ABSTRACT

Palatini (first order) formalism In general relativity, in the framework of the first order formalism the metric gµν and the affine connection αβγ are considered as independent variables for the purpose of obtaining the Einstein equations from a variational principle. Taking variational derivatives of the otherwise standard Hilbert form of the variational action for general relativity with respect to those variables, one arrives at the dynamical Einstein equations in first order formalism. In the action

SEH = ∫

dnx √−ggµνRµν()

where Rµν() depends on the connection only, but not on the metric, the equation of motion for αβγ gives the same result as the metricity condition, that is, it fixes connection αβγ to be the Cristoffel symbol

} . Taking this constraint

into account, the equations for the metric become usual Einstein equations. One may apply similar concepts in gauge theories or in other variations of gravity theories. See christoffel symbol, curvature tensor, metricity of covariant derivative.