Spherically symmetric solution of the Weyl-Dirac theory of gravitation and possible influence of dark matter on the interplanetary spacecraft motion

Abstract

The Poincar'e and Poincar'e-Weyl gauge theories of gravitation with Lagrangians quadratic on curvature and torsion in post-Riemannian spaces with the Dirac scalar field is discussed in a historical aspect. The various hypothesizes concerning the models of a dark matter with the help of a scalar field are considered. The new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static spherically symmetric solution of the field equations in vacuum for a central compact mass is obtained as the metrics conformal to the Yilmaz-Rosen metrics. On the base of this solution one considers a radial movement of an interplanetary spacecraft starting from the Earth. Using the Newton approximation one obtains that the asymptotic line-of-sight velocity in this case depends from the parameters of the solution, and therefore one can obtain on basis of the observable data the values of these parameters.

Approximate axially symmetric solution of the Weyl-Dirac theory of gravitation and the spiral galactic rotation problem

“…As well as in the spherically symmetric case, [8][9][10] we adopt that the torsion 2-form has the form,…”

“…where s and q are arbitrary numbers determined by the coupling constants of the Lagrangian density (1). By analogy to the spherically symmetric case, [8][9][10] we search the metrics of the axially symmetric solution of these equations in the form…”

“…which is the same condition as in the spherically symmetric case. [8][9][10] If we introduce the condition,…”