@matthiasludewig Yes, I understand about the delta function; physicist's sloppiness :) Sorry for muddying the waters.

Thanks, but I really did mean a delta function rather than a step function and @MatthiasLudewig has resolved the question.

I also note that your points should carry through to more general spacetimes where the metric is smooth except for a surface of discontinuity as described.

Thank you. (a) was my intuition, but what may be "trivial" is absolutely dependent on knowing you know the correct approach - thanks for being definitive ;) (b) The Christoffel issue generally I had considered, but not from the $\gamma(t) \in \Sigma$ not discretem geodesic equation at $\Sigma$ case - except for being distracted by null sections. I don't follow the weak formulation concept, but insofar as the answer remains -ve I'm content.

@matthiasludewig oops, forgot to @ you

Re constant metric I meant that, except at the discontinuity, the space is Minkowski space; the metric is continuous everywhere except at the discontinuity, so the d.e. would, I thought be meaningful by default except there, and the question asks whether the natural insolubility of the equation across the surface of discontinuity could be overcome by a piecewise approach, taking one side derivatives up to the surface on either side. I'm sorry I don't know how to express it more precisely. NB I'm afraid I didn't understand the last sentence.

Thanks for the question but no, I meant the differentiability conditions; hence part II where I wondered whether single sided derivatives, and limit points of geodesic sections meeting at a point (etc.) might succeed in producing a valid solution despite the lack of suitable differentiability everywhere (hence the curve crossing constraints). My thinking: if the metric is "greater" on $\Sigma$ there could be a solution, as moving even infinitesimally along $\Sigma$ would produce a longer path; but... I'm a physicist and not a mathematician hence the request for an assist!