Dust in the York canonical basis of ADM tetrad gravity: The problem of vorticity

Abstract

Brown's formulation of dynamical perfect fluids in Minkowski space-time is extended to ADM tetrad gravity in globally hyperbolic, asymptotically Minkowskian space-times. For the dust, we get the Hamiltonian description in closed form in the York canonical basis, where we can separate the inertial gauge variables of the gravitational field in the non-Euclidean 3-spaces of global non-inertial frames from the physical tidal ones. After writing the Hamilton equations of the dust, we identify the sector of irrotational motions and the gauge fixings forcing the dust 3-spaces to coincide with the 3-spaces of the non-inertial frame. The role of the inertial gauge variable York time (the remnant of the clock synchronization gauge freedom) is emphasized. Finally, the Hamiltonian Post-Minkowskian linearization is studied. This formalism is required when one wants to study the Hamiltonian version of cosmological models (for instance back-reaction as an alternative to dark energy) in the York canonical basis.