Given a spacelike 2-surface σ in a spacetime N and a constant future timelike unit vector T0 in R3,1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(σ,X, T0) for a given isometric embedding X of σ into a flat 3-slice in R3,1. The quantity E(σ,X, T0) itself depends on the choice of X; however, the infimum of E(σ,X, T0) over T0 does not. In particular, when σ bounds a compact domain ω in a time symmetric 3-slice in N and has nonnegative Brown-York quasilocal mass mBY(σ, ω), our estimates show that infT0 E(σ,X, T0) equals mBY(σ, ω). We also study the spatial limit of infT0 E(Sr,Xr, T0), where Sr is a large coordinate sphere in a fixed end of an asymptotically flat initial data set (M, g, p) and Xr is an isometric embedding of Sr into R3 ∪ R3,1. We show that if (M, g, p) has future timelike ADM energy-momentum, then lim r→∞ infT0 E(Sr,Xr, T0) equals the ADM mass of (M, g, p).
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)