EFE and Local Minkowski Suppose we view the Einstein Field Equations (EFE) in the context of a boundary value problem with a given stress-energy tensor and boundary conditions. The problem is solved by finding a pseudo-metric.
Is there an unspoken condition that the metric to be found is locally Minkowski or is this implied by the EFE?
Specifically, do you get a different resulting pseudo-metric if you choose a different signature such as (+---) rather than (-+++)?
 A: Given mild differentiability conditions on the metric (I thought this might be $C^\infty$ which is not very mild, but see comments to this answer by 0celo7 below) then, for any point $p$, you can always pick a coordinate system $\left\{x^i\right\}$ which is locally flat, which means that


*

*$g_{ij}(p) = \pm\delta_{ij}$ -- tangent vectors along coordinate curves are orthonormal at $p$;

*$\partial g_{ij} / \partial x_k \rvert_p = 0$ -- it's a good approximation;

*$\partial^2 g_{ij} / \partial x_k \partial x_l \rvert_p \ne 0$ in general -- but not that good an approximation.


In addition it is a theorem that the metric signature is invariant (this is because you can pick a basis for the whole manifold where the metric components are $\pm\delta_{ij}$ although this is not a coordinate basis in general of course, and it follows from continuity  of the metric that its signature can therefore not vary).
Between them these two results are sufficient to show that, on sufficiently small neighbourhoods, things look like Minkowski space in GR.  Note that this result just depends on differentiability conditions, not on the particular form of the field equations.
