The Einstein-Maxwell-Particle System in the York Canonical Basis of ADM Tetrad Gravity: I) The Equations of Motion in Arbitrary Schwinger Time Gauges

TL;DR

This paper develops a Hamiltonian formulation of Einstein-Maxwell-charged particles in ADM tetrad gravity using the York basis, analyzing inertial and tidal variables, gauge choices, and implications for dark energy and dark matter.

Contribution

It introduces a canonical transformation to the York basis in ADM tetrad gravity with charged particles, separating inertial and tidal variables and analyzing gauge effects on energy and inertial phenomena.

Findings

Explicit Hamilton equations in Schwinger time gauges.

Identification of the role of York time in inertial effects.

Potential link between gauge choices and dark energy/matter.

Abstract

We study the coupling of N charged scalar particles plus the electro-magnetic field to ADM tetrad gravity and its canonical formulation in asymptotically Minkowskian space-times without super-translations. We make the canonical transformation to the York canonical basis, where there is a separation between the {\it inertial} (gauge) variables and the {\it tidal} ones inside the gravitational field and a special role of the Eulerian observers associated to the 3+1 splitting of space-time. The Dirac Hamiltonian is weakly equal to the weak ADM energy. The Hamilton equations in Schwinger time gauges are given explicitly. In the York basis they are naturally divided in four sets: a) the contracted Bianchi identities; b) the equations for the inertial gauge variables; c) the equations for the tidal ones; d) the equations for matter. Finally we give the restriction of the Hamilton equations and of the constraints to the family of {\it non-harmonic 3-orthogonal} gauges, in which the instantaneous Riemannian 3-spaces have a diagonal 3-metric. The non-fixed inertial gauge variable ${}^3K$ (the freedom in the clock synchronization convention) gives rise to a negative kinetic term in the weak ADM energy vanishing only in the gauges with ${}^3K = 0$: is it relevant for dark energy and back-reaction? In the second paper there will be the linearization of the theory to obtain Hamiltonian post-Minkowskian gravity with asymptotic Minkowski background, non-flat instantaneous 3-spaces and no post-Newtonian expansion. This will allow to explore the inertial effects induced by the York time ${}^3K$ in non-flat 3-spaces and to check how much dark matter can be explained as an inertial aspect of Einstein's general relativity.