I'm wondering how does one write the energy momentum tensor of a fermion (like an electron)? I've seen the formula for deriving it through the action. What I'm specifically asking is how would each of the 16 different components of the $T_{\mu\nu}$ of the fermion look like and what is the physics meaning for each of those components?
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$\begingroup$ For curved spacetime see physics.stackexchange.com/q/161821/2451 $\endgroup$– Qmechanic ♦Commented Jul 20, 2017 at 5:17
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The energy-momentum tensor for a spinor is given by $T^{\mu\nu}=i\bar\psi\gamma^\mu\partial^\nu\psi$. Its components have the usual meaning, e.g., $T_{00}$ is energy density, etc.
To obtain it, just follow the usual definition: it is the Noether current of spacetime translations, given the spinor Lagrangian ${\cal L}=i\bar\psi\gamma^\mu\partial_\mu\psi$.
More details can be found in every basic field theory textbook (Peskin & Schroeder...)
