pp. 113-129 in U. Majer and H.-J. Schmidt, eds., Semantical aspects of spacetime theories, BI-Wissenschaftsverlag, Mannheim 1994

Relativistic and nonrelativistic physical
theories on five-dimensional space-time

H.P. Künzle
Department of Mathematics, University of Alberta
Edmonton, Canada T6G 2G1
e-mail:HP.Kunzle@UAlberta.ca

and

C. Duval
Centre de Physique Théorique
C.N.R.S. Luminy, Case 907
F 13288 Marseille Cedex 9, France
e-mail: Christian.Duval@cptsu5.univ-mrs.fr

Abstract

A five-dimensional Lorentzian manifold (M,g) together with a covariantly constant vector field X has proved to be a convenient framework in which to describe, in a unified way, relativistic physical theories as well as their nonrelativistic limits. If X is a space-like vector field the quotient manifold M/{orbits of X} becomes the standard four-dimensional general relativistic space-time, but if X is a nullvector the corresponding quotient space inherits a full Newton-Cartan structure. In this paper a survey of applications is presented which include, apart from the Einsteinian and Newtonian gravitation theory, other classical field theories (Dirac field, perfect fluids, Yang-Mills fields) as well as a some remarks about possible models of a particle systems. For field theories one starts with a five-dimensional Lagrangian similar in structure to the standard relativistic one. The field equations and a stress-energy tensor can be derived in a straightforward way. The additional components of this tensor on five-dimensional space-time lead to some interesting questions of interpretation.