Unified fields in pentadimensional theory

Abstract

The theory of Jordan-Thiry is investigated by using a five-dimensional Riemannian manifold V 5 which admits a one-parameter group of isometries. The set of trajectories is supposed to represent the space-time of Relativity. The use of the induced metric in the quotient space leads to essential difficulties. It is necessary to consider a conformal metric and to modify the energy tensor in order to obtain the classical results of relativistic celestial mechanics. Moreover, the conformal metric brings out the evident interpretation of the fifteenth potential like a massless scalar field. A mass term referring to the scalar field is introduced; then the gravitational, electromagnetic, and mesonic scalar fields are unified through the metric of V 5. Several results make the new theory very coherent; in particular, the exact relativistic equations of motion are obtained asymptotically when the matter density vanishes. Exact solutions are searched. The classical Schwarzschild solution and neighbouring solutions are valid in the interior of the matter. Special non-static solutions are also obtained; some of these may be interpreted locally as describing the “collapse” of neutron stars; others ones, analogous to Robertson's metric, can be used to build a cosmology of the unified field.