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.
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.