Yes, the components are constant, so all the derivatives will be zero. In curved space-time you need the theorem that there is a unique connection with the given properties, one of which is that the ##\epsilon## has zero covariant derivative.
Is there a reason why you expect that the answer shouldn't depend on what exacly the formation is, and on what time coordinate you use? Or do you have something more specific in mind?
The question shouldn't be about between what entities the influence is. It should be about what entity carries the influence. If you don't have a field that propagates, from one to the other, faster than light, then talking about FTL causal influences is a missleading choice of words.