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I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as $$\mathscr{L}= \frac{1}{2}\eta_{\mu\nu}(\partial^{\mu}\phi)(\partial^{\nu}\phi)-\frac{1}{2}m^2\phi^2$$ and I want to compute the energy-momentum tensor $T^{\mu\nu}$. Beginning with the relation $$ T^{\mu\nu} = \frac{\partial \mathscr{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-\eta^{\mu\nu}\mathscr{L}$$ one can find ${T_{\mu}}^{\nu} = \eta_{\mu\nu} T^{\mu\nu}$ and then afterwards go back to $T^{\mu\nu}$ through $T^{\mu\nu} = \eta^{\nu\mu} {T_{\mu}}^{\nu}$.

My questions are the following:

  • Is there a difference between $\frac{\partial}{\partial(\partial_{\mu}\phi)}$ and $\frac{\partial}{\partial(\partial^{\mu}\phi)}$, and if so how does one compute $\frac{\partial}{\partial(\partial_{\mu}\phi)} \frac{1}{2}\eta_{\mu\nu}(\partial^{\mu}\phi)(\partial^{\nu}\phi)$?

  • What happens to the term $\frac{\partial \mathscr{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi$ when multiplying by $\eta_{\mu\nu}$? Should any indices be moved? My current understanding is that I first need to find an expression for this not involiving $\frac{\partial}{\partial(\partial_{\mu}\phi)}$ and then just move all upper indices $\mu$ down.

As you probably can guess I am very new to this type of notation, and I would very much appreciate any help.

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  • When contracting with the metric $\eta_{\mu\nu}$ one has explicitly,

$$\eta_{\mu\nu} \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)} \partial^\nu \phi = \frac{\partial \mathcal L}{\partial (\partial_0 \phi)} \partial^0 \phi - \sum_{i=1}^3\frac{\partial \mathcal L}{\partial (\partial_i \phi)} \partial^i \phi$$

  • For the other term, one has,

$$\frac12 \eta_{\mu\nu} \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)} \partial^\mu \phi \partial^\nu \phi = \frac12 \left(\frac{\partial \mathcal L}{\partial (\partial_0 \phi)} (\partial^0 \phi)^2 - \sum_{i=1}^3 \frac{\partial \mathcal L}{\partial (\partial_i \phi)} (\partial^i \phi)^2 \right)$$

Note: the metric convention is $\eta = \mathrm{diag}(1,-1,-1,-1)$.

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