Spacetime symmetries vs. diffeomorphism invariance

[PAllen]

Here is a way of thinking about theories that perhaps helps to clarify the sense in which GR is simpler than Newtonian gravity, in terms of covariance.

As people have pointed out, any theory of physics can be written in a covariant form. However, when one writes Newtonian physics in a covariant form, one finds that there is a "nondynamic" scalar field--Newton's universal time. It's nondynamic, in the sense that matter and energy and so forth don't affect it.

If you write Special Relativity in a covariant form, you will find that it has no nondynamic scalar fields. But it has a nondynamic tensor field, namely the metric tensor.

In General Relativity, there are no nondynamic scalar, vector or tensor fields. All the geometric fields are dynamic, they are affected by matter and energy.

I agree with this, and it is what the reference WNB gave in post#2 is formalizing (based on the approach given by Anderson in 1967 - where I first encountered it).

I realized there is a further terminological confusion going on, which I have contributed to (). Within covariantly formulated theories, one may produce scalar invariants. However this sense of an invariant is different from formulating a principle of general invariance that is distinct from covariance.

I have always preferred direct terminology like "no absolute objects" as the way way to formulate a theory filtering principle as opposed to a language for formulating any theory (general covariance) - rather than using the historically confused 'general invariance' for this. Some have used that the symmetry group of a theory ( as defined by Anderson - equivalently Straumann ) is the diffeomorphism group as the substantive (theory filtering) principle. To me, though, it all gets back to absolute objects, because absolute objects underpin this definition of the symmetry group of a theory.

 

[PAllen]

Why "NOT"? From particle interaction to galaxy formation, we see "nature uses as little as possible".
I would say that nature uses Occam's in the same way it uses the principle of least action. Is the action principle a law of nature?

Why are there three particle generations not one? Why are there 4 forces, not fewer? Occam's razor is, at best, a guide, just another way of saying simpler, more elegant theories are usually right.

Least action is a very different case. From classical, pre-relativity physics, to GR, to QFT, all experience, without exception, is consistent with "theories of nature are described by action principles".

 

[dEdt]

Well, this is exactly what I was referring to as needing to add to the requirement of coordinate independence. The law is definitely coordinate invariant if you express it appropriately (and quite naturally). What you want to add is an additional requirement: that there is no simpler expression of the law that is true in some coordinate systems. To me, that is not coordinate invariance or general covariance, but something else. Anderson and others have worked on formulating this something else, but they admit that it is something beyond coordinate invariance.

It seems like we're not actually disagreeing with each other, just that we're using different definitions for "coordinate invariant".

 

[samalkhaiat]

Why are there three particle generations not one? Why are there 4 forces, not fewer?

Maybe you should ask Occam. Still 3 generations is more natural than 3 millions resonances. Laws of nature show decoupling at different length scales, and this exactly why we are still working towards unification.

Least action is a very different case. From classical, pre-relativity physics, to GR, to QFT, all experience, without exception, is consistent with "theories of nature are described by action principles".

Isn’t this what I was trying to tell you? So, Is the action principle a law of nature?

Sam

 

[PAllen]

Maybe you should ask Occam. Still 3 generations is more natural than 3 millions resonances. Laws of nature show decoupling at different length scales, and this exactly why we are still working towards unification.




Isn’t this what I was trying to tell you? So, Is the action principle a law of nature?

Sam

Yes, I'm agreeing with that (action principle), emphatically. While I think Occam's razor is not a law of nature but only a guide, effectively the same as 'seek the simplest possible laws', with a lot of judgment about what that means.