In an asymptotically hyperbolic (AH) manifold, the notion of total energy-momentum (mass) is well-defined and was proved to be positive (positive mass theorem) by X.D. Wang. On the other hand, energy density is not well-defined in general relativity, so physicists propose the notions of quasi-local masses which are expected to recover the total mass in domains
exhausting to infinity.
On an AH manifold, we propose a quasi-local mass integral on $/Omega$, defined by $/displaystyle /int_{/partial /Omega} (H_0-H)X$, where $X$ is the isometric embedding of $/partial
/Omega$ into the hyperbolic space $/mathbb{H}^3/subset /mathbb{R}^{3,1}$ and $H_0$ (resp. $H$) is the mean curvature of $/partial /Omega$ after embedding (resp. before embedding). We
will show that as the coordinate sphere goes to infinity, this will tend to the total mass. This is the analogue of the well-known result that the Brown-York mass of coordinate spheres will tend to the ADM mass in an asymptotically Euclidean manifold. This is joint work with Luen-Fai Tam.