Suppose I have a lab frame that is freely falling in a gravitational field of the Earth -- assume non-homogeneity-- and a uniform constant electric field in the vertical direction. There are 2 test particles in the frame -- both of mass $m$, but one is of charge $e$ and the other neutral. They are initially separated by by a vertical distance $h$. I would like to model how their distance evolves. Could anyone help me?
Things I've thought of (but may not be entirely right): I shall assume that the interaction between the particles is negligible.
Then the geodesic equation for the neutral particle is $$u^a\nabla_a u^b=0$$ where $u^a$ is its 4-velocity.
The worldline of a charged particle is $$u'^a\nabla_a u'^b=\frac{e}{m}F^b{}_au'^a$$ where $F^b{}_a$ is the electromagnetic tensor.
And then...?