Palatini variation

In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection.

In fact, as is well known, the Einstein–Hilbert action for general relativity was first formulated purely in terms of the spacetime metric ${\displaystyle \scriptstyle {g_{\mu \nu }}}$. In the Palatini variational method one takes as independent field variables not only the ten components ${\displaystyle \scriptstyle {g_{\mu \nu }}}$ but also the forty components of the affine connection ${\displaystyle \scriptstyle {\Gamma _{\,\beta \mu }^{\alpha }}}$, assuming, a priori, no dependence of the ${\displaystyle \scriptstyle {\Gamma _{\,\beta \mu }^{\alpha }}}$ from the ${\displaystyle \scriptstyle {g_{\mu \nu }}}$ and their derivatives.

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