Asymptotically Flat Spacetimes at Spatial Infinity: II. Gauge Invariance of the Integrals of Motion in the Field Approach

Abstract

In this paper we complete the work which was started in an earlier paper where asymptotically flat spacetimes at spatial infinity were examined in the framework in the field approach to general relativity (GR) in the Lagrangian description. In the field formulation of GR, the Hamiltonian description of a real isolated system for which all the physical fields are concentrated in a confined domain of space is developed. The integrals of motion are constructed and examined under the weakest fall-off conditions for gravitational potentials found earlier. It is shown that these integrals of motion are exactly the Arnowitt–Deser–Misner (ADM) integrals. The gauge invariance of the integrals of motion in both the Lagrangian and the Hamiltonian descriptions in studied from the point of view of the field approach to GR. It is found that the invariance of the ADM integrals of motion, which follows from the Stokes theorem, is explained as a gauge invariance of the linear field of the spin 2. The ambiguities in the ADM angular momentum can be explained as a violation of the weakest fall-off conditions for the gauge transformations found earlier.