Why doesn't Diffeomorphism Invariance lead to Scale Invariance?

[lugita15]

But as soon as sources get involved scale-invariance breaks down.

So for instance, how would you define an absolute length scale using Maxwell's equations alone, without referencing known physical objects like electrons?

 

[Sam Gralla]

So for instance, how would you define an absolute length scale using Maxwell's equations alone, without referencing known physical objects like electrons?

I can't; that's the point. Maxwell's equations (without sources) are scale-invariant. But if you allow sources (like electrons), then you can.

 

[lugita15]

I can't; that's the point. Maxwell's equations (without sources) are scale-invariant. But if you allow sources (like electrons), then you can.

I think you misunderstood me. I'm asking for how Maxwell's Equations WITH sources can be used to communicate an absolute length to aliens. What I don't want reference to is objects that you and them know. So it's cheating to say "that black hole we're both familiar with," just as it's cheating to say "that subatomic particle we're both familiar with." How can you use the laws alone ?

 

[TrickyDicky]

I think you misunderstood me. I'm asking for how Maxwell's Equations WITH sources can be used to communicate an absolute length to aliens. What I don't want reference to is objects that you and them know. So it's cheating to say "that black hole we're both familiar with," just as it's cheating to say "that subatomic particle we're both familiar with." How can you use the laws alone ?

According to relativity and the equivalence principle the covariant derivative of the metric must vanish there too, this allows the aliens to parametrize their geodesics with a proper time parameter that should be equally valid here and there, knowing this it is just a matter of ascertaining the distance covered by EM waves in vacuum for a certain previously agreed time.

 

[PAllen]

I see a big difference between referring to a unique object versus referring to feature of the universe which can be independently measured anywhere. Thus .001 times the distance light travels in one vibration of atom x is very different from half the height of the tallest mountain on planet x.

 

[Sam Gralla]

I think you misunderstood me. I'm asking for how Maxwell's Equations WITH sources can be used to communicate an absolute length to aliens. What I don't want reference to is objects that you and them know. So it's cheating to say "that black hole we're both familiar with," just as it's cheating to say "that subatomic particle we're both familiar with." How can you use the laws alone ?

I'm claiming that the size of the electron really should have the status of a "law" and not a "particular state of the universe", because electrons are the same everywhere. In particular the equations that describe them will not be scale-invariant. (Since these equations aren't classical, I can't use this example to relate to the definition we've been discussing in a precise sense. But most if not all realistic smoothed-out equations describing classical sources to be put on the RHS of Maxwell's equations will inherit the dependence on scale. For example, take Feynman's matchsticks and make them charged.)

But if you want to disagree about what's a "law" and what's a "state", that's totally fine. The main point I'm making is that once the laws are given in the form of classical equations of motion, then there is A) a precise definition of scale-invariance and B) a couple of pretty good physical interpretations.