On the Definition of Averagely Trapped Surfaces

TL;DR

The paper proposes a natural, well-defined mathematical criterion for averagely trapped surfaces based on the product of null expansions and relates it to an isoperimetric inequality involving area and Hawking mass.

Contribution

It introduces a new, consistent definition of averagely trapped surfaces and establishes a precise inequality linking area, Hawking mass, and null expansions.

Findings

Averagely trapped surfaces satisfy the inequality 16π M^2 > A.

Previous definitions were not well-defined and included flat space surfaces.

The new definition aligns with quasi-local energy concepts.

Abstract

Previously suggested definitions of averagely trapped surfaces are not well-defined properties of 2-surfaces, and can include surfaces in flat space-time. A natural definition of averagely trapped surfaces is that the product of the null expansions be positive on average. A surface is averagely trapped in the latter sense if and only if its area $A$ and Hawking mass $M$ satisfy the isoperimetric inequality $16\pi M^2 > A$, with similar inequalities existing for other definitions of quasi-local energy.