In GR there is the equivalence principal, stating that in an arbitrary gravitational field, no local non-gravitational experiment can distinguish a freely falling, non-rotating system from a uniformly moving system in absence of the gravitational field. Or in mathematical terms, we can always choose a reference system in which the Christoffel symbols vanish.
In gauge theory, we can always find a coordinate system in the internal space, i.e. a section in the principal bundle, such that the connection 1-form vanishes. Let's assume electromagnetism, but it should hold also for the strong and weak interaction. Can we state an 'electromagnetic'-equivalence principle, i.e. that there in no local (i.e. at a specific point) electromagnetic experiment which can distinguish a charge 'falling' in an electromagnetic field from a charge outside every electromagnetic field?
My doubts: In GR the fact that we can choose a reference frame in which the Christoffel symbols vanish is equivalent to the statement of the equivalence principle. I have doubts about the equivalence of the vanishing connection 1-form and the 'electromagnetic equivalence principal', since accelerated charges radiate and their is self interaction. However I only require the equivalence principle locally, i.e. at one point (maybe this saves it). But a spherical mass, e.g. a point mass does not radiate gravitational waves when accelerated, so I assume their is also no self interaction. This is a big difference to electromagnetism.