Invariant definition of acceleration in Newtonian physics vs proper acceleration in GR

[PeterDonis]

So, as you pointed out in post #24, in section 2 in that paper (in particular in 2.1) a global condition on $\tau$ is actually implicitly assumed.

I don't know if it is, because I don't know if that entire section was intended to just cover standard Newtonian gravity, or whether it was intended to cover a more general category of models that all use the same general formalism.