GR Noether Current: Confirm Pseudo Tensor?

[fhenryco]

Dear,

If i start from the Einstein Hilbert ACtion and apply the usual Noether rules (as we use them on flat spacetime ie treating the metric tensor g_munu as any other tensor assuming the existence of another hidden tensor eta_munu describing a flat spacetime non dynamical background, though the latter appears actually nowhere in the action which makes this approach appear quite metaphysical though it might make more sense if the true dynamical metric g_munu describes an asymptotically minkowskian spacetime)

Considering translations (the action is general covariant so also invariant under translations of course) the Noether current computing is expected to be tedious (in part because the Lagrangien involves second order derivatives) ... but i guess this has been done already a long time ago and my question is:

- can anyone confirm that i will get the GR pseudo tensor (the one used to compute the energy-momentum carried by GW) ?

the answer is difficult to find in most textbook because there is a much easier way to introduce the pseudo energy-momentum tensor starting from the Einstein equation and making use of the linearised Bianchi identities...

So i need confirmation that the pseudo tensor is also the same that would be obtained by applying the Noether theorem.

Thank you in advance

Fred

 

[samalkhaiat]

Yes you get the same pseudo-tensor. In fact you can show that without using the explict form of the Einstein-Hilbert action.

 

[pervect]

Dear,

If i start from the Einstein Hilbert ACtion and apply the usual Noether rules (as we use them on flat spacetime ie treating the metric tensor g_munu as any other tensor assuming the existence of another hidden tensor eta_munu describing a flat spacetime non dynamical background, though the latter appears actually nowhere in the action which makes this approach appear quite metaphysical though it might make more sense if the true dynamical metric g_munu describes an asymptotically minkowskian spacetime)

Considering translations (the action is general covariant so also invariant under translations of course) the Noether current computing is expected to be tedious (in part because the Lagrangien involves second order derivatives) ... but i guess this has been done already a long time ago and my question is:

- can anyone confirm that i will get the GR pseudo tensor (the one used to compute the energy-momentum carried by GW) ?

the answer is difficult to find in most textbook because there is a much easier way to introduce the pseudo energy-momentum tensor starting from the Einstein equation and making use of the linearised Bianchi identities...

So i need confirmation that the pseudo tensor is also the same that would be obtained by applying the Noether theorem.

Thank you in advance

Fred

My understanding (from Wald, "General Relativity", pg 457 - in appendix E) is that when you apply Noether's therom to the Einstein-Hilbert action, you get the canonical energy-momentum tensor $S^{ab}$.

Which isn't the same as the pseudotensor.

For fields without spin (Klein Gordon fields), this cannonical tensor is the same as the stress energy tensor $T^{ab}$

For fields with spin, it is not the same as $T^{ab}$- it's also not gauge invariant, and it's not symmetric.

The stress-energy tensor $T^{ab}$ is determined by the functional derivative of the Lagrangian with respect to the metric (see Wald, 450-451). This gives a "local" form of the conservation of energy $\nabla_a T^{ab} = 0$, which amounts to the continuity equation, that the energy momentum in an infinitesimal volume is conserved. It doesn't provide a definition of energy that's conserved in a system with finite volume, however. Pseuedo-tensors and other approaches do give a notion of energy that's conserved in a system with finite volume - there are approaches that give you this notion of conserved energy in terms of Noether's theorem, but you need to consider things like asymptotic time translation symmetries "at infinity" to get there.

Wald has a discussion of how this works, while I could and have tried to summarize it, it's technical enough that you'd be better off getting it from the text.