Relativistic and non-relativistic classical field theory on five-dimensional spacetime

Abstract

This paper is a sequel to earlier ones, (1984), in which, on the one hand, classical field theories were described on a curved Newtonian spacetime, and, on the other hand, the Newtonian gravitation theory was formulated on a five-dimensional spacetime with a metric of signature (++++-) and a covariantly constant vector field. Here the authors show that Lagrangians for matter fields are easily formulated on this extended spacetime from simple invariance arguments and that stress-energy tensors can be derived from them in the usual manner so that four-dimensional spacetime expressions are obtained that are consistent in the relativistic as well as the Newtonian case. In the former the theory is equivalent to general relativity. When the magnitude of the distinguished vector field vanishes equations for the (covariant) Newtonian limit follow. They demonstrate this here explicitly in the case of the Klein-Gordon/Schrodinger and the Dirac field and its covariant non-relativistic analogue, the Levy-Leblond field. In especially the latter example the covariant Newtonian theory simplifies dramatically in this five-dimensional form.