First what we know: Gμν=Rμν−12gμνR and Tμν are tensors in that they transform properly under coordinate transformations (Gμν by construction and Tμν because of the EFEs), so it doesn't matter which frame we do our measurements in, this tensor equation will always hold.
Suppose that a comoving observer takes careful measurements in his frame and finds the first equation to be true. This is a special case of how we determine what an observer with an arbitrary four-velocity will measure, which is the contraction with that four-velocity Gμνuμuν=8πTμνuμuν Then imagine other observers with four velocities of the form uαi=Aeα0+Beαi, where e0 denotes the unit vector in the time direction, e1 denotes the unit vector in the 1/x/r/whathaveyou direction, etc., and A and B are normalization factors. The above equation is an invariant scalar equation and from this fact and a plethora of observers we can build up the rest of the relations. The same procedure can be applied to the energy conservation equation, only now we are contracting uν∇μTμν
This post imported from StackExchange Physics at 2014-06-21 08:52 (UCT), posted by SE-user Jordan