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I want to show that on a geodesic we have for a particle $$\nabla_mT^{mn}=0$$ where $$T^{mn}=\frac{m}{\sqrt{-g}}\int d\tau~ u^mu^n\delta^4(x-x(\tau)).$$

So I found

$$\nabla_mT^{mn} =\partial_mT^{mn}+\Gamma^m_{ml}T^{ln}+\Gamma^m_{ml}T^{ln}\\ \partial_m\left(\frac{m}{\sqrt{-g}}\int d\tau ~u^mu^n\delta^4(x-x(\tau))\right) =\frac{m}{\sqrt{-g}} \times \\ \int d\tau \left[(\partial_mu^m)u^n\delta^4(x-x(\tau))+ u^m(\partial_mu^n)\delta^4(x-x(\tau))+u^m u^n \partial_m\delta^4(x-x(\tau))\right]$$

How do I continue from here? I know that I will eventually need to use the geodesic equation and the $\delta^4(x-x(\tau))$ term will cancel stuff. But how do I use these things here?

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