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Any chance you could point me to an online reference that's not paywalled and that that rigorously proves a bound? I didn't find anything in Wald, and in MTW there is a claim of a bound, but no proof that I was able to locate. At this point I think I have basically pieced together my own argument that works in 1+1 dimensions, as sketched in this thread, but I need to clean it up and polish it. I would like to see the generalization to 3+1 dimenions, but for pedagogical use I actually like the simplicity of 1+1. It would also be interesting to compare with other people's techniques and see if there's something slicker I could do.The thing to look at here is the extensive literature on Fermi-Normal coordinates. The bounds on these are exactly that same as this problem.