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. 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 terms coming from 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,
where formula_2 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
Where formula_4 is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metric formula_5). We define a curvature via
We obtain
We introduce the covariant derivative which annihilates the tetrad,
The connection is completely determined by the tetrad. The action of this on the generalized tensor formula_9 is given by
We define a curvature formula_11 by
This is easily related to the usual curvature defined by
via substituting formula_14 into this expression (see below for details). One obtains,
for the Riemann tensor, Ricci tensor and Ricci scalar respectively.
The tetradic Palatini action.
The Ricci scalar of this curvature can be expressed as formula_16 The action can be written
where formula_18 but now formula_19 is a function of the frame field.
We 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, formula_20 The connection associated with this covariant derivative is completely determined by the tetrad. The difference between the two connections we have introduced is a field formula_21 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 of formula_24 and formula_25 and noting that the variation with respect to formula_26 is the same as the variation with respect to formula_25 (when keeping the tetrad fixed). The action becomes
We first vary with respect to formula_25. The first term does not depend on formula_25 so it does not contribute. The second term is a total derivative. The last term yields 
We show below that this implies that formula_32 as the prefactor formula_33 is non-degenerate. This tells us that formula_24 coincides with formula_35 when acting on objects with only internal indices. Thus the connection formula_35 is completely determined by the tetrad and formula_37 coincides with formula_38. To compute the variation with respect to the tetrad we need the variation of formula_39. From the standard formula
we have formula_41. Or upon using formula_42, this becomes formula_43. We compute the second equation by varying with respect to the tetrad,
One gets, after substituting formula_45 for formula_11 as given by the previous equation of motion,
which, after multiplication by formula_48 just tells us that the Einstein tensor formula_49 of the metric defined by the tetrads vanishes. We have therefore proved that the Palatini variation of the action in tetradic form yields the usual Einstein equations.
Generalizations of the Palatini action.
We change the action by adding a term
This modifies the Palatini action to
where
This action given above is the Holst action, introduced by Holst and formula_53 is the Barbero-Immirzi parameter whose role was recognized by Barbero and Immirizi. The self dual formulation corresponds to the choice formula_54.
It is easy to show these actions give the same equations. However, the case corresponding to formula_55 must be done separately (see article self-dual Palatini action). Assume formula_56, then formula_57 has an inverse given by
(note this diverges for formula_55). As this inverse exists the generalization of the prefactor formula_33 will also be non-degenerate and as such equivalent conditions are obtained from variation with respect to the connection. We again obtain formula_32. 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.
Details of calculation.
Relating usual curvature to the mixed index curvature.
The usual Riemann curvature tensor formula_62 is defined by
To find the relation to the mixed index curvature tensor let us substitute formula_64
where we have used formula_8. Since this is true for all formula_67 we obtain
Using this expression we find
Contracting over formula_70 and formula_71 allows us write the Ricci scalar
Difference between curvatures.
The derivative defined by formula_73 only knows how to act on internal indices. However, we find it convenient to consider a torsion-free extension to spacetime indices. All calculations will be independent of this choice of extension. Applying formula_74 twice on formula_75,
where formula_77 is unimportant, we need only note that it is symmetric in formula_70 and formula_71 as it is torsion-free. Then
Hence:
Varying the action with respect to the field formula_25.
We would expect formula_83 to also annihilate the Minkowski metric formula_84. If we also assume that the covariant derivative formula_85 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 to formula_88
formula_89
or
or
where we have used formula_92. This can be written more compactly as
Vanishing of formula_25.
We will show following the reference "Geometrodynamics vs. Connection Dynamics" that
implies formula_96 First we define the spacetime tensor field by
Then the condition formula_98 is equivalent to formula_99. Contracting Eq. 1 with formula_100 one calculates that
As formula_102 we have formula_103 We write it as
and as formula_105 are invertible this implies
Thus the terms formula_107 and formula_108 of Eq. 1 both vanish and Eq. 1 reduces to
If we now contract this with formula_110, we get
or
Since we have formula_99 and formula_114, 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 formula_105 are invertible, we get formula_119. This is the desired result.