When we start from the Polyakov action, we can choose to work in the conformal gauge $h^{\alpha\beta}=\eta^{\alpha\beta}$ where $h^{\alpha\beta}$ is the metric on the world-sheet and $\eta^{\alpha\beta}$ is the 2D Minkowiski metric. The action becomes $$S_p= -\dfrac{T}{2} \int d\sigma d\tau\ \ \partial_\alpha X^m\ \partial^\alpha X^n \ g_{mn} $$ where $g_{mn}$ is the metric of the background spacetime and English indices refer to this background spacetime metric and Greek indices refer to the world-sheet metric (Minkowski).
For choosing a gauge, we have to also impose the constraint $$T_{\alpha\beta}=-\dfrac{2}{T}\dfrac{1}{\sqrt{-h}}\dfrac{\delta S}{\delta h^{\alpha\beta}}=0$$ Most of the resources that I use to study string theory, say that this condition is the vanishing of the stress-energy tensor. But, we can also have a stress-energy tensor given by $$T^{'}_{\mu\nu}=-\dfrac{2}{T}\dfrac{1}{\sqrt{-g}}\dfrac{\delta S}{\delta g^{\mu\nu}}$$ What I am confused about is what is the metric that, say, if we take $\alpha=\beta=0$ or $\mu=\nu=0$ will give the energy density of the string? And, on a deeper level, what is the conceptual, physical difference between the two?
