The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. It is called a first order formulation as the variables to vary over involve only up to first derivatives in the action and so doesn't overcomplicate the Euler–Lagrange equations with higher derivative terms. The tetradic Palatini action is another first-order formulation of the Einstein–Hilbert action in terms of a different pair of independent variables, known as frame fields and the spin connection. The use of frame fields and spin connections are essential in the formulation of a generally covariant fermionic action (see the article spin connection for more discussion of this) which couples fermions to gravity when added to the tetradic Palatini action. Not only is this needed to couple fermions to gravity and makes the tetradic action somehow more fundamental to the metric version, the Palatini action is also a stepping stone to more interesting actions like the self-dual Palatini action which can be seen as the Lagrangian basis for Ashtekar's formulation of canonical gravity (see Ashtekar's variables) or the Holst action which is the basis of the real variables version of Ashtekar's theory. Another important action is the Plebanski action (see the entry on the Barrett–Crane model), and proving that it gives general relativity under certain conditions involves showing it reduces to the Palatini action under these conditions. Here we present definitions and calculate Einstein's equations from the Palatini action in detail. These calculations can be easily modified for the self-dual Palatini action and the Holst action.
Some definitions We first need to introduce the notion of tetrads. A tetrad is an orthonormal vector basis in terms of which the space-time metric looks locally flat,
g α β = e α I e β J η I J {\displaystyle g_{\alpha \beta }=e_{\alpha }^{I}e_{\beta }^{J}\eta _{IJ}}
where η I J = diag ( − 1 , 1 , 1 , 1 ) {\displaystyle \eta _{IJ}={\text{diag}}(-1,1,1,1)} is the Minkowski metric. The tetrads encode the information about the space-time metric and will be taken as one of the independent variables in the action principle. Now if one is going to operate on objects that have internal indices one needs to introduce an appropriate derivative (covariant derivative). We introduce an arbitrary covariant derivative via
D α V I = ∂ α V I + ω α I J V J . {\displaystyle {\mathcal {D}}_{\alpha }V_{I}=\partial _{\alpha }V_{I}+{\omega _{\alpha I}}^{J}V_{J}.}
Where ω α I J {\displaystyle {\omega _{\alpha I}}^{J}} is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metric η I J {\displaystyle \eta _{IJ}} ). We define a curvature via
Ω α β I J V J = ( D α D β − D β D α ) V I {\displaystyle {\Omega _{\alpha \beta I}}^{J}V_{J}=({\mathcal {D}}_{\alpha }{\mathcal {D}}_{\beta }-{\mathcal {D}}_{\beta }{\mathcal {D}}_{\alpha })V_{I}}
We obtain
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