On the Existence of Spacetime Structure: Technical Appendices† (original) (raw)

I sketch here the construction of Geroch (1969) (whose exposition I closely follow), which grounds the arguments of section 3 in Curiel (2016). (I simplify his construction in non-essential ways for our purposes, and gloss over unnecessary technicalities.) Consider a 1-parameter family of relativistic spacetimes, by which I mean a family {(Mλ, g(λ))}λ∈(0,1], where each (Mλ, g(λ)) is a relativistic spacetime with signature (+, −, −, −) for g(λ). (It will be clear in a moment why I work with the contravariant form of the metric tensor.) In particular, I do not assume that Mλ is diffeomorphic to Mλ′ for λ 6= λ′. The problem is to find a limit of this family, in some suitable sense, as λ → 0. To solve the problem in full generality, we will use a geometrical construction, gluing the manifolds Mλ of the family together to form a 5-dimensional manifold M, so that each Mλ is itself a 4-dimensional submanifold of M in such a way that the collection of all of them foliate M. λ becomes a scal...