# Energy-momentum tesor

Science Advisor

## 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

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hellfire
Science Advisor
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.

Science Advisor
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.