Area deficits and the Bel-Robinson tensor

TL;DR

This paper investigates the second-order area deficit of small balls in vacuum spacetimes, examining its relation to gravitational energy and the Bel-Robinson tensor, revealing deviations from expected energy measures.

Contribution

It analyzes the second-order area deficit in vacuum, exploring its connection to gravitational energy and addressing shape ambiguity issues in defining gravitational energy.

Findings

The second-order area deficit is not simply proportional to the Bel-Robinson energy density.

Shape ambiguities affect the relation between area deficit and gravitational energy.

Proposals are provided to resolve shape ambiguity issues.

Abstract

The first law of causal diamonds relates the area deficit of a small ball relative to flat space to the matter energy density it contains. At second order in the Riemann normal coordinate expansion, this energy density should receive contributions from the gravitational field itself. In this work, we study the second-order area deficit of the ball in the absence of matter and analyze its relation to possible notions of gravitational energy. In the small ball limit, any proposed gravitational energy functional should evaluate to the Bel-Robinson energy density $W$ in vacuum spacetimes. A direct calculation of the area deficit reveals a result that is not simply proportional to $W$. We discuss how the deviation from $W$ is related to ambiguities in defining the shape of the ball in curved space, and provide several proposals for fixing these shape ambiguities.