  The detection of gravitational waves from binary black holes sources has
opened the possibility to search for electric charges and "dark" charges on
black holes, the latter being candidates for dark matter. This requires
theoretical predictions about the effect of charges on the inspiral of binary
black holes in order to place constraint on charges. The effects of these
charges on the inspiral of binary black holes can be described using
Einstein-Maxwell theory. They have previously been derived up to first
post-Newtonian (1PN) order, and the results were recently used to place bounds
on the charge-to-mass ratio on black holes. Here we use the effective field
theory approach with a metric parameterization based on a temporal Kaluza-Klein
decomposition with non-relativistic gravitational fields to arrive at the
Lagrangian for binary motion under the influence of charges up to 2PN order.
