Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces

TL;DR

This paper proves that Hawking quasi-local energy is nonnegative and rigid on certain critical surfaces under various conditions, confirming its physical consistency in general relativity.

Contribution

It establishes the first nonnegativity and rigidity theorems for Hawking energy on area-constrained critical surfaces in both time-symmetric and dynamical cases, extending previous results.

Findings

Hawking energy is nonnegative on Hawking surfaces under the dominant energy condition.

Rigidity holds: Hawking energy vanishes only in flat regions.

Results extend to charged, cosmological constant, and higher-dimensional cases.

Abstract

A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, when evaluated on its natural area-constrained critical surfaces, henceforth called "Hawking surfaces", satisfies both properties under the dominant energy condition. In the time-symmetric case, where Hawking surfaces coincide with area-constrained Willmore surfaces, we extend positivity and rigidity to include electric charge, a nonzero cosmological constant, and higher dimensions. In the fully dynamical (non-time-symmetric) case, we establish the first nonnegativity and rigidity theorems for the Hawking energy in this general setting. These results confirm the Hawking energy consistency with basic physical principles and address several longstanding ambiguities and criticisms.