Angular momentum and an invariant quasilocal energy in general relativity

TL;DR

This paper introduces an invariant quasilocal energy (IQE) in general relativity, based on angular momentum considerations, which addresses embedding ambiguities and connects to holographic energy concepts in AdS/CFT.

Contribution

It proposes a geometrically natural, invariant definition of quasilocal energy that incorporates angular momentum and resolves embedding ambiguities using curvature properties.

Findings

IQE is invariant under local boosts of observers on a two-surface.

Embedding ambiguity can be addressed via curvature of tangent and normal bundles.

Reference IQE in AdS spacetime resembles counterterm energy in AdS/CFT.

Abstract

Owing to its transformation property under local boosts, the Brown-York quasilocal energy surface density is the analogue of E in the special relativity formula: E^2-p^2=m^2. In this paper I will motivate the general relativistic version of this formula, and thereby arrive at a geometrically natural definition of an `invariant quasilocal energy', or IQE. In analogy with the invariant mass m, the IQE is invariant under local boosts of the set of observers on a given two-surface S in spacetime. A reference energy subtraction procedure is required, but in contrast to the Brown-York procedure, S is isometrically embedded into a four-dimensional reference spacetime. This virtually eliminates the embeddability problem inherent in the use of a three-dimensional reference space, but introduces a new one: such embeddings are not unique, leading to an ambiguity in the reference IQE. However, in this codimension-two setting there are two curvatures associated with S: the curvatures of its tangent and normal bundles. Taking advantage of this fact, I will suggest a possible way to resolve the embedding ambiguity, which at the same time will be seen to incorporate angular momentum into the energy at the quasilocal level. I will analyze the IQE in the following cases: both the spatial and future null infinity limits of a large sphere in asymptotically flat spacetimes; a small sphere shrinking toward a point along either spatial or null directions; and finally, in asymptotically anti-de Sitter spacetimes. The last case reveals a striking similarity between the reference IQE and a certain counterterm energy recently proposed in the context of the conjectured AdS/CFT correspondence.