Some Estimates of Wang-yau Quasilocal Energy

Given a spacelike 2-surface Σ in a spacetime N and a constant future timelike unit vector T0 in R 3,1 , 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 R 3,1 . 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 lies in a time symmetric 3-slice in N and has nonnegative Brown-York quasilocal mass m BY (Σ), our estimates show that inf T0 E(Σ, X, T0) equals mBY(Σ). We also study the spatial limit of inf T0 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 embeddings of Sr into R 3 ⊂ R. We show that if (M, g, p) has future timelike ADM energy-momentum, then lim r→∞ inf T0 E(Sr, Xr, T0) equals the ADM mass of (M, g, p).