The high pericenter velocities (up to a few percent of light) of the S stars around the Galactic-center black hole suggest that general relativistic effects may be detectable through the time variation of the redshift during pericenter passage. Previous work has computed post-Newtonian perturbations to the stellar orbits. We study the additional redshift effects due to perturbations of the light path (what one may call "post-Minkowskian" effects), a calculation that can be elegantly formulated as a boundary-value problem. The post-Newtonian and post-Minkowskian redshift effects are comparable: both are \mathcal O(\beta ^3) and amount to a few km s-1 at pericenter for the star S2. On the other hand, the post-Minkowskian redshift contribution of spin is \mathcal O(\beta ^5) and much smaller than the \mathcal O(\beta ^4) post-Newtonian effect, which would be ~0.1 km s-1 for S2.