Hi all, Just a question i was wondering about. We know that in electrodynamics the Lagrangian is invariant under a gauge transformation of the potential, and this is equivalent to the law of conservation of charge. Concerning relativity, what is the quantity that is conserved and are the gauge transformations Poincare transformations?
The gauge group is the smooth coordinate transformations (diffeomorphisms, technically), which includes the Poincare group as a subgroup. Penrose has a brief discussion of how Noether's theorem relates (or doesn't relate) to this on p. 489 of The Road to Reality: "For example, it is not at all a clear-cut matter to apply these ideas to obtain energy-momentum conservation in general relativity, and strictly speaking, the method does not work in this case. The apparent gravitational analogue [of EM gauge symmetry] is 'invariance under general coordinate transformations' [...] but the Noether theorem does not work in this situation, giving something of the nature '0=0'." I don't understand the technical details of N's theorem well enough to know what exactly it is that fails in this case. I believe that what she published in 1918 is actually a very restricted version of the theorem, whereas what physicists refer to today as "Noether's theorem" is actually a loosely defined set of generalizations of the 1918 version. There are, however, some pretty simple and fundamental reasons why it *can't* apply to GR. If it did apply to GR, then the conserved quantity it gave would have to be the energy-momentum four-vector. But Gauss's theorem fails in a curved spacetime when the conserved quantity isn't a scalar, basically because any attempt to define the total flux through a surface is subject to the ambiguity introduced by having to parallel-transport the flux (which is a vector in this case) from one part of the surface to another.
Or, for another way to treat GR as a gauge theory, you can recast it in Ashtekar's variables: http://arxiv.org/abs/gr-qc/9312032