In Choquet-Bruhat's solution to the Cauchy problem for Einstein's equation, one reduces the Einstein equations to a quasidiagonal quasilinear hyperbolic system on $ M := [0, T] \times \bar M$ where $T > 0$ and $\bar M$ is some initial spacelike 3-manifold for unknowns $g\_{\alpha\beta}$ on $M$. Hyperbolic PDE theory then shows that one can, given first-order initial data, solve these equations forward in time to find smooth solutions $g\_{\alpha\beta}$. Thus we get a metric $g$ on $M$.


1. How do we know that the resulting metric $g = (g\_{\alpha\beta})$ is Lorentzian? Is there a reason why the signature cannot change, or that it cannot become degenerate?


At first I thought continuity of the metric would solve this problem - if it is Lorentzian at $t = 0$ then it should be Lorentzian shortly afterwards. However if $\bar M$ is noncompact (i.e. in the natural case of $\bar M = \mathbb{R}^3$) then we may not get a uniform $T > 0$ on which $g$ is Lorentzian, as given $x \in \bar M$ the time interval $I\_x$ on which $g(\cdot, x)$ is Lorentzian could get small as $x$ varies, making $|I\_x| \to 0$ as $x \to \infty$. But from what I can tell, the results on this problem *do* give us a uniform $T > 0$, so I am guessing that this is not the reason.


I have not studied carefully the proof of well-posedness to hyperbolic systems of this form; perhaps the answer is part of this theorem. Any help is much appreciated.



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.


