The structure of the curvature tensor of plane gravitational waves

Plane gravitational waves in the Riemann space of General Relativity is considered. The criterion of plane gravitational waves is used based on the analogy between plane gravitational and electromagnetic waves. The Theorem is proved that the action of the Lie derivative on the plane wave curvature 2-form in the direction of the vector generating the invariance group of this wave in the Riemann space is equal to zero. It is justified that the gravitational waves can be used to transmit information in the Riemann space.

Dirac particle in the external coulomb field on the background of the Lobachevsky–Riemann space models

The known systems of the radial equations describing the hydrogen atom on the basis of the Dirac equation in the Lobachevsky–Riemann spaces of constant curvature are investigated. In the both geometrical models, the differential equations of second order with six regular singular points are found, and their exact solutions of Frobenius type are constructed. To produce the quantization rule for energy values we use the known condition which separates the transcendental Frobenius solutions. This provides us with the energy spectra that are physically interpretable and are similar to those for the Klein–Fock–Gordon particle in these space models. These spectra are similar to those that previously have appeared in studying the same systems of the equations with the use of the semi-classical approximation.

Relativistic Rotation of the Rigid Body in the Rodrigues – Hamilton Parameters: Lagrange Function and Equations of Motion

The main purposes of this research are to obtain Lagrange function for the relativistic rotation of the rigid body, which is generated by metric properties of Riemann space of general relativity and to derive the differential equations, determining the rigid body rotation in the terms of the Rodrigues - Hamilton parameters. The Lagrange function for the relativistic rotation of the rigid body is derived from the Lagrange function of the nonrotation point of masses system in the relativistic approximation.

A gauge theory of nucleonic interactions by contact

A gauge theory of contact is presented, based on the general idea that the local deformation of the nucleon surface at contact should be gauged by the variation of curvature. A contact force is then defined so as to cope with both the variation of curvature and the deformation. This force generalizes the classical definition of surface tension, in that it depends on the mean curvature, but also depends on the variance of the second fundamental form of surface, considered as a statistical variable over the ensemble of contact spots. It turns out that the variance of the second fundamental form does not depend but on the metric of the space of curvature parameters, organized as Riemann space. This result compels us to review the definition of physical surface of a nucleon.