Some estimates of Wang-Yau quasilocal energy

Abstract

Given a spacelike 2-surface $\Sigma$ in a spacetime $N$ and a constant future timelike unit vector $T_0$ in ${\mathbb{R}}^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(\Sigma ,X,T_0)$ for a given isometric embedding $X$ of $\Sigma$ into a flat 3-slice in ${\mathbb{R}}^{3,1}$. The quantity $E(\Sigma ,X,T_0)$ itself depends on the choice of $X$, however the infimum of $E(\Sigma ,X,T_0)$ over $T_0$ does not. In particular, when $\Sigma$ lies in a time symmetric 3-slice in $N$ and has nonnegative Brown-York quasilocal mass $\mby(Σ)$, our estimates show that $\underset{T_0}{\inf }E(\Sigma ,X,T_0)$ equals $ \mby (Σ)$. We also study the spatial limit of $\underset{T_0}{\inf }E(S_r,X_r,T_0)$, where $S_r$ is a large coordinate sphere in a fixed end of an asymptotically flat initial data set $(M,g,p)$ and $X_r$ is an isometric embeddings of $S_r$ into ${\mathbb{R}}^3\subset {\mathbb{R}}^{3,1}$. We show that if $(M,g,p)$ has future timelike ADM energy-momentum, then $\underset{r\to \infty }{\lim }\underset{T_0}{\inf }E(S_r,X_r,T_0)$ equals the ADM mass of $(M,g,p)$.

17 pages

References in corpus (3)

• Positivity of Quasilocal Mass
• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass