Tetradic Palatini action
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.[1] 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.Here we present definitions and calculate Einstein's equations from the Palatini action in detail.A tetrad is an orthonormal vector basis in terms of which the space-time metric looks locally flat, whereThe tetrads encode the information about the space-time metric and will be taken as one of the independent variables in the action principle.is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metricby This is easily related to the usual curvature defined by via substitutingWe will derive the Einstein equations by varying this action with respect to the tetrad and spin connection as independent quantities.As a shortcut to performing the calculation we introduce a connection compatible with the tetrad,[2] The connection associated with this covariant derivative is completely determined by the tetrad.defined by We can compute the difference between the curvatures of these two covariant derivatives (see below for details), The reason for this intermediate calculation is that it is easier to compute the variation by reexpressing the action in terms ofWe compute the second equation by varying with respect to the tetrad, One gets, after substitutingWe have therefore proved that the Palatini variation of the action in tetradic form yields the usual Einstein equations.is the Barbero-Immirzi parameter whose role was recognized by Barbero[4] and Immirizi.It is easy to show these actions give the same equations.must be done separately (see article self-dual Palatini action).will also be non-degenerate and as such equivalent conditions are obtained from variation with respect to the connection.While variation with respect to the tetrad yields Einstein's equation plus an additional term.However, this extra term vanishes by symmetries of the Riemann tensor.is defined by To find the relation to the mixed index curvature tensor let us substituteallows us write the Ricci scalar The derivative defined byHowever, we find it convenient to consider a torsion-free extension to spacetime indices.annihilates the Minkowski metric (then said to be torsion-free) we have, Implying From the last term of the action we have from varying with respect toThis can be written more compactly as We will show following the reference "Geometrodynamics vs.First we define the spacetime tensor field by Then the condition, we can successively interchange the first two and then last two indices with appropriate sign change each time to obtain, Implying or and since the