Area deficits and the Bel-Robinson tensor

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.

31 pages. v2: Corrected a mistake that occurred in the calculation of the extrinsic curvature of the ball at first order in Riemann normal coordinates. Equations (65), (67-70) have been corrected in the present version, and an appendix summarizing the changes has been added

References in corpus (2)

• Quasilocal mass in general relativity
• Entanglement equilibrium for higher order gravity

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