1)-dimensional anti-de Sitter space, providing a proof for this specific case and offering insights into the original conjecture.

Contribution

It formulates and proves a toy version of the Penrose inequality in (2+1)-dimensional anti-de Sitter space, illustrating the inequality's validity in this simplified setting.

Findings

The toy Penrose inequality holds in (2+1)-dimensional anti-de Sitter space.

The proof relies on established assumptions and geometric arguments.

Provides a simplified context for understanding the original Penrose inequality.

Abstract

We formulate and prove a toy version of the Penrose inequality. The formulation mimics the original Penrose inequality in which the scenario is the following: A shell of null dust collapses in Minkowski space and a marginally trapped surface forms on it. Through a series of arguments relying on established assumptions, an inequality relating the area of this surface to the total energy of the shell is formulated. Then a further reformulation turns the inequality into a statement relating the area and the outer null expansion of a class of surfaces in Minkowski space itself. The inequality has been proven to hold true in many special cases, but there is no proof in general. In the toy version here presented, an analogous inequality in (2+1)-dimensional anti-de Sitter space turns out to hold true.