Area deficits and the BelRobinson 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 secondorder 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, a reasonable expectation for any proposed gravitational energy functional is that it evaluate to the BelRobinson 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.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 April 2018
 DOI:
 10.1088/13616382/aab06e
 arXiv:
 arXiv:1710.07379
 Bibcode:
 2018CQGra..35h5005J
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 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), (6770) have been corrected in the present version, and an appendix summarizing the changes has been added