On a Generalization of the Einstein Gravitational Equations Based on Weyl Geometry

Abstract

In the recent years, the interest in modifications of the Einstein theory of gravitation has seriously increased due to the unsolved problem of dark energy. One of them was suggested in our earlier publications where a generalization of the Einstein gravitational theory with Weyl’s connection was studied. In the generalization, the Weyl vector potentials were regarded as a weak field giving small corrections to the Einstein gravitational equations and which could be associated with dark energy. However, in these publications only uncharged dustlike matter was considered as a source of gravitation. In the present paper, we consider the generalized Einstein gravitational equations with Weyl’s connection in the important case in which gravitation is caused by charged matter consisting of particles interacting by means of gravitational and electromagnetic forces. In Weyl’s theory and in a number of other gravitational theories based on Weyl’s geometry, gauge-invariant Lagrangians of second order in the curvature were used, which gave gravitational equations of fourth order in the derivatives of the metric, in contrast to the second order of the Einstein equations. That is why we choose another way to investigate the Einstein gravitational equations with Weyl connection. We study the consequences of our equations and obtain conditions of their consistency. Using these conditions, we come to second-order differential equations for the Weyl vector field and to generalized dynamic equations for charged matter.