# Noether currents associated with diffeomorphism invariance

1. May 23, 2010

### GargleBlast42

Having some generic curved spacetime, what are the Noether currents that are guaranteed to exist by diffeomorphism invariance? Is the energy-momentum tensor such a current?

2. May 25, 2010

### Staff: Mentor

I wanted to bump this since I have a similar question. Poincare symmetry includes spacetime translations (energy and momentum), spatial rotations (angular momentum), and boosts. What is the Noether current associated with symmetry under boosts?

3. May 25, 2010

### George Jones

Staff Emeritus
4. May 25, 2010

### GargleBlast42

Yeah, right, but this is clear to me.
I'm asking more generally, if you have just diffeomorphism invariance, do you always have some associated currents/charges?

5. May 27, 2010

### samalkhaiat

Like local gauge invariance, diffeomorphism invariance of the action integral is the subject of the 2nd Noether theorem. The conservation statements of this theorem are nothing but the twice contracted Bianchi identities. However, one can use the (gravitational) field equations to obtain “genuine” conservation laws from the twice contracted Bianchi identities. That is;

$$\partial_{a}(T^{ab} + t^{ab}) = 0$$

where $t^{ab}$ is the gravitational energy-momentum pseudotensor. It follows from this that (in curved spacetime) neither matter nor gravitational fields obey separate conservation laws. Also, it is a mistake to associate $T^{ab}$ solely with the matter field and $t^{ab}$ with the “pure” gravitational field, because the theory is highly nonlinear; the T depends on the metric (geometry) as well as the matter field quantities, and the t depends on the matter distribution through the metric. Further complications come from the fact that t is a frame dependent (non covariant) object.

An excellent, possibly the best (old) survey article discussing Noether theorems and conservation laws in curved spacetime is given by A.Trautman, “Foundations and Current Problems of General Relativity”, In Gravitation: An Introduction to Current Research, Ed, L. Witten, John Wiley & Sons Inc. N.Y.

Regards

sam

6. May 27, 2010

### haushofer

I have a related question.

I have an action which is invariant under a certain symmetry group, but the Lagrangian transforms as a total derivative. I get the feeling that this changes my Noether current, but because the Noether charges obey the symmetry algebra I would suspect that such a "Lagrangian transforming as a total derivative" introduces some sort of a central charge in my symmetry algebra.

Is this true, and where could I find some information about these things?

7. May 28, 2010

### haushofer

By the way, a nice article to read here is "Black hole entropy is Noether charge" by Robert Wald, where the Noether charge of diffeomorphism invariance is coupled to the entropy of stationary black holes.

8. May 28, 2010

### haushofer

I found the answer on my own question. It's about the Galilei group :)

The action of a free, nonrelativistic particle is invariant under the Galilei group, but its Lagrangian is not; it changes by a total derivative. Thus the corresponding Noether charge has to be adjusted. As such the Poisson brackets of the Noether charges will change; if you now calculate the Poisson bracket of the Noether charges corresponding to spatial translations and boosts a central element will appear.

And because the algebra of the Poisson brackets of the Noether charges is isomorphic to the algebra of the global symmetry group, this central element will also pop up in the commutator of boosts and translations.

9. Aug 2, 2010

### jdstokes

I don't believe this is correct. The covariant conservation of energy-momentum for any field theory follows directly from diffeomorphism invariance of the matter action. The twice contracted Bianchi identity follows from diffeomorphism invariance of the kinetic term in the Einstein Hilbert action.