Einstein tensor as a conserved current?

Question

As is well-known, the ``traditional" conserved quantities (energy, momentum...) are Noether currents whose conservation depends on the existence of various Killing fields in Minkowski space. In general spacetimes, we can expect to find no such symmetries, and thus no conserved quantities.

Now in some spacetimes we find the existence of conserved quantities arising from more complicated transformations than parallel drag along a vector field; for example, the Carter constant of the Kerr spacetime. Interestingly the resulting conservation law seems to be in some senses weaker than those of the ``traditional" quantities; see e.g. http://arxiv.org/abs/1503.05164.

One more-complicated transformation one might consider is that of a general diffeomorphism. Indeed the basic idea of GR is that all spacetimes related by a diffeomorphism are to be physically identified (see http://relativity.livingreviews.org/Articles/lrr-2014-1/). Mathematically this does not exclude any spacetime, but instead binds together an equivalence class of Einstein tensors $G_{ab}$ with one of stress-energy tensors $T_{ab}$, precisely via the Einstein field equation $$G_{ab} = 8\pi T_{ab}.$$

According to this point of view, the above equation is simply the conservation law associated with general covariance. My question is whether this can be made more precise: can the Einstein field equations be viewed as an expression of Noether's theorem?

Answer 1

The analogous gauge theory statement is the equations of motion of the gauge field which takes the form $$ \nabla^\mu F_{\mu\nu} = j_\nu $$ The weakest mathematical statement of the Noether's theorem is the conservation of a charge $$ \frac{dQ}{dt} = 0 $$ or more generally in local quantum field theories $$ \partial_\mu j^\mu = 0 $$ The first equation is to be understood as equations that govern the dynamics of the gauge field, not as a statement of Noether's theorem itself. Notice however that, as is obviously required, the equations are consistent with the statement of Noether's theorem.

Similarly, in GR, the Noether's current corresponding to space-time isometries is the stress tensor and the statement of Noether's theorem is $$ \nabla^\mu T_{\mu\nu} = 0 $$ The Einsteins equation is then seen as the set of equations that govern the dynamics of the metric (perturbations).