I've been meaning to ask this for some time, and now I've plucked up the courage! It is puzzling to me that many fundamental relationships in GR are explained in terms of euclidean space. Taking for example the geodesic deviation equation, it occurs to me that if defined in 3+1 spacetime there is at least the possibility that u, v or w could represent something non-causal. What is the basic argument for translating Riemannian concepts to pseudo-Reimannian situations? Is there a sense that we could bake causality into things like geodesic deviation, or is this a non-issue? I don't recall seeing this issue discussed in any of my other investigations . . .