In Wald's book General Relativity, Wald gives the energy-momentum tensor of the scalar field that leads to the Klein-Gordon equation, $$ \begin{align} T_{\mu\nu} =\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\eta_{\mu\nu}\left(\phi_{,\sigma}\phi^{,\sigma}+m^2\phi^2\right),\tag{1} \end{align} $$ wherein I used the notation $\phi_{,\mu}:=\partial_\mu\phi$. Eq. (1) is eq. (4.2.20) on p. 63; Wald uses the $(-1,+1,+1,+1)$ convention for the Minkowski metric. The Lagrangian density of the real free and massive scalar field that satisfies the Klein-Gordon equation using said metric convention is given by, $$ \begin{align} \mathcal{L}=-\frac{1}{2}\phi_{,\sigma}\phi^{,\sigma}-\frac{1}{2}m^2\phi^2=-\frac{1}{2}\phi_{,\sigma}\phi^{,\sigma}-V(\phi).\tag{2} \end{align} $$ The former expression can also be found on p. 451 as Eq. (E.1.6). We now write Eq. (1) as, $$ \begin{align} T_{\mu\nu} =\phi_{,\mu}\phi_{,\nu}+\eta_{\mu\nu}\mathcal{L}.\tag{3} \end{align} $$ The energy-momentum tensor with raised indices can be obtained from Eq. (3), $$ \begin{align} T^{\mu\nu} =\eta^{\mu\sigma}\eta^{\nu\rho}T_{\sigma\rho}=\phi^{,\mu}\phi^{,\nu}+\eta^{\mu\sigma}\eta^{\nu\rho}\eta_{\sigma\rho}\mathcal{L} =\eta^{\mu\sigma}\eta^{\nu\rho}T_{\sigma\rho}=\phi^{,\mu}\phi^{,\nu}+\eta^{\mu\nu}\mathcal{L}.\tag{4} \end{align} $$ From now on, we will refer to the energy-momentum tensor of the Klein-Gordon scalar field as represented in Eq. (4).

Okay, so now get to the actual confusion. In the appendix, Wald derives the energy-momentum tensor in eq. (E.1.36) $$ \begin{align} T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\phi_{,\mu})}\phi^{,\nu}-\eta^{\mu\nu}\mathcal{L}\tag{5} \end{align} $$ from Noether's theorem on p. 457 and calls it the canonical energy-momentum tensor. Let us first evaluate, $$ \begin{align} \frac{\partial\mathcal{L}}{\partial(\phi_{,\mu})} =-\frac{1}{2}\eta^{\sigma\rho}\frac{\partial(\phi_{,\sigma}\phi_{,\rho})}{\partial(\phi_{,\mu})} =-\frac{1}{2}\eta^{\sigma\rho}\delta_{\sigma\mu}\phi_{,\sigma}-\frac{1}{2}\eta^{\sigma\rho}\delta_{\sigma\rho}\phi_{,\rho} =-\phi^{,\mu}, \tag{6} \end{align} $$ and insert Eq. (6) into (5). We find, $$ \begin{align} T^{\mu\nu}=-\phi^{,\mu}\phi^{,\nu}-\eta^{\mu\nu}\mathcal{L}. \tag{7} \end{align} $$ It is obvious that Eq. (7) does not equal (4), however, both equations only differ by a sign.

Can it be that the sign difference has something to do with Does metric signature affect the stress energy tensor?

If not, how can one explain the sign difference?


1 Answer 1


OP is right: Wald's book has a sign mistake in eq. (E.1.36).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.