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.
Contents
- Some definitions
- The tetradic Palatini action
- Generalizations of the Palatini action
- Relating usual curvature to the mixed index curvature
- Difference between curvatures
- Varying the action with respect to the field C I J displaystyle Calpha IJ
- Vanishing of C I J displaystyle Calpha IJ
- References
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
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
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
We define a curvature
This is easily related to the usual curvature defined by
via substituting
for the Riemann tensor, Ricci tensor and Ricci scalar respectively.
The tetradic Palatini action
The Ricci scalar of this curvature can be expressed as
where
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,
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
We first vary with respect to
we have
One gets, after substituting
which, after multiplication by
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
It is easy to show these actions give the same equations. However, the case corresponding to
(note this diverges for
Relating usual curvature to the mixed index curvature
The usual Riemann curvature tensor
To find the relation to the mixed index curvature tensor let us substitute
where we have used
Using this expression we find
Contracting over
Difference between curvatures
The derivative defined by
where
Hence
Varying the action with respect to the field C α I J {displaystyle C_{alpha }^{;;IJ}}
We would expect
Implying
From the last term of the action we have from varying with respect to
or
or
where we have used
Vanishing of C α I J {displaystyle C_{alpha }^{;;IJ}}
We will show following the reference "Geometrodynamics vs. Connection Dynamics" that
implies
Then the condition
As
and as
Thus the terms
If we now contract this with
or
Since we have
Implying
and since the