Scalar-force gravitational redshift (${\nu }_r/{\nu }_e=e^{{\Phi }_e-{\Phi }_r}$)

For a photon in the flat-background scalar-force theory of gravity, $e^{\Phi }{p}^i$ is constant along its path. In a static field, energies measured by stationary laboratories therefore satisfy the displayed ratio. At first order, $\Delta \nu /\nu =-\Delta \Phi$ in units $c=1$, matching the weak-field Pound-Rebka experiment. This agreement does not imply light bending: a momentum change parallel to the original null tangent changes its scale, not its direction.