Initial Data for General Relativity: EFE

[HomogenousCow]

I'm getting pretty lost with how initial data works in general relativity.
Assuming the energy momentum tensor is known (coupled with the metric somehow), what is initial data required to determine the metric at all later times?

 

[WannabeNewton]

The initial value problem is an extremely standard topic in most (all?) advanced GR texts. You would benefit much more from a thorough reading of a chapter on the Cauchy problem from a GR text and subsequently asking more specific questions here. See for example chapter 10 of Wald. You can also find a lot of readings online on the Cauchy problem e.g.

http://www.icm2006.org/proceedings/Vol_III/contents/ICM_Vol_3_22.pdf

http://arxiv.org/pdf/gr-qc/0002074v1.pdf

http://projecteuclid.org/download/pdf_1/euclid.cmp/1103841822

See also chapter 27 of Geroch's GR notes: http://home.uchicago.edu/~geroch/Links_to_Notes.html