Angular momentum and an invariant quasilocal energy in general relativity

A key feature of the Brown-York definition of quasilocal energy is that, under local boosts of the fleet of observers measuring the energy, the quasilocal energy surface density transforms as one would expect based on the equivalence principle, namely, like E in the special relativity formula: $E^2-p^2{=m}^2.$ This paper provides physical motivation for the general relativistic analogue of this formula, and thereby arrives at a geometrically natural definition of an “invariant quasilocal energy” (IQE). In analogy with the invariant mass m, the IQE is invariant under local boosts of the fleet of observers on a given two-surface S in spacetime. A reference energy subtraction procedure is required, but in contrast with the Brown-York procedure, S is isometrically embedded in a four-dimensional reference spacetime of one’s choosing. For example, it is well known that any sphere, round or not, can always be isometrically embedded into Minkowski space, even if its scalar curvature is not everywhere positive. So rather than embeddability being a concern, the problem now is that 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 curvature of its tangent bundle, and the curvature of its normal bundle. Taking advantage of this fact suggests 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. The IQE is analyzed in the following cases: both the spatial and future null infinity limits of a large sphere in asymptotically flat spacetimes; a small sphere shrinking to a point along either spatial or null directions; and finally, a large sphere 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.