The limiting behavior of the Liu-Yau quasi-local energy

Abstract

The small- and large-sphere limits of the quasi-local energy recently proposed by Liu and Yau are carefully examined. It is shown that in the small-sphere limit, the non-vacuum limit of the Liu-Yau quasi-local energy approaches the expected value \frac{4\pi}{3} r^3 T(e_0, e_0)$. Here, T is the energy-stress tensor of matter, e_0 \in T_p M is unit time-like and future-directed at the point p located at the center of the small sphere of radius $r$ in the limit r \to 0. In vacuum, however, the limiting value of the Liu-Yau quasi-local energy contains the desired limit \frac{r^5}{90} B(e_0, e_0, e_0, e_0), where B is the Bel-Robinson tensor, as well as an extra term. In the large-sphere limit at null infinity, for isolated gravitational sources, the Liu-Yau quasi-local energy is shown to recover the Bondi mass and Bondi news flux, in space-times that are asymptotically empty and flat at null infinity. The physical validity of the Liu-Yau model in view of these results is discussed.