Preservation of metric signature in Cauchy problem for the Einstein equations

  1. The dynamic metric is the metric for the quasidiagonal system; and so for self consistency the solution can only be proven to exist (and be unique) when the metric (i.e. the solution itself) is hyperbolic/Lorentzian.

  2. Noncompactness has zero impact whatsoever. Hyperbolic equations have finite speed of propagation and hence "local local uniqueness" (one local for in time, and one local for in space).

  3. Finally, about $T$: time functions in Relativity theory is not god given. With diffeomorphism invariance you can always re-coordinate an open spacetime neighborhood of the spacelike hypersurface $M$ as $[0,T]\times M$. On the other hand, there are times where you can prove that on your coordinate system $|g(\partial_t, \partial_t)|$ has a lower bound; in this case usually what happened is that you assumed enough decay on the initial data (in some weighted Sobolev space, say) that you can in fact prove using energy estimates that you have good global foliations. In situations like this you automatically also get uniform boundeness on, say, the connection coefficients, and then your continuity argument will have no problem.

I will add a pessimistic answer. You are right that Choque-Bruhat's (and any related local-in-time) existence result only guarantees that the solution metric exists and is sufficiently regular (including remaining of Lorentzian signature) only in some open neighborhood of the Cauchy surface $\bar{M}$, without any guarantee that this neighborhood will be of uniform thickness over $\bar{M}$ with respect to a pre-determined time coordinate, if $\bar{M}$ is non compact.

To change signature, say from Lorentzian to Riemannian, the metric must first become degenerate (non-invertible). Such a degeneration of the metric is considered a singularity, so in the PDE approach is is treated as a "blow up" of the solution. As we know, blow ups are just a fact of life that we must live with for some equations. For example, a cosmological metric $\mathrm{d}s^2 = -\mathrm{d}t^2 + f(t) \mathrm{d}\mathbf{x}^2$ changes signature when $f(t)$ changes sign, or more conservatively you can only say that it degenerates when $f(t)=0$. There are certainly solutions with a Big Bang singularity that have this feature. If you choose your Cauchy surface $\bar{M}$ to asymptote to this Big Bang singularity in some directions, then you produce exactly the situation you've described: the solution metric exists and remains Lorentzian for a a positive time that depends on where you are on $\bar{M}$, but the metric degenerates arbitrarily quickly in those asymptotic directions.