Question about the derivation of the energy momentum tensor

  • #1
Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf

Homework Statement


As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. [itex]L=-1/4 F^{μν} F_{μν}[/itex] where [itex]F_{μν}=∂_μA_ν-∂_νA_μ [/itex]
I'm stuck at equation (11): [itex] \frac{δL}{δ(∂_μA_λ)}=-F^{μν} [/itex].
Could someone help me understand how they get this result?

Homework Equations


Some tensor product rule maybe?

The Attempt at a Solution


I would get the result if I use the usual product rule. But how do I treat the indices?
Any tips or redirects to where I can read about it is very appreciated :)

Kind regards
Alex
 

Answers and Replies

  • #2
MarcusAgrippa
Gold Member
152
36
Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf

Homework Statement


As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. [itex]L=-1/4 F^{μν} F_{μν}[/itex] where [itex]F_{μν}=∂_μA_ν-∂_νA_μ [/itex]
I'm stuck at equation (11): [itex] \frac{δL}{δ(∂_μA_λ)}=-F^{μν} [/itex].
Could someone help me understand how they get this result?

Homework Equations


Some tensor product rule maybe?

The Attempt at a Solution


I would get the result if I use the usual product rule. But how do I treat the indices?
Any tips or redirects to where I can read about it is very appreciated :)

Kind regards
Alex
Write the Lagrangian density in terms of the derivatives of the vector potential.
Then use the fact that
## \frac{\partial (\partial_\mu A_\nu)}{\partial (\partial_\sigma A_\tau)} = \delta^\mu_\sigma \delta^\nu_\tau ##
 

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