Princeton University Gravity Seminar - ADDED

I shall discuss the notion of conserved quantities, including mass, energy, linear momentum, angular momentum, and center of mass at both the quasi-local and total level (for isolated systems). In particular, a definition of quasi-local energy-momentum introduced by S.-T. Yau and myself in 2009 shall be reviewed. This was derived from Hamilton-Jacobi analysis of the Einstein-Hilbert action and a gravitational conservation law. An underlying mathematical tool is the isometric embedding equation for surfaces. In recent joint work with P.-N. Chen and S.-T. Yau, we complement this by the definitions of angular momentum and center of mass at the quasi-local level. The limits of these definitions at spatial and null infinity provide new definitions of total angular momentum and total center of mass that are free of ambiguities and difficulties found in existing definitions, and satisfy some highly desirable properties. For example, the new center of mass $C^i$ at spatial infinity evolves along the vacuum Einstein equation by $p^i=e \cdot{C^i}$ where $p^i$ is the ADM linear momentum and $e$ is the ADM energy.