Equivalent of the geodesic equation for fields Mass tells spacetime how to curve using the Einstein field equations and spacetime tells mass how to move using the geodesic equation.
The effect of the electromagnetic field on space-time is given by the Einstein field equation. But what equation gives the effect of spacetime curvature on the electromagnetic field? Is there no replacement of the geodesic equation?
 A: Maxwell's equations (albeit in curved spacetime).
The geodesic equation is, in some sense, just the equations of motion you get for a massive point-particle coupled to the gravitational field. The action would read
$$S = S_{EH} + m \int \sqrt{-\frac{\mathrm{d} x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d} x^\nu}{\mathrm{d}\tau} g_{\mu\nu}} \mathrm{d}\tau,$$
and varying the action leads to the Einstein equations and to the geodesic equation (there are issues with self-field corrections, but I won't dive into that in here).
For the electromagnetic field, you'll have a similar setup, but the additional term in the action will be the action for the electromagnetic field. In the end of the day, you'll get to Maxwell's equations, which read
\begin{align}
\nabla_a F^{ab} &= - 4 \pi j^b, \\
\nabla_{[a}F_{bc]} &= 0,
\end{align}
in Gaussian units. See, e.g., Wald's General Relativity, Sec. 4.3.
For other field contents, one would replace Maxwell's equations with the equations of motion for the field in question. A scalar field has the Klein–Gordon equation, spin-1/2 fields have the Dirac equation, and so on.
