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 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 or the Holst action which is the basis of the real variables version of Ashtekar's theory. Another important action is the Plebanski action, 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 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. We introduce an arbitrary covariant derivative via
Where is a Lorentz connection. 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 is given by
We define a curvature by
This is easily related to the usual curvature defined by
via substituting into this expression. 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 The action can be written
where but now 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, 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 defined by
We can compute the difference between the curvatures of these two covariant derivatives,
The reason for this intermediate calculation is that it is easier to compute the variation by reexpressing the action in terms of and and noting that the variation with respect to is the same as the variation with respect to . The action becomes
We first vary with respect to. The first term does not depend on so it does not contribute. The second term is a total derivative. The last term yields
We show below that this implies that as the prefactor is non-degenerate. This tells us that coincides with when acting on objects with only internal indices. Thus the connection is completely determined by the tetrad and coincides with. To compute the variation with respect to the tetrad we need the variation of. From the standard formula
we have. Or upon using, this becomes. We compute the second equation by varying with respect to the tetrad,
One gets, after substituting for as given by the previous equation of motion,
which, after multiplication by just tells us that the Einstein tensor 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 is the Barbero-Immirzi parameter whose role was recognized by Barbero and Immirizi. The self dual formulation corresponds to the choice.
It is easy to show these actions give the same equations. However, the case corresponding to must be done separately. Assume, then has an inverse given by
. As this inverse exists the generalization of the prefactor will also be non-degenerate and as such equivalent conditions are obtained from variation with respect to the connection. We again obtain. 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 is defined by
To find the relation to the mixed index curvature tensor let us substitute
where we have used. Since this is true for all we obtain
Using this expression we find
Contracting over and allows us write the Ricci scalar

Difference between curvatures

The derivative defined by 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 twice on,
where is unimportant, we need only note that it is symmetric in and as it is torsion-free. Then
Hence:

Varying the action with respect to the field {C_\alpha}^{IJ}

We would expect to also annihilate the Minkowski metric. If we also assume that the covariant derivative annihilates the Minkowski metric we have,
Implying
From the last term of the action we have from varying with respect to
or
or
where we have used. This can be written more compactly as

Vanishing of {C_\alpha}^{IJ}

We will show following the reference "Geometrodynamics vs. Connection Dynamics" that
implies First we define the spacetime tensor field by
Then the condition is equivalent to. Contracting Eq. 1 with one calculates that
As we have We write it as
and as are invertible this implies
Thus the terms and of Eq. 1 both vanish and Eq. 1 reduces to
If we now contract this with, we get
or
Since we have and, 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 are invertible, we get. This is the desired result.