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In the following question the OP asked what is meant in chapter 27-4 of Feynman's Lectures on Physics Vol.II by the ambiguity in the location of electromagnetic field energy: Why is there ambiguity of the field energy?

@AccidentalFourierTransform answers the question with an example of how u and S may be redefined and how the location of the energy is ambiguous. It is also mentioned that u and S are contained in $T_{\mu \nu}$ which leads me to believe that their form somehow appears in or is derived from the matter Lagrangian.

My question is: When u and S are redefined, how does that change the E&M Lagrangian? Said differently, what changes in the E&M Lagrangian leave the equations of motion the same (in the absence of gravity) while redefining u and S?

My initial guess was that Feynman was talking about some kind of gauge transformation. However, as Feynman mentioned, the redefinitions would have measurable effects in a full theory of gravity. Therefore, I believe these redefinitions are not gauge transformations.

Thanks in advance for your help!

speckofdust
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Redefining $u$ and $S$ does not even have to change the Lagrangian. One way of seeing this is by considering $T_{\mu \nu}$. If you don't demand it to be symmetric (which is an additional condition that you may need for full covariant consistency) you are free to redefine it by adding a term like $\partial_\sigma f^{\sigma \mu \nu}$ such that $f^{\sigma \mu \nu} = - f^{\mu \sigma \nu} $. In the case of electromagnetism, you can do this explicitly, by taking, for instance, $f^{\sigma \mu \nu} = F^{\sigma \mu} A^{\nu}$. You can check the details in section 12.10 of Jackson's book, Classical Electrodynamics. There you see that a term as this makes the difference between the symmetric stress energy tensor (which reproduces $u$ and $S$ as usual) and the canonical stress energy tensor, which reproduces these quantities up to a four divergence. The equations of motion will also stay the same.

secavara
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