  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.
