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)?

  • $\begingroup$ Do you really want to use a quantum particle spin to demonstrate the affine connection in GR? Won't the discrete measurements for spin components (and the non commutativity of different components) mess the explanation up? $\endgroup$
    – twistor59
    Jun 21 '13 at 15:26
  • $\begingroup$ @twistor59: you could use expectation values rather than discrete experimental values and still be fine (but then I don't see the advantage of saying 'quantum' beyond just not losing generality) $\endgroup$ Jun 21 '13 at 15:31
  • $\begingroup$ Re geodesics for test particles, see Ehlers and Geroch, arxiv.org/abs/gr-qc/0309074v1 . You might want to consider taking a more physical approach. E.g., I assume that some subset of your mathematical assumptions amounts to the equivalence principle, but which subset? Re measurement, it suffices to have a clock plus test particles; for some nice elementary presentations of this, see Laurent, Introduction to spacetime, or Geroch, General Relativity from A to B. $\endgroup$
    – user4552
    Jun 21 '13 at 15:53
  • $\begingroup$ @BenCrowell: isn't one of the versions of the eqivalence principle just stateable as "test particles follow the geodesics of a semi-Riemannaian manifold" ? $\endgroup$ Jun 21 '13 at 16:10
  • 1
    $\begingroup$ @JerrySchirmer: You can have the equivalence principle in lots of different flavors. Strong, weak, ... See arxiv.org/abs/0707.2748 . $\endgroup$
    – user4552
    Jun 22 '13 at 3:42

I know that this isn't quite a "minimal" set of assumptions to add in, but if you're going to interpret $G_{ab} = 8\pi G T_{ab}$ as having something to do with "ordinary matter", you should be starting with a Lagrangian formulation where you have:

$$S = \int \sqrt{|g|}d^{4}x \frac{1}{16\pi G}R + 2\Lambda + L_{M}$$

where $L_{M}$ represents the Lagrangian density of the ordinary matter. Then, the calculus of variations gives you Einstein's equation, and the "ordinary matter" interpretation of $T_{ab}$ is trivial.

As for the comparison with the Newtonian limit, the only real way to do it is the adored and dreaded Post Newtonian Formalism--where you perturbatively expand in relativistic corrections to Newton's laws, as seen in this article in addition to any GR textbook. It quickly gets very ugly, as you start getting effects like self-force showing up in terms that have factors like $\frac{567849}{98478433}$, but people studying gravitational waves use these techniques pretty regularly.

  • $\begingroup$ And I should add that Weinberg's (old) book has a very physically motivated discussion of PPN, where he carefully shows what the parameters mean and compares GR with newtonian theory. $\endgroup$ Jun 21 '13 at 15:33
  • $\begingroup$ I have also been thinking of using the Euler-Lagrangian approach to GR (which gives notions of momentum, current, etc. for free), but this approach is less based on physical interpretation but more mathematics, isn't it? I think there is also a philosophical difference: when we define the stress-energy of matter simply to be the Einstein tensor (up to constants), the field equations simply become a definition (like one can make the gravitational potential in Newtonian physics the main object and define mass density as its Laplacian) and the question is not how to solve them, but to... $\endgroup$
    – Marc
    Jun 21 '13 at 18:45
  • $\begingroup$ ... find further physical theories that interpret the resulting stress-energy tensor. In the usual Lagrangian approach, we get an equation, the EL equation, which we have to solve (and we had to put the right matter Lagrangian into the equation from the beginning). (Well, maybe the difference is less than I have been thinking... In the Lagrangian formulation, we can simply take the matter Lagrangian as an indeterminate, which won't give an equation to solve but instructions how that matter term will have to look for a given affine connection.) $\endgroup$
    – Marc
    Jun 21 '13 at 18:47
  • $\begingroup$ @Marc: in a real theory, the matter lagrangian will be a real thing defined as well as it is anywhere else. It's perfectly cool to study, for instance, the Einstein-Klein gordon equation, by setting $L_{M} = \nabla^{a}\phi\nabla_{a}\phi - m^{2}\phi^{2}$ $\endgroup$ Jun 21 '13 at 19:38
  • $\begingroup$ And in this case, the matter will have it's own equation of motion to satisfy. You just can't write that down without specifying the matter content. $\endgroup$ Jun 22 '13 at 12:33

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