Kinematics of tensor fields in hyperspace. II

Abstract

Various kinematical relations, holding between hypersurface projections of spacetime tensor fields in an arbitrary Riemannian spacetime, are studied in terms of differential geometry in hyperspace. A criterion is given that a collection of hypertensor fields is generated by the projections of a single spacetime tensor field intersected by the embeddings. From here, it is shown that the super-Hamiltonian of an arbitrary tensor field splits into two parts, Hφ↑ and Hφ−, Hφ↑ being local in the field momenta and Hφ− containing their first derivatives. The form of Hφ− for an arbitrary tensor field is determined from the field behavior under hypersurface tilts. The kinematical equations for the intrinsic metric and the extrinsic curvature are written in a quasicanocical form, and their connection with the closing relations for the gravitational super-Hamiltonian is exhibited. The conservation laws of charge, energy and momentum, and the contracted Bianchi identities, are written as hypertensor equations.