This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were helpful in that they answered my question using modern methods, but my quest to re-derive the solution using only the methods available to relativists in the 1970’s resulted in a dead end. Basically, I wanted to know why a certain gluing procedure, put forth by Robert Wald in his book General Relativity, but which was originated by Yvonne Choquet-Bruhat and James W. York, Jr. in their paper The Cauchy Problem (published in the book General Relativity and Gravitation - One Hundred Years After the Birth of Albert Einstein. Volume 1) produces a Hausdorff space-time.
The theorem that I am interested in is the following, taken from the Bruhat-York paper mentioned above.
Theorem. Every initial data set $ (\Sigma,\gamma,K) $, where $ \gamma \in {H^{\text{loc}}_{s}}(\Sigma) $ and $ K \in {H^{\text{loc}}_{s - 1}}(\Sigma) $, admits a vacuum development $ (V,g) $, where $ g \in {H^{\text{loc}}_{s}}(V) $.
Note: A tensor field on a manifold $ V $ is said to be in $ {H^{\text{loc}}_{s}}(V) $ if its norm is square-integrable, together with its generalized derivatives of order $ \leq s $ on every compact subset of $ V $.
The proof by the two authors is as follows.
Proof: Let $ (\Sigma_{a},\varphi_{a}) $ and $ (\Sigma_{b},\varphi_{b}) $ be two coordinate patches on $ \Sigma $ compatible with its smooth structure, where $ \varphi_{a}: \Sigma_{a} \to \mathbb{R}^{3} $ and $ \varphi_{b}: \Sigma_{b} \to \mathbb{R}^{3} $. Let $ (U_{a},g_{a}) $ and $ (U_{b},g_{b}) $ be $ {H^{\text{loc}}_{s}} $-vacuum developments of $ (\Sigma_{a},\gamma,K) $ and $ (\Sigma_{b},\gamma,K) $ respectively. By local uniqueness, there exists, if $ U_{a} $ and $ U_{b} $ are conveniently reduced, an admissible isometry $ \psi_{ab} $ between the largest development $ u_{ab} \subseteq U_{a} $ of $ {\varphi_{a}}[\Sigma_{a} \cap \Sigma_{b}] $ with the metric $ g_{a} $, and $ u_{ba} \subseteq U_{b} $ of $ {\varphi_{b}}[\Sigma_{a} \cap \Sigma_{b}] $ with the metric $ g_{b} $. We consider now the quotient of $ (U,g) $, the disjoint union of the $ (U_{a},g_{a}) $’s, by the equivalence relation $ x_{a} \sim x_{b} $ if $ x_{a} \in u_{ab} $ and $ x_{b} = {\psi_{ab}}(x_{a}) $. One can show that $ U $ is a smooth manifold$ ^{\dagger} $ and that $ (U,g) $ is a vacuum development of $ (\Sigma,\gamma,K) $. $ \quad \blacksquare $
$ ^{\dagger} $ A smooth identification between open sets $ U_{1} $ and $ U_{2} $ of two smooth manifolds $ V_{1} $ and $ V_{2} $ respectively leads to a (Hausdorff) smooth manifold if no $ x_{1} \in \partial U_{1} $ is an accumulation of points whose images in $ V_{2} $ also have an accumulation point $ x_{2} \in \partial U_{2} $.
I understand the importance of $ ^{\dagger} $, but I am utterly unable to prove that in the proof above, no $ x_{a} \in \partial u_{ab} $ is an accumulation of points whose images in $ U_{b} $ also have an accumulation point $ x_{b} \in \partial u_{ba} $. I also have no idea why, in the proof, there is a need to conveniently reduce $ U_{a} $ and $ U_{b} $.
I am starting to believe that the proof is woefully incomplete or even wrong, so I would appreciate it if someone could offer any suggestions. Thank you!
