In the lecture notes of David Tong on String Theory he defines the energy momentum tensor of the polyakov action as \begin{align*} T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\alpha\beta}}.\tag{1} \end{align*} Since the equation of motion for the dynamical metric $g_{\alpha\beta}$ can be obtained by $$\frac{\delta S}{\delta g^{\alpha\beta}}=0\tag{2}$$ of course this energy momentum tensor vanishes on-shell. I'm a bit confused about this statement. In GR we define the energy momentum tensor exactly the same way, but there of course the EMT does not vanish, but its derivative $\nabla_\mu T^{\mu\nu}$ does. So why don't we conclude in GR that the EMT also vanishes on-shell? What am I getting wrong here?
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$\begingroup$ The partial derivative $\partial_\mu T^{\mu\nu}$ does not vanish! The covariant derivative $\nabla_\mu T^{\mu\nu}$ vanishes due to energy-momentum conservation. $\endgroup$– Samuel Adrian AntzJun 5 at 10:52
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$\begingroup$ Einstein's equation is basically the statement that the stress tensor of GR does vanish. We only get a non-zero stress tensor for the worldsheet of a string because of gauge fixing. $\endgroup$– Connor BehanJun 5 at 11:03
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$\begingroup$ But we get a vanishing EMT on the worldsheet of a string right? I'm also not sure if einstein's field equation imply that the EMT is zero in non vacuum cases. $\endgroup$– AralianJun 5 at 11:09
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2$\begingroup$ In GR, we define the EM tensor as $T_{\alpha\beta} = - \frac{2}{T} \frac{1}{\sqrt{-g}} \frac{\delta S_{matter}}{\delta g^{\alpha\beta}}$. Notice that only $S_{matter}$ appears here, NOT the whole action $S$. DTs statement applies if you define $T$ using the whole action. $\endgroup$– PraharJun 5 at 11:43
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There are at least 2 differences:
The metric $g_{\alpha\beta}$ in the Polyakov action $S$ of string theory is a worldsheet metric, not the metric of spacetime (=target space).
The action $S$ in OP's eq. (1) in the context of GR is a matter action (and the $T_{\alpha\beta}$ is a matter SEM tensor); it does not include the gravitational sector. In other words, the corresponding Euler-Lagrange (EL) equations $\frac{\delta S}{\delta g_{\alpha\beta}}=0$ are incomplete; they are not the full equations of motion for $g_{\alpha\beta}$. We need to include the gravitational Einstein tensor to form EFE.