Charges of the gravitational field and (3+1) decomposition of CYK tensors part 2

Abstract

The work describes the method of construction of charges (conserved quantities) for the gravity field in the (3 + 1) decomposition. The presented construction uses tensors of the electrical and magnetic parts of the Weyl tensor and conformal Killing vectors. In the case of conformally flat spatial hypersurfaces, we get twenty local charges, which can be expressed in terms of the initial data (three-dimensional metric and extrinsic curvature tensor). The work shows the relationships between charges which are obtained by this method and the usual ADM approach. In traditional ADM approach Killing vectors are used to construct corresponding charges e.g. time translation corresponds to ADM mass, spatial translations give linear momentum and rotations correspond to angular momentum. Gravito-electric and gravito-magnetic charges need conformal Killing vectors e.g. mass corresponds to dilation, linear momentum is related to rotation and angular momentum needs conformal acceleration. The analyzed example of the Schwarzschild--de Sitter spacetime suggests that in some cases the mass calculated by the presented method has better properties than the traditional ADM mass. Next we discuss asymptotic charges which are no longer rigidly conserved but rather approach finite value at spatial infinity.