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In the variation of Einstein-Hilbert action the integral of the term $d^4x\sqrt{-g}g^{\mu\nu}\delta R_{\mu\nu}$ over the spacetime manifold is equal to zero. $$I=\underset{\mathcal M}\int d^4x\sqrt{-g}g^{\mu\nu}\delta R_{\mu\nu}=\underset{\mathcal M}\int d^4x\sqrt{-g}D_{\mu}(g^{\alpha\beta}\delta\Gamma^{\mu}_{\alpha\beta}-g^{\alpha\mu}\delta\Gamma^{\beta}_{\alpha\beta})$$

Denoting $g^{\alpha\beta}\delta\Gamma^{\mu}_{\alpha\beta}-g^{\alpha\mu}\delta\Gamma^{\beta}_{\alpha\beta}=\delta\mathcal V^{\mu}:$ $$ \underset{\mathcal M}\int d^4x\sqrt{-g}D_{\mu}\mathcal \delta V^{\mu}=\underset{\partial \mathcal M}\oint d\Sigma_{\mu}\delta\mathcal V^{\mu}=0 $$ where $d\Sigma_{\mu}=n_{\mu}\sqrt{|\gamma|}d^3\xi$, $\gamma$ is $3$-dimensional induced metric. What is the reason of being zero of the integral?

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    $\begingroup$ Usually when one is performing the variation of the action to get the equations of motion of the theory and suddenly throws boundary terms away, the non-stated assumption is that the variations of the fields are supposed to be of compact support, meaning that they vanish outside of a compact set. In that case these boundary integrals end up being set to zero. But if the variations are not compactly supported one cannot throw these terms away. In particular they are relevant in defining the sympletic form following Wald's method. Please see e.g. aip.scitation.org/doi/10.1063/1.528801. $\endgroup$ – user1620696 Jan 29 at 18:37
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This is the divergence of a vector. One can use Stokes theorem and say that this is equal to a boundary contribution at infinty which is what we can set to be equal to zero.

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  • $\begingroup$ At the infinity/boundary of a hypersurface, variation is of the metric is zero due to negligible variation of metric? $\endgroup$ – Constantin Jan 29 at 12:03
  • $\begingroup$ We consider that the deformations of the field vanish at the boundary. $\endgroup$ – ApolloRa Jan 29 at 12:14
  • $\begingroup$ The field is here is the metric field? $\endgroup$ – Constantin Jan 29 at 12:23
  • $\begingroup$ Yes, i refer to the field we vary with respect to. $\endgroup$ – ApolloRa Jan 29 at 12:24
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    $\begingroup$ @ApolloRa - It is clear that these lecture notes are a very basic introduction to GR and that Carroll is therefore sweeping a lot of things under the rug. This is obviously fine since boundary issues are not something an introductory course is expected to teach. Instead, if you read his textbook (Spacetime and Geometry by Carroll) (the paragraph below eq (4.65)), you will see that he has an extra sentence about the boundary issues that I mentioned. Again, he doesn't discuss it in detail since it derails the main discussion that the author is interested in. $\endgroup$ – Prahar Jan 29 at 18:34

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