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. 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 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 is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metric ). 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 (see below for details). One obtains,
for the Riemann tensor, Ricci tensor and Ricci scalar respectively.
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
We would expect to also annihilate the Minkowski metric . If we also assume that the covariant derivative 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
or
or
where we have used . This can be written more compactly as
Vanishing of
We will show following the reference "Geometrodynamics vs. Connection Dynamics"[6] 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.