Classical tests for Weyl gravity: Deflection of light and time delay

Abstract

Weyl gravity has been advanced in the recent past as an alternative to General Relativity (GR). The theory has had some success in fitting galactic rotation curves without the need for copious amounts of dark matter. To check the viability of Weyl gravity, we propose two additional classical tests of the theory: the deflection of light and time delay in the exterior of a static spherically symmetric source. The result for the deflection of light is remarkably simple: besides the usual positive (attractive) Einstein deflection of $4GM/r_{0}$ we obtain an extra deflection term of $-\gamma r_{0}$ where $\gamma$ is a constant and $r_0$ is the radius of closest approach. With a negative $\gamma$, the extra term can increase the deflection on large distance scales (galactic or greater) and therefore imitate the effect of dark matter. Notably, the negative sign required for $\gamma$ is opposite to the sign of $\gamma$ used to fit galactic rotation curves. The experimental constraints show explicitly that the magnitude of $\gamma$ is of the order of the inverse Hubble length something already noted by Mannheim and Kazanas as an interesting numerical coincidence in the fitting of galactic rotation curves.