Geodesic Deviation in Spacetime: Exploring the Possibilities

[m4r35n357]

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 . . .

 

[WannabeNewton]

It's a non-issue. There is certainly nothing non-causal present. We simply have a congruence of time-like geodesics with some associated 4-velocity field and we Lie transport a space-like vector field along this 4-velocity field in order to define infinitesimal spatial displacements between neighboring observers of the geodesic congruence. We then look at the second covariant derivative of this space-like vector field along the geodesic congruence and from that we get the geodesic deviation equation which just measures the second order rate of change of the infinitesimal spatial displacements between the neighboring observers of the congruence. It's a completely local equation.