@idontgetoutmuch You don't have an institutional logon? I can guarantee that Geroch does not personally care about finding other less reputable sources. Perhaps you would be interested in the work Alexandra Elbakyan? It's also a 56 year old paper - it is very unlikely to contain anything interesting.

Yes that's correct. That is why I said "Lorentzian representative".

The literature on the representation of Lorentzian geometry on posets is deep and wide. You need to be looking for the "causal set" approach to quantum gravity. I'm sure that your questions are answered in the literature. There is a lot of literature though...

The formula for a family of hypersurfaces is the same, except that the initial conditions of the Jacobi tensor is different. See Section 12.3 of Beem, Ehrlich and Easley. Their book does a much better job of explaining all this than I can / am willing to do. The book is called "Global Lorentzian Geometry" any reasonable uni library will have it. I do not recommend using libgen to get a copy of this out of print book. Second hand physical copies are expensive...

I mean... yes, but I'm not sure it'd be helpful because there's quite a bit of required context. The formula you are looking for are in Definition 12.2 of Beem, Ehrlich and Easley. Given a Jacobi tensor J the associated divergence is the trace of B = J' J^{-1} where the ' indicates covariant differentiation along the give geodesic. Vorticity is 0.5 (B - B^*), where * indicates the adjoint (watch out for null geodesics here) and the shear is the remaining stuff. The Jacobi equation can now be rearranged into the Raychaudhuri equation. Literally a rewrite with new notation, see equation 12.1ff.

So... just to resurrect an old question. If the manifold is four dimensional then the existence of a spin structure is equivalent to parallelization. So the "local" calculations are enough in this case. aip.scitation.org/doi/10.1063/1.1664507

I'll add to Igor and say not so elementary, but more that most of the people who work in GR aren't interested, firstly because of the "failure" of things like Kaluza-Klein and secondly because of Global Hyperbolicity (Igor is spot on).