Why doesn't Diffeomorphism Invariance lead to Scale Invariance?

[PAllen]

So how does the presence of a stress-energy term break scale invariance? To use your language, how would we communicate a length to aliens using GR, without reference to common objects?

The particular definition of scale invariance Sam was using in the appendix of the paper he referenced (and co-authored) was:

Multiply the metric by a constant. Is it still a solution of vacuum field equations (if original metric had G(tensor)=0, new metric will have G=0). This is trivially true.

With matter solution, multiply the metric, and the resultant G will be different. Then you can ask the nature of the consequent change to the tensor T (=G). This change will not be trivial, and may even violate an energy condition satisfied by the original T.

 

[Sam Gralla]

I am a little skeptical of this point. Just because the equations are scale-invariant does not mean that solutions to those equations have to be scale-invariant. As Finbar mentions, the Schwarzschild solution has a scale (and the Kerr solution has two scales).

I agree. What I meant (without saying it) was that if you considered some kind of "generic universe with black holes and gravitational radiation" (i.e., a sort of ensemble over all possible solutions), then you wouldn't be able to decide that "black holes are X big", or say anything about an overall scale. You could always take a specific solution, pick a black hole (maybe it only has one black hole anyway), and call that your "scale". But somehow in a generic universe it would be clear that there isn't any "absolute" scale.

This is different from our universe and its theory, where (e.g.) the Compton wavelength of the electron sets a real length scale.

 

[Sam Gralla]

Surely you can refer to a specific black hole. Or send them gravitational waves of a specific wavelength (assuming you have the means to create and detect them, and you know the aliens' relative motion to yours with sufficient accuracy).

Well, if you get to refer to a specific black hole, you might as well refer to your specific spaceship. Similarly, if you get to send the aliens gravitational waves, you might as well send them your spaceship.

I guess I wasn't clear about this, but the idea is not to use any shared knowledge of the particular state of the universe, but only to use the fact that you and the aliens both know the theory of the universe. With a gravity-only (or otherwise scale-invariant) theory, telling them the size of garage to make seems impossible. But with our real universe, you can always say "just measure the Compton wavelength of the electron and make the garage 10^15 times as big".

Of course, none of this is a physical example; it has in mind being able to talk to the aliens by some means that does not use the physical universe. Still, I find it a useful way to think about scale-invariance.

 

[lugita15]

I guess I wasn't clear about this, but the idea is not to use any shared knowledge of the particular state of the universe, but only to use the fact that you and the aliens both know the theory of the universe. With a gravity-only (or otherwise scale-invariant) theory, telling them the size of garage to make seems impossible.

So are you saying nothing in theory of general relativity (as opposed to a particular solution) allows you to communicate a length to aliens without referring to commonly known objects? Could you communicate a length using classical electromagnetic theory?

If you cannot communicate a length using either one of these theories, then how does Feynman's example of a matchstick house make sense? Surely the only forces at play here are gravity and electromagnetism.

 

[Sam Gralla]

So are you saying nothing in theory of general relativity (as opposed to a particular solution) allows you to communicate a length to aliens without referring to commonly known objects? Could you communicate a length using classical electromagnetic theory?

If you cannot communicate a length using either one of these theories, then how does Feynman's example of a matchstick house make sense? Surely the only forces at play here are gravity and electromagnetism.

If there are no sources, Classical electromagnetism is scale-invariant too. (I don't think this has been discussed yet, but in the definition of scale-invariance you get to rescale all fields by some fixed power of lambda. If you can find a power of lambda that preserves the equations of motion, then one says the theory is scale invariant. This corresponds to the physical interpretation we have been discussing.) So I claim you can't tell the aliens a length scale with just source-free electromagnetism and gravity (even if you let them interact; electrovacuum Einstein-Maxwell theory is scale-invariant too). But as soon as sources get involved scale-invariance breaks down. In principle you might be able to find equations for the sources that are scale-invariant, but in any case the types of sources in our real world (including Feynman's matchsticks) are not described by scale-invariant equations. To harp my my previous example, the Compton wavelength of the electron is definitely relevant for the description of wood. It's probably easiest to think about shrinking the matchsticks; at some point you reach the scale of the Compton wavelength of the electron and (whoops!) there aren't going to be matchstick-like solutions any more.

 

[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.