Next semester, I am going to lecture about (the mathematics of) general relativity and I am still thinking hard how to organize and even more importantly how to motivate all the stuff.
I am wondering what minimal assumptions I have to make about the objects and their relations to be able to interpret the formulae and their relation to classical Newtonian physics. I should explain further:
I think the assumption that spacetime is modeled by a four-dimensional differentiable manifold M is fine and is easily to be motivated. I am also fine with assuming that we have an affine connection on the manifold because it can be measured by moving a (quantum) particle with spin along a closed loop and comparing spin direction (and relative position/phase for torsion) before and after going through the loop.
Then we may assume that the holonomy of the affine connection lies inside the Poincaré group (because we measure no other holonomy). Using this, we can parallel transport a chosen Lorentz metric in one tangent space to each other tangent space, so we get a Lorentz manifold. (Usually texts on general relativity start with a Lorentzian manifold, but they do not explain where the measuring of lengthes and angles should come from — a rod is itself a complicated physical object).
Now having such a manifold, we can write down the Riemannian curvature and the torsion tensor. For simplicity, let us assume that torsion vanishes for the moment. Given the Riemannian curvature, we can contract it and write down the Einstein tensor G. Now the Einstein field equations can be stated as a definition: "The Einstein tensor G is the stress-energy tensor", that is G tells us where we measure matter.
Mathematically this is fine (and actually of no content). From the viewpoint of physics, however, we want to be able to interpret the so defined matter (or stress-energy tensor to be more precise) as what is usually consider to be matter (or mass density or pressure or stress). What other inputs do I need to achieve this?
Do I have to add the geodesic equation for free-falling test particles, for example, or does this already follow from my definitions (that is the field equations) above (of course, one has to relate a test particle to the matter term)?
I am aware of the geometric interpretation of the Einstein field equation which relates the trace of the stress-energy tensor to the second derivative of the change of the volume of a ball of free-falling test particles. In order to use this, one has to know the equations of motions for free-falling test particles first. Further, one has to compare with the change of volume in the Newtonian limit. But how would we then get the pressure dependent parts in the trace of the stress-energy tensor, because Newtonian gravity depends only on the mass (the 00-part)?