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I'm having troubles to rigorously understand what a development (or maximal development) of a solution is in General Relativity. I was reading a paper by Burnett and Rendall and they write "By maximal development of a globally hyperbolic space-time, we mean the maximal development of an initial data set induced on a Cauchy surface in the spacetime".

This was not of much help. What I'm trying to understand is, mathematically, how do I construct the development, given a solution?

The questions/answers I browsed so far were all of non-mathematitcal nature (i.e. how to "morally" understand what a maximal development is) but those were not the answers I was looking for.

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Could you link to the paper you tried to read? This could help people find an answer for you. –  Dilaton Jun 4 '13 at 14:59
There is a theorem (Choquet-Bruhat and Geroch) that states that given initial data satisfying certain conditions there exists a unique maximal development. Should be in many books in the part on initial value formulation. For example Wald page 264. –  MBN Jun 4 '13 at 18:44

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