Editing Einstein–Hilbert action (section)

===Variation of the Ricci scalar===
The variation of the [[Ricci scalar]] follows from varying the [[Riemann curvature tensor]], and then the [[Ricci curvature tensor]].

The first step is captured by the [[Palatini identity]]

:<math>
  \delta R_{\sigma\nu} \equiv \delta {R^\rho}_{\sigma\rho\nu} =
  \nabla_\rho \left( \delta \Gamma^\rho_{\nu\sigma} \right) - \nabla_\nu \left( \delta \Gamma^\rho_{\rho\sigma} \right)</math>.

Using the [[product rule]], the variation of the Ricci scalar <math>R = g^{\sigma\nu} R_{\sigma\nu}</math> then becomes

:<math>\begin{align}
\delta R &= R_{\sigma\nu} \delta g^{\sigma\nu} + g^{\sigma\nu} \delta R_{\sigma\nu}\\
         &= R_{\sigma\nu} \delta g^{\sigma\nu} + \nabla_\rho \left( g^{\sigma\nu} \delta\Gamma^\rho_{\nu\sigma} - g^{\sigma\rho} \delta \Gamma^\mu_{\mu\sigma} \right),
\end{align}</math>

where we also used the [[Metric connection#Riemannian connection|metric compatibility]] <math>\nabla_\sigma g^{\mu\nu} = 0</math>, and renamed the summation indices <math>(\rho,\nu) \rightarrow (\mu,\rho)</math> in the last term.

When multiplied by <math>\sqrt{-g}</math>, the term <math>\nabla_\rho \left( g^{\sigma\nu} \delta\Gamma^\rho_{\nu\sigma} - g^{\sigma\rho}\delta\Gamma^\mu_{\mu\sigma} \right)</math> becomes a [[total derivative]], since for any [[Ricci calculus|vector]] <math>A^\lambda</math> and any [[tensor density]] <math>\sqrt{-g}\,A^\lambda</math>, we have 
:<math>
        \sqrt{-g} \, A^\lambda_{;\lambda} =
  \left(\sqrt{-g} \, A^\lambda\right)_{;\lambda} =
  \left(\sqrt{-g} \, A^\lambda\right)_{,\lambda}
</math> or <math>
  \sqrt{-g} \, \nabla_\mu A^\mu =
    \nabla_\mu\left(\sqrt{-g} \, A^\mu\right) =
  \partial_\mu\left(\sqrt{-g} \, A^\mu\right)
</math>.

By [[Stokes' theorem]], this only yields a boundary term when integrated. The boundary term is in general non-zero, because the integrand depends not only on <math>\delta g^{\mu\nu},</math> but also on its partial derivatives <math>\partial_\lambda\, \delta g^{\mu\nu} \equiv \delta\, \partial_\lambda g^{\mu\nu}</math>; see the article [[Gibbons–Hawking–York boundary term]] for details. However, when the variation of the metric <math>\delta g^{\mu\nu}</math> vanishes in a neighbourhood of the boundary or when there is no boundary, this term does not contribute to the variation of the action. Thus, we can forget about this term and simply obtain

{{NumBlk|:|<math>\frac{\delta R}{\delta g^{\mu\nu}} = R_{\mu\nu}</math>.|{{EquationRef|3}}}}

at [[event (relativity)|events]] not in the [[closure (topology)|closure]] of the boundary.