# Energy-momentum tensor for dust

#### dEdt

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0 v^\alpha v^\beta$, where $\rho_0$ is the mass density in the dust's rest frame and $v^\alpha$ is the dust's four-velocity.

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 $$S_M=\int -\rho_0 \sqrt{-g} d^4x.$$ Thus $$\frac{\delta S_M}{\delta g^{\alpha\beta}}=\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}\sqrt{-g}-\rho_0\frac{\delta \sqrt{-g}}{\delta g^{\alpha\beta}}.$$

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
$$T_{\alpha\beta}=\rho_0 v_\alpha v_\beta -\rho_0 g_{\alpha\beta}.$$

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

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