I'm trying to derive the dust energy momentum tensor from the equation ##T_{\alpha \beta}=-\frac{2}{\sqrt{-g}}\frac{\delta S_M}{\delta g^{\alpha\beta}}## but I'm getting the wrong answer.

The action for dust is [tex]S_M=\int -\rho_0 \sqrt{-g} d^4x.[/tex] Thus [tex]\frac{\delta S_M}{\delta g^{\alpha\beta}}=\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}\sqrt{-g}-\rho_0\frac{\delta \sqrt{-g}}{\delta g^{\alpha\beta}}.[/tex]

To evaluate ##\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}##, I define ##K_\alpha=\rho_0 v_\alpha##. Then ##\rho_0=\sqrt{g^{\alpha\beta}K_\alpha K_\beta}## and thus ##\delta \rho_0=\frac{1}{2\rho_0}K_\alpha K_\beta \delta g^{\alpha \beta}##. It follows that

[tex]T_{\alpha\beta}=\rho_0 v_\alpha v_\beta -\rho_0 g_{\alpha\beta}.[/tex]

There's an extra term that I can't get rid of. Any idea where I went wrong?