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.