Quasilocal center-of-mass

Gravitating systems have no well-defined local energy-momentum density. Various quasilocal proposals have been made, however the center-of-mass moment (COM) has generally been overlooked. Asymptotically flat gravitating systems have 10 total conserved quantities associated with the Poincaré symmetry at infinity. In addition to energy-momentum and angular momentum (associated with translations and rotations) there is the boost quantity: the COM. A complete quasilocal formulation should include this quantity. Obtaining good values for the COM is a fairly strict requirement, imposing the most restrictive fall off conditions on the variables. Here we take a covariant Hamiltonian approach, associating Hamiltonian boundary terms with quasilocal quantities and boundary conditions. Unlike several others, our covariant symplectic quasilocal expressions do have the proper asymptotic form for all 10 quantities.