Minkowski space has both translational and Lorentz symmetry, which together give Poincare symmetry. (It also has some discrete symmetries like parity and time-reversal that I won't be concerned with.) In some senses, it seems natural to think of the diffeomorphism invariance/general covariance of general relativity as the "gauged" version of some of these symmetries. But which ones?
1) The equivalence principle is often stated as "spacetime always looks locally like Minkowski space," or "the value of a scalar contracted from Lorentz-covariant tensors at the same point in spacetime is coordinate-invariant," or something along those lines. It seems to me that if you only look at an infinitesimal patch of spacetime, then you can't really talk about translational invariance (which would move you outside the patch), so the symmetry group of that tiny region of spacetime should be thought of as the Lorentz group rather than the Poincare group. If Lorentz symmetry now holds locally at each point in spacetime, then you can say that we have "gauged" the Lorentz group.
2) On the other hand, the "conserved current" in GR is the local conservation of the stress-energy tensor $\nabla_\mu T^{\mu \nu} = 0$.
(I know this will unleash a torrent of commentary about whether GR is a gauge theory, and Noether's first vs. second theorem, and conservation laws that are mathematical identities vs. those only hold under the equations of motion, and so on. Those question have all been beaten to death on this site already, and I don't want to open up that Pandora's box. Let's just say that there is a formal similarity between $J^\mu$ in E&M and $T_{\mu \nu}$ in GR, in that their conservation is trivially true under the equations of motion $\partial_\nu F^{\mu \nu} = J^\mu$ and $G_{\mu \nu} \propto T_{\mu \nu}$ and leave it at that.)
But this is just the diffeomorphism-covariant version of the result $\partial_\mu T^{\mu \nu} = 0$ in Minkowski-space field theory, which is the Noether current corresponding to translational symmetry (As opposed to generalized angular momentum, which corresponds to Lorentz symmetry.) This seems to imply that the natural interpretation of diffeomorphism invariance is as the gauged version of translational symmetry.
Is it more natural to think of diffeomorphism invariance (or the general covariance of GR, which depending on your definitions may or may not be the same thing) as the gauged version of (a) Lorentz symmetry or (b) translational symmetry? Or (c) both (i.e. Poincare symmetry)? Or (d) neither, and these are just vague analogies that can't be made rigorous? If (a) or (b), then why does only a proper subgroup of the Poincare group get gauged? And if (c), then why does only the translational part of the gauge group seem to correspond to a conserved current?
(BTW, I'm looking for a high-level, conceptual answer, rather that one with a lot of math jargon.)
No comments:
Post a Comment