It is shown that the nonrotating coordinates wherein the energy-momentum is globally conserved share the experimental features of the inertial frames. The falling of matter in a spherically symmetric gravitational field is studied in the light of the energy-momentum conservation valid in these coordinates.
A tensorial expression for localized gravitational energy-momentum is delineated as an integral part of the energy-momentum tensor. A bona fide conservation law of the total energy-momentum tensor is obtained in the geodesic-nonrotating coordinates, in which the covariant divergencelessness of the energy-momentum tensor reads, globally, as ordinary divergencelessness. The integral gravitational energy in the exterior of a spherically symmetric source is calculated based on this tensorial relativistic expression. For an ordinary star, such as the sun, it coincides with the Newtonian value up to six digits.
It is shown that there exists a family of coordinate systems in which the energy-momentum tensor is globally conserved. Furthermore, this preferred class of frames includes geodesic systems with respect to any arbitrary point or timelike geodesic line. This implies a physically satisfactory conservation law with no need to introduce an extraneous pseudotensor.
The theories of the neutrino and the electron are studies from a five-dimensional point of view. The concept of periodic 5-spinors is examined, and the formalism of five-dimensional spinor calculus is applied to the manipulations of spinor equations. It is shown that the five-dimensional approach reproduces the results of the familiar four-dimensional approach in a unified and coherent manner. Furthermore, the present approach predicts in a natural way the correct coupling to the electromagnetic field, without ad hoc assumptions.
The concept of periodic fields in the framework of five-dimensional relativity is introduced in a 5-covariant manner. An analysis is given of periodic scalar, vector and bivector fields. The resulting 4-dimensional theories are examined, and it is found that some of them correspond to familiar physical theories in space-time, whereas others require further investigation.
A consistent spinor calculus is developed within the framework of five-dimensional relativity. The formalism is manifestly five-covariant, and in special coordinate systems and special spin frames it reduces to a familiar spinor formalism in curved space-time. The five-dimensional formulation is free of the difficulties involving the coupling of the electromagnetic field, which characterize the four-dimensional approach to spinor calculus. Some theorems and useful identities in the five-dimensional spinor calculus are proved.
The five-dimensional relativity theory proposed by Kaluza is formulated covariantly for a Riemannian space containing a Killing geodesic vector field. From this five-dimensional space a four-dimensional physical space is extracted. The field equations in empty 5-space are essentially uniquely determined and correspond to the Einstein-Maxwell equations in 4-space. In the presence of a field in 5-space the field equations involve a tensor which is associated with energy, momentum, charge and current densities in 4-space. For a 5-space containing dust the field equations lead to particle motion described by the geodesic equations. The latter correspond in 4-space to the Lorentz equations of motion for particles with arbitrary ratios of charge to mass and also for certain entities (tachyons and luminons) unobserved hitherto.
