Palatini variation

Remarks

For a torsion-free connection, the variation of the Ricci tensor obeys the Palatini identity $\delta R_{\mu\nu} = \nabla_\lambda \delta\Gamma^\lambda_{\mu\nu} - \nabla_\nu \delta\Gamma^\lambda_{\lambda\mu}$, where $\nabla$ is the covariant derivative of $\Gamma$. The equation from varying $\Gamma$ reduces to $\nabla_\lambda\left(\sqrt{-\det g}\, g^{\mu\nu}\right) = 0$. Since $n \ge 3$, this gives $\nabla g = 0$, so $\Gamma$ is the Levi-Civita connection of $g$. The equation from varying $g$ is $R_{\mu\nu}(\Gamma) - \tfrac12 g_{\mu\nu} g^{\rho\sigma} R_{\rho\sigma}(\Gamma) = 8\pi G\, T_{\mu\nu}$, where $T_{\mu\nu}$ is the stress–energy tensor of $S_{\mathrm{m}}$. With $\Gamma$ the Levi-Civita connection, these are the Einstein field equations.

Example: in vacuum on $\R^4$ with the Minkowski metric $\eta$ and the flat connection $\Gamma = 0$ in Cartesian coordinates, both equations hold, since $\partial_\lambda \eta^{\mu\nu} = 0$ and $R_{\mu\nu} = 0$.

Non-example: the usual metric variation of the Einstein–Hilbert action is not a Palatini variation. There $\Gamma$ is fixed as the Levi-Civita connection of $g$ and is not varied on its own. The two give the same field equations in general relativity, but they can differ in modified theories such as $f(R)$ gravity.