Diffeomorphism invariance and Noether's theorem

[cuallito]

I've read that GR is diffeomorphism invariant, I asked a math buddy of mine and I have a VERY BASIC idea of what that means in this case - the theory is the same regardless of your choice of coordinates?

Noether's theorem states that for every symmetry there's a corresponding conservation law. Would diffeomorphism invariance count as a symmetry in this case and what would be the conserved quantity? Would it just be mass-energy?

 

[atyy]

As I understand, but am not sure, diffeomorphism invariance is a gauge symmetry, so there is no conserved quantity associated with it. Rather the gauge symmetry can be used as a form of minimal coupling, just like generating the minimal coupling of the Maxwell and Dirac fields. Universal minimal coupling in general relativity is an expression of the principle of equivalence. http://arxiv.org/abs/hep-th/0009058

However, there are apparently a bunch of conserved non-gauge invariant quantites that follow http://arxiv.org/abs/hep-th/9310025

 

[bcrowell]

As I understand, but am not sure, diffeomorphism invariance is a gauge symmetry, so there is no conserved quantity associated with it.

Hmm...gauge symmetry of electromagnetism leads to conservation of charge. (If charge were not conserved, then the amount of energy required in order to create a charge q at a certain point in space would depend on the potential at that location, so you could determine the potential absolutely, not just up to an additive constant. This would violate gauge invariance.) So it seems to me that gauge symmetries *can* lead to conservation laws.

 

[bcrowell]

I've read that GR is diffeomorphism invariant, I asked a math buddy of mine and I have a VERY BASIC idea of what that means in this case - the theory is the same regardless of your choice of coordinates?

Yes, that's right.

Noether's theorem states that for every symmetry there's a corresponding conservation law. Would diffeomorphism invariance count as a symmetry in this case and what would be the conserved quantity? Would it just be mass-energy?

This is a good question, and I don't know enough to give a really satisfying answer. However, diffeomorphism invariance definitely does not lead to conservation of mass-energy in general relativity, because mass-energy is not conserved in general relativity. There is not even any good way to define mass-energy in general relativity: http://en.wikipedia.org/wiki/Mass_in_general_relativity

One thing to be careful about is that Noether's theorem is actually a lot of different theorems, which say things that are much more specific and limited than "for every symmetry there's a corresponding conservation law." I believe she actually had two theorems in her original paper http://www.physics.ucla.edu/~cwp/pubs/noether.trans/english/mort186.html , and those have to be generalized in order to deal with continuous fields.

This may be relevant: http://en.wikipedia.org/wiki/Stress-energy_tensor#Conservation_law

 

[bcrowell]

I found a discussion of this in Penrose's "The Road to Reality," on p. 489.

For example, it is not at all a clear-cut matter to apply these ideas to obtain energy-momentum conservation in general relativity, and strictly speaking, the method does not work in this case. The apparent gravitational analogue [of EM gauge symmetry] is 'invariance under general coordinate transformations' [...] but the Noether theorem does not work in this situation, giving something of the nature '0=0'.

There are some fairly general reasons why GR can't have nonlocal conservation laws. This is discussed in Misner, Thorne, and Wheeler, "Gravitation," p. 457.

 

[atyy]

Hmm...gauge symmetry of electromagnetism leads to conservation of charge. (If charge were not conserved, then the amount of energy required in order to create a charge q at a certain point in space would depend on the potential at that location, so you could determine the potential absolutely, not just up to an additive constant. This would violate gauge invariance.) So it seems to me that gauge symmetries *can* lead to conservation laws.

Well, there are two "gauge" symmetries. There is exp(i.theta) which is a global symmetry and gives charge conservation. But exp(i.theta(x)) is a local or "gauge" symmetry, and this can lead to the minimal coupling or "gauge principle" with the electron field, but there is no additional conserved quantity from having more symmetry here than the global symmetry.

 

[bcrowell]

Well, there are two "gauge" symmetries. There is exp(i.theta) which is a global symmetry and gives charge conservation. But exp(i.theta(x)) is a local or "gauge" symmetry, and this can lead to the minimal coupling or "gauge principle" with the electron field, but there is no additional conserved quantity from having more symmetry here than the global symmetry.

Ah, I see. Thanks for explaining that!

 

[atyy]

There's a pretty clear discussion on p 126 of http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf , where the (local) gauge symmetry is a redundancy of description, and leads to equations of motion that don't have unique evolution - unless we accept that the multiple possible evolutions have the same physical meaning - exactly the same as in GR.