A Jackson justification of the Poynting vector by GR

[coquelicot]

The Poynting vector is a definition, that is supposed to represent the energy flow at each point. Unfortunately, the only observable effect caused by the Poynting vector is through the energy variation in a volume subject to an energy flux through its surface, that is, the Poynting theorem.

As a curl could be added to the Poynting vector without changing the Poynting theorem, it can not be decided by EM only that this should be the actual flow of energy at each point. Feynman, commenting other proposed forms of energy flow, just said that "no one has ever found something bad with the Poynting vector". Jackson seemed to be aware of this issue, because he said, and apparently demonstrated in his book, that "the Poynting vector is the only expression of the energy flow compatible with GR".

I cannot follow Jackson in his argument. Could someone try to simplify/explain Jackson argument? Is there a better argument for the form of the Poynting vector?

Jackson 3d edition, section 12.10 (claim about the uniqueness of the energy flow representation in sec 6.7 relative to the Poynting vector, where the author sends us to sec 12.10).

 

[JimWhoKnew]

I'm currently away from my copy of Jackson's, so I can't look it up. However, note that in classical physics, QM and QFT we can add a complete time derivative to the Lagrangian without affecting the physics (equations of motion). Consequently, we can derive by Noether theorem many expressions for the energy-momentum tensor. They are all equivalent, since the mentioned theories don't care about the absolute values of the components - only the differences affect measurable quantities. This is not the case in GR, where the EMT is the "source of curvature" (RHS of Einstein's field equations), and its absolute values do matter(!). Only one specific form can be compatible.

 

[coquelicot]

I'm currently away from my copy of Jackson's, so I can't look it up. However, note that in classical physics, QM and QFT we can add a complete time derivative to the Lagrangian without affecting the physics (equations of motion). Consequently, we can derive by Noether theorem many expressions for the energy-momentum tensor. They are all equivalent, since the mentioned theories don't care about the absolute values of the components - only the differences affect measurable quantities. This is not the case in GR, where the EMT is the "source of curvature" (RHS of Einstein's field equations), and its absolute values do matter(!). Only one specific form can be compatible.

Thank you for your answer. I would appreciate something more detailed though, as I am not that good at GR.