Uncharged Particle near Charged Mass

[DarthVader]

I know that the curvature of space-time around an uncharged mass is different from that around a charged mass (specifically, one situation is characterized by the Schwarzschild metric and the other by the Reissner-Nordstorm metric). But consider an uncharged particle in each situation. Wouldn't this imply that this particle follows a different geodesic trajectory depending on whether the source is charged or uncharged? So wouldn't this mean that there is an electromagnetic force on this particle even though it is uncharged?

 

[Orodruin]

So wouldn't this mean that there is an electromagnetic force on this particle even though it is uncharged?

No. There is no force on the uncharged particle, it is following a geodesic with no proper acceleration in both cases. Do not mistake the different geometry of space-time in the two cases for an electromagnetic force.

 

[Ibix]

It means the gravitational field (i.e. the geometry of spacetime) is different due to the energy stored in the EM field. It doesn't mean that there is any electromagnetic interaction between the two bodies.