# Does an electromagnetic field affect neutral particles via the metric because of the EM stress-energy tensor?

I'm just starting to learn general relativity (GR), and I'm a beginner, but I came out with this situation which is unclear to me: The trajectory of a charged particle in GR is given from the equation:

$$\dot{u}^{\mu} + \Gamma^{\mu}_{\alpha \beta} u^{\alpha} u^{\beta} = \frac{q}{m} F^{\mu}_{\; \nu} \, u^{\nu}$$

So, if I have a neutral particle $q=0$ the equation reduces to the geodesic equation for a free particle, but because of the Einstein-Maxwell equations:

$$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = T^{EM}_{\mu \nu}$$

the EM stress-energy tensor determines the form of the metric, and consequently the Christoffel symbols that appears in the geodesic equation for the neutral particle. So would the trajectory of this neutral particle in an EM field be different from the case of a space-time with a null EM field?