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From the Linearized Einstein Field Equation, we have $\Box\bar{h}_{\mu \nu} =-16\pi GT_{\mu \nu}$. How can I obtain conservation of energy and momentum, $T_{\mu \nu},^{\nu}=0$, from the previous equation? Is there any extra gauge condition needed?

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  • $\begingroup$ Yes, corrected now. Thanks $\endgroup$ – D_dm Apr 17 at 13:41
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Yes, you need the Lorentz gauge $\partial^\mu \bar{h}_{\mu \nu}=0$ to show energy-momentum conservation. In fact, in order to derive the linearized equation of motion that you wrote, you have already used the Lorentz gauge.

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  • $\begingroup$ Oh, so using Lorentz gauge to get linearized theory and automatically $\Box (\partial ^{\mu} \bar{h}_{\mu \nu} ) =0 \rightarrow \partial ^{\mu} T_{\mu \nu}=0$ right ? $\endgroup$ – D_dm Apr 17 at 13:49
  • $\begingroup$ Yes, that's right. $\endgroup$ – Avantgarde Apr 17 at 13:51

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