Diffeomorphism Invariance of General Relativity

Question

I'm sorry I know this has been asked before, but I'm still a bit confused. I understand that an active diffeomorphism $\varphi:M\to M$ can be equivalently viewed as a coordinate transformation so that since the equations of general relativity are tensorial $\varphi^*g$ will be a solution to Einstein's equations if $g$ is. However I don't see how that same reasoning doesn't imply that other physical theories are diffeomorphism invariant. What's the difference between general relativity and other physical theories, like classical mechanics? Why can't diffeomorphisms be viewed as coordinate transformations in both (or am I confused?).

Answer 1

The diffeomorphism invariance of GR means we're operating in the category of natural fiber bundles, where for any bundle $Y\to X$ of geometric objects that appear in the theory, we have a monomorphism $$ \mathrm{Diff} X \hookrightarrow \mathrm{Aut} Y $$ Any diffeomorphism of space-time $X$ needs to lift to a general covariant transformation of $Y$, which are not mere coordinate transformations.

These transformations play the role of gauge transformations of GR, but are different from the gauge transformations of Yang-Mills theory: The latter are related to the inner automorphisms of the group and are vertical, ie they leave space-time alone.

I'm not sure about the naturalness of the various geometric formulations of classical mechanics - I'd be interested in that as well (but am too lazy to look into it right now).

Answer 2

ben's answer is exactly correct. The idea of diffeomorphism invariance (or "general covariance") was extremely important to Einstein for developing GR, but that fact has led to the unfortunately common misconception that it's somehow special to GR. Under the current way of thinking, all physical theories are diffeomorphism invariant - this follows more or less trivially from the definition of a "physical theory."

What is special about GR is this: for most other theories, there is a natural system of coordinates (usually Cartesian coordinates) in which the fundamental equations that define the theory take on a particularly simple form. Indeed, it's so natural to work in those coordinates that we rarely explicitly specify that the forms of the equations that we write down only apply in that coordinate system. In GR, on the other hand, there's no preferred system of coordinates - e.g. writing Einstein's equations out in Cartesian coordinates would decrease rather than increase the physical intuition - so we almost always go ahead and work with manifestly coordinate-independent equations.

(Technical note: in GR, we often consider spacetimes that are topologically different from Minkowski space, so there would be issues with using global Minkowski coordinates anyway.)

(Also, the extent to which you agree with the second paragraph above may depend on whether consider the metric to be a field on spacetime, or "spacetime itself." See here for a discussion.)

Answer 3

You are correct that any theory can be written in diffeomorphism invariant language. Diffeomorphism invariance does not have anything to do with GR, except that it is hard (impossible?) to write down the theory in a gauge-fixed way.

What sets GR apart from other theories, aside from the fact that the metric is dynamical, is that it does not allow for "prior spacetime," in the language of Misner, Thorne, and Wheeler. This means it cannot be consistently coupled to non-dynamical background fields. I do not see how this feature is related to diffeomorphism invariance.