Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 61 / 2 / c / Solution

The null momentum equation gives $p^i=C^i{e}^{-\Phi }$, with a constant null vector $C^i$. Since the tangent is parallel to $p^i$, every spatial direction ratio is constant:The changing potential changes the scale of the four-momentum, not its direction. Photon paths are therefore straight lines in the background Minkowski spacetime, up to reparametrization. This is also consistent with conformal preservation of null geodesic paths: the particle motion law can be expressed through a conformally flat metric, which cannot change the unparametrized null trajectories of the background.

Hence no: the theory predicts zero solar light deflection, in disagreement with the observed nonzero bending of light. For comparison, the leading Schwarzschild light deflection is $4GM/(b{c}^2)$ at impact parameter $b$. The scalar theory's nonzero gravitational redshift does not rescue this failed directional prediction.