Einstein field equations and SEM tensor + Alcubierre I wonder how I can find, using the Einstein field equations, the SEM tensor in a region of space with a function $k(x,y,x)$ that describes the curvature of space in that region at a moment (so it is not a function of time). Is there a way to do it? 
An example would be with the Alcubierre metric, we can fing $T_{00}$ by defining the metric and working out the Ricci tensor. Is defining the metric a little bit like defining my function $k$?
 A: If your goal is to consider the Alcubierre metric then you'll notice the energy density is sometimes negative. The classical vacuum has an energy density that is zero.
So this tells you that you either need quantum theory (which is wishful thinking since if you have quantum gravity you need a different theory than GR and so there might not be Alcubierre metrics in that theory) or else you need to have matter.
Exotic matter. Matter with a negative energy density.
A: When you talk about the energy density in the vacuum I assume you mean the energy of the gravitational field. Obviously the stress-energy tensor is zero because otherwise it wouldn't be a vacuum.
In that case the object you need is the stress-energy-momentum pseudotensor. This is constructed entirely from the metric tensor (which is sort of your function $k$ depending on what exactly you mean by $k(x,y,z$) so it is purely geometrical/gravitational.
However defining the energy of the gravitational field this way is a tricky business. Since it's always possible to choose coordinates that make spacetime locally flat it always possible to choose coordinates that make the energy of the gravitational field locally zero. The SEM pseudotensor is, as the name suggests, not a tensor and it isn't coordinate independant. So the energy you get depends on the coordinate system you choose.
