Question about noether's theorem

[geoduck]

If you have purely a coordinate transformation whose Jacobian equals 1, and your Lagrangian density has no explicit coordinate dependence (just a dependence on the fields and their first derivatives), then is it true that the transformation is a symmetry transformation?

It looks like it is, that under these conditions the action is unchanged. I just want some verification.

 

[jambaugh]

Yes, I believe this is the gauge symmetry of unimodular gravity. The corresponding conserved quantity is the traceless part of the stress-energy tensor.

 

[geoduck]

Yes, I believe this is the gauge symmetry of unimodular gravity. The corresponding conserved quantity is the traceless part of the stress-energy tensor.

Thanks.

It seemed trivially true to me that the action is unchanged under any pure coordinate transformation so long as your Lagrangian is not coordinate dependent, but I had forgotten about the Jacobian, and the Jacobian allows you to get a non-trivial result.

When I realized I forgot the Jacobian, then it occurred to me that if the transformation has Jacobian 1, then the trivial result would still follow. Of course translations and rotations have Jacobian 1 and the action is unchanged leading to conservation of energy and angular momentum, but it seems you can dilate each coordinate in such a way that the Jacobian is still 1.