GR Noether Current: Confirm Pseudo Tensor?

In summary, the pseudotensor is not the same as the canonical energy-momentum tensor, and it's not gauge invariant.
  • #1
fhenryco
63
5
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
 
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  • #2
Yes you get the same pseudo-tensor. In fact you can show that without using the explict form of the Einstein-Hilbert action.
 
  • #3
fhenryco said:
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.
 

Related to GR Noether Current: Confirm Pseudo Tensor?

1. What is a GR Noether Current?

The GR (General Relativity) Noether Current is a mathematical concept in the field of physics that describes the conservation of energy and momentum in a system governed by Einstein's theory of general relativity.

2. How is the GR Noether Current related to the Pseudo Tensor?

The GR Noether Current is derived from the Pseudo Tensor, which is a mathematical construct used to describe the distribution of energy and momentum in a gravitational field. The two are closely related and are used together to understand the dynamics of a system in general relativity.

3. Why is it called a Pseudo Tensor?

The term "pseudo" refers to the fact that the Pseudo Tensor behaves differently from a traditional tensor in terms of its transformation properties. It is not a true tensor, but a mathematical entity that behaves like one in certain ways. This is necessary in general relativity due to the non-Euclidean nature of space-time.

4. How is the GR Noether Current confirmed?

The GR Noether Current is confirmed through mathematical calculations and equations that are based on the principles of general relativity. These calculations can be used to predict the behavior of a system and can be tested through experiments and observations. When the predicted results match with the observed results, it confirms the validity of the GR Noether Current.

5. What are the implications of the GR Noether Current?

The GR Noether Current has important implications in understanding the conservation laws of energy and momentum in a system governed by general relativity. It also helps in predicting the behavior of objects in a gravitational field, such as the motion of planets, stars, and galaxies. Additionally, it has been used to develop other theories and concepts in physics, such as the theory of black holes and gravitational waves.

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