Hamiltonian boundary term and quasilocal energy flux

Abstract

The Hamiltonian for a gravitating region includes a boundary term which determines not only the quasi-local values but also, via the boundary variation principle, the boundary conditions. Using our covariant Hamiltonian formalism, we found four particular quasi-local energy-momentum boundary term expressions; each corresponds to a physically distinct and geometrically clear boundary condition. Here, from a consideration of the asymptotics, we show how a fundamental Hamiltonian identity naturally leads to the associated quasi-local energy flux expressions. For electromagnetism one of the four is distinguished: the only one which is gauge invariant; it gives the familiar energy density and Poynting flux. For Einstein's general relativity two different boundary condition choices correspond to quasi-local expressions which asymptotically give the ADM energy, the TrautmanBondi energy and, moreover, an associated energy flux (both outgoing and incoming). Again there is a distinguished expression: the one which is covariant.

Momentum flow in black-hole binaries. I. Post-Newtonian analysis of the inspiral and spin-induced bobbing

“…1), sinusoidally and synchronously. After merger the com- 1 Although as Chen, Nester and Tung have shown, that various formulations of pseudotensors could also be motivated from quasilocal points of view [4]. 2 For binaries in non-circular orbits, larger kick velocities have been observed by Healy et al [8].…”

“…4 Correspondingly, the gravitational field's velocity (as "seen" in our auxiliary flat spacetime) points in the direction of +g × H and has a magnitude of order |H|/|g|. The direction of this field velocity is the same as the direction of motion of an inertial point mass (relative to our Harmonic coordinates) that is induced by a brief joint action of g followed by H: The geodesic equation, in Harmonic coordinates and for low particle velocities v takes the Lorentz-force form…”

Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism symmetry

“…Similar constructions for the computations of conserved quantities in the gauge theories of gravity were used in [4,[25][26][27][28]47,48].…”

“…These aspects are discussed in a review [3] (see also the references therein). For a covariant Hamiltonian formulation see, in particular, [25][26][27][28].…”

Positive small-vacuum-region gravitational-energy expressions

“…The Hamiltonian approach, however, gives these ambiguities a clear physical and geometric meaning. Specifically, the freedom in the choice of expression is associated (via the boundary term in the Hamiltonian variation) with the freedom to choose the type of boundary conditions, and the choice of quasi-local reference (which determines the zero energy or ground state) is effectively just the choice of coordinates on the boundary for the holonomic pseudotensor [8,9,10,11,12].Various criteria have been proposed for quasi-local quantities (see, e.g., [7,13]), including having the correct asymptotic limits at infinity and positivity. Positivity has been regarded as a strong condition.…”