Quasilocal Center-of-Mass

TL;DR

This paper develops a covariant Hamiltonian framework to define and compute the center-of-mass (COM) as a quasilocal quantity in gravitating systems, ensuring proper asymptotic behavior for all conserved quantities.

Contribution

It introduces a covariant symplectic quasilocal formulation that includes the COM, addressing a gap in previous proposals and satisfying strict fall-off conditions.

Findings

Proper asymptotic form for all 10 conserved quantities achieved

Covariant Hamiltonian approach successfully incorporates the COM

Provides a consistent method for quasilocal COM in gravitating systems

Abstract

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 graviating systems have 10 total conserved quantities associated with the Poincar{\'e} 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. Getting good values for the COM is a fairly strict requirement, imposing the most restrictive fall off conditions on the variables. We take a covariant Hamiltonian approach, associating Hamiltonian boundary terms with quasilocal quantities and boundary conditions. Unlike several others, our {\it covariant symplectic} quasilocal expressions do have the proper asymptotic form for all 10 quantities.