Abstract
It is shown that, given any set of initial data for Einstein's equations which satisfy the constraint conditions, there exists a development of that data which is maximal in the sense that it is an extension of every other development. These maximal developments form a well-defined class of solutions of Einstein's equations. Any solution of Einstein's equations which has a Cauchy surface may be embedded in exactly one such maximal development.
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References
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Choquet-Bruhat, Y., Geroch, R. Global aspects of the Cauchy problem in general relativity. Commun.Math. Phys. 14, 329–335 (1969). https://doi.org/10.1007/BF01645389
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DOI: https://doi.org/10.1007/BF01645389