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I have the following ansatz $$T^{\mu\nu}=AF^{\mu\alpha}F_{\alpha}^\nu+B\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}$$ for some constants $A,B.$ Here, $F_{\mu\nu}$ is the electromagnetic field tensor, and $\eta^{\mu\nu}$ is the (flat?) metric with signature $(-,+,+,+).$ I need to show that energy and momentum conservation of the EM field (i.e. $\partial_\mu T^{\mu\nu}=0$ for $\nu=0,1,2,3$) implies that $B=A/4$.

The only idea I have is to expand everything and try to eventually compare coefficients. But this is super tedious and unenlightening. Is there a better, smarter approach to this? I am just hoping for a nudge in the right direction.

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  • $\begingroup$ Use $F^{\nu}_{\alpha} = \eta_{\alpha \beta} F^{\nu \beta}$ or the same transformation for covariant components. And then do direct computation of derivatives $\endgroup$
    – basics
    Oct 9, 2022 at 21:59
  • $\begingroup$ A calculation that takes a few lines doesn’t count as “super tedious”. $\endgroup$
    – Ghoster
    Oct 10, 2022 at 6:52
  • $\begingroup$ a nudge in the right direction You have to make use of Maxwell’s equations in covariant form. $\endgroup$
    – Ghoster
    Oct 10, 2022 at 6:55

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Your formula does not make sense. The left-hand side has indices $\mu,\nu$, and the first term of the right-hand side has indices $\alpha,\nu$.

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  • $\begingroup$ indices look fine to me $\endgroup$
    – basics
    Oct 9, 2022 at 21:37
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    $\begingroup$ Thanks. I had a typo. Corrected now! (Shouldn't that have just been a comment though?) $\endgroup$
    – Stuck
    Oct 9, 2022 at 21:37
  • $\begingroup$ @basics : Because the OP edited his/her question a minute before your comment:-) $\endgroup$
    – akhmeteli
    Oct 9, 2022 at 21:43

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