Weyl geometric effects on the propagation of light in gravitational fields

We consider the effects of Weyl geometry on the propagation of electromagnetic waves and on the gravitational spin Hall effect of light. It is usually assumed that in vacuum the electromagnetic waves propagate along null geodesics, a result which follows from the geometrical optics approximation. However, this model is valid only in the limit of infinitely high frequencies. At large but finite frequencies, the ray dynamics is affected by the wave polarization. Therefore, the propagation of the electromagnetic waves can deviate from null geodesics, and this phenomenon is known as the gravitational spin Hall effect of light. On the other hand, Maxwell's equations have the remarkable property of conformal invariance. This property is a cornerstone of Weyl geometry and the corresponding gravitational theories. As a first step in our study, we obtain the polarization-dependent ray equations in Weyl geometry, describing the gravitational spin Hall effect of light in the presence of nonmetricity. As a specific example of the spin Hall effect of light in Weyl geometry, we consider the case of the simplest conformally invariant action, constructed from the square of the Weyl scalar, and the strength of the Weyl vector only. The action is linearized in the Weyl scalar by introducing an auxiliary scalar field. In static spherical symmetry, this theory admits an exact black hole solution, which generalizes the standard Schwarzschild solution through the presence of two new terms in the metric, having a linear and a quadratic dependence on the radial coordinate. We numerically study the polarization-dependent propagation of light rays in this exact Weyl geometric metric, and the effects of the presence of the Weyl vector on the magnitude of the spin Hall effect are estimated.