Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.




---


**Definition 1:** A triplet $ (\Sigma,h,k) $ is called an **initial data set** if $ (\Sigma,h) $ is a Riemannian $ 3 $-manifold, $ k $ is a symmetric $ 2 $-form on $ \Sigma $, and both $ h $ and $ k $ satisfy the so-called Einstein constraint equations for vacuum (we do not need to know what these equations are). $ \quad \spadesuit $


**Definition 2:** Let $ (\Sigma,h,k) $ be an initial data set. A triplet $ (M,g,i) $ is called a **vacuum development of $ (\Sigma,h,k) $** if


* $ (M,g) $ is a spacetime manifold (i.e., a Lorentzian $ 4 $-manifold) that satisfies the vacuum Einstein Field Equations;
* $ i $ is a smooth embedding of $ \Sigma $ into $ M $ such that $ {i^{\*}}(g) = h $ and $ {i^{\*}}(\Pi) = k $, where $ \Pi $ is the second fundamental form (i.e., the extrinsic curvature) of $ \Sigma $ in $ M $; and
* $ i[\Sigma] $ is a smooth Cauchy surface in $ M $. $ \quad \spadesuit $




---


The Cauchy Problem in General Relativity is to prove the existence of a vacuum development for every initial data set. The proof proceeds by working locally on small patches of the initial data set and ends by performing some gluing to obtain a global solution.


It is a well-known result from the theory of hyperbolic PDE’s that if $ U $ is a local coordinate patch in $ \Sigma $ (i.e., $ U $ is homeomorphic to an open subset of $ \mathbb{R}^{3} $), then we can find a vacuum development $ (M\_{U},g\_{U},i\_{U}) $ of the initial data set $ (U,h|\_{U},k|\_{U}) $ such that $ M\_{U} $ is an open neighborhood of $ U \times \lbrace 0 \rbrace $ in $ U \times \mathbb{R} $ and $ {i\_{U}}[U] = U \times \lbrace 0 \rbrace $.


Knowing now that local vacuum developments exist, we have to glue them together in order to obtain a global-in-space vacuum development of $ (\Sigma,h,k) $. Once the Hausdorff property is established, paracompactness is for free by a 1968 result of Robert Geroch, which states: If a smooth $ 4 $-manifold is Hausdorff and admits a smooth Lorentz metric, then it is automatically paracompact.


In Robert Wald’s book *General Relativity*, the following method for gluing is proposed:


1. By the paracompactness of $ \Sigma $ as a $ 3 $-manifold, we can find a locally finite open cover $ \mathcal{U} $ of $ \Sigma $ consisting of coordinate patches. As mentioned above, for each $ U \in \mathcal{U} $, we have a vacuum development $ (M\_{U},g\_{U},i\_{U}) $ of $ (U,h|\_{U},k|\_{U}) $.
2. For each $ p \in \Sigma $, define $ \mathcal{U}\_{p} := \lbrace U \in \mathcal{U} ~|~ p \in U \rbrace $. We have from (1) that $ \mathcal{U}\_{p} $ is a finite collection for each $ p \in \Sigma $.
3. For each $ p \in \Sigma $, let $ W\_{p} \subseteq \Sigma $ be a neighborhood of $ p $ such that $ W\_{p} $ is contained in all the open sets belonging to $ \mathcal{U}\_{p} $, i.e., $ W\_{p} \subseteq \bigcap \mathcal{U}\_{p} $. Using a result known as ‘local geometric uniqueness’, we can then find a vacuum development $ (M\_{p},g\_{p},i\_{p}) $ of $ (W\_{p},h|\_{W\_{p}},k|\_{W\_{p}}) $ such that there exists an isometric embedding $ j\_{p,U}: (M\_{p},g\_{p}) \to (M\_{U},g\_{U}) $ for each $ U \in \mathcal{U}\_{p} $.
4. Using the embeddings $ j\_{p,U} $ to make identifications, we can glue the $ (M\_{p},g\_{p}) $’s together in order to obtain a vacuum development of $ (\Sigma,h,k) $. Although Wald does not clearly describe this step, this is how I interpret it. For each $ U \in \mathcal{U} $, consider the union $ \tilde{M}\_{U} := \displaystyle \bigcup\_{p \in U} {j\_{p,U}}[M\_{p}] \subseteq M\_{U} $. We wish to glue the $ (\tilde{M}\_{U},g\_{U}|\_{\tilde{M}\_{U}}) $’s together by identifying points as follows. Given $ U\_{1},U\_{2} \in \mathcal{U} $, $ x\_{1} \in \tilde{M}\_{U\_{1}} $ and $ x\_{2} \in \tilde{M}\_{U\_{2}} $, we say that $ x\_{1} \sim x\_{2} $ if and only if there exist a $ p \in U\_{1} \cap U\_{2} $ and a $ q \in M\_{p} $ such that $ {j\_{p,U\_{1}}}(q) = x\_{1} $ and $ {j\_{p,U\_{2}}}(q) = x\_{2} $. Finally, define $ \displaystyle M := \bigsqcup\_{U \in \mathcal{U}} \tilde{M}\_{U} \bigg/ \sim $.




---


**Problem:** How do we show that $ M $ is Hausdorff?


This has eluded my best efforts for the past month, so I would appreciate it if anyone on MathOverflow could offer any insight. Thank you very much!



Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?
Your construction won't work as stated. The problem is that $M\_p$ can be too small. Let me give a slightly silly example of why it doesn't work. 


Cover $\mathbb{R}^3$ with the two local coordinate charts $U = \lbrace x\_1 < 1\rbrace$ and $V = \lbrace x\_1 > -1\rbrace$, and prescribe on it trivial initial data. An admissible "local solution" has $M\_U$ and $M\_V$ being the following subsets of the Minkowski space $\mathbb{R}^{1,3}$:


$$ \begin{align}
M\_U &= \lbrace x\_1 < 1 - |t|\rbrace \newline
M\_V &= \lbrace x\_1 > -1 + |t| \rbrace \end{align} $$


Let $W\_p = \lbrace |x\_1| < 1\rbrace$ for every $p \in U\cap V$. And let 
$$ M\_p = M\_U \cap M\_V \cap \lbrace |t| < \frac12\rbrace $$


So far all the objects are admissible ones that could, in principle, be the ones you picked out from steps 1 through 3. 


By your step 4, you have two distinct points corresponding to what would've been the point $t = \frac12, x\_1 = x\_2 = x\_3 = 0$, since the ones in $M\_U$ and $M\_V$ are not identified since that point technically lies outside of $M\_p$ for every $p$. Those two points are *not* separated by neighborhoods in the space-time you constructed, which violates Hausdorff axiom.




---


If I am not mistaken, this would be the primary obstruction. Possible resolutions are to either make $M\_p$ larger or allow us to make $M\_U$ and $M\_V$ smaller. (With the caveat that I haven't written down a full argument saying that this will work.) In either case you need to get yourself down to something analogous to Ringström's case where the images of gluing maps $j\_{p,U}$ cover the neighborhood of intersection. 




---


(Edit 19.06.2013) Ah, I found a reference where the argument is given in slightly more detail. 


* Choquet-Bruhat and York, "The Cauchy Problem". [MathSciNet relay link](http://www.ams.org/mathscinet-getitem?mr=583716)


It depends on a slightly stronger local uniqueness theorem than you supposed. (See Hawking-Ellis, *Large Scale Structure...* section 7.5.) And some details are still missing. But it appears to be more easily filled-in. And considering the footnote on page 106, they certainly did worry about Hausdorff-ness in their construction. 


