Energy-momentum tesor

samalkhaiat
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Main Question or Discussion Point

How can you go about and prove the following :
The energy-momentum tensor for any classical field theory = -2 X the functional derivative of the action with respect to the metric tensor.
 

Answers and Replies

hellfire
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Basically, if you have an action for matter interacting with gravity S = SG + SM and perform the functional derivative of S with respect to the metric tensor, you will get a first part corresponding to SG, which will be Einstein’s equations in vacuum, and a second part corresponding to SM, which will be the source of gravity. This is the covariant energy momentum tensor.
 
samalkhaiat
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You are right, this is how text books "define" the mater field E-M tensor in GR. This was not what I asked. I wanted a proof for the following;
given any "free" matter field Lagrangian, show that the symmetric E-M tensor is equal to -2 X the functional derivative of the "free" matter action with respect to the metric tensor.(I am asking for proof with no reference to gravity). thanks.
 

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