Ashtekar variables, which were a new canonical formalism of general relativity, raised new hopes for the canonical quantization of general relativity and eventually led to loop quantum gravity.
Smolin and others independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the Tetradic Palatini action principle of general relativity.
A purely tensorial proof of the new variables in terms of triads was given by Goldberg[4] and in terms of tetrads by Henneaux et al.[5] The Palatini action for general relativity has as its independent variables the tetrad
Much more details and derivations can be found in the article tetradic Palatini action.
The spin connection defines a covariant derivative
The space-time metric is recovered from the tetrad by the formula
The Palatini action for general relativity reads where
implies that the spin connection is determined by the compatibility condition
Variation with respect to the tetrad gives Einstein's equation We will need what is called the totally antisymmetry tensor or Levi-Civita symbol,
, we define its dual as The self-dual part of any tensor
, Then An important object is the Lie bracket defined by it appears in the curvature tensor (see the last two terms of Eq.
We have the results (proved below): and That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts.
2 it is easy to see that the curvature of the self-dual connection is the self-dual part of the curvature of the connection, The self-dual action is As the connection is complex we are dealing with complex general relativity and appropriate conditions must be specified to recover the real theory.
One can repeat the same calculations done for the Palatini action but now with respect to the self-dual connection
Varying the tetrad field, one obtains a self-dual analog of Einstein's equation: That the curvature of the self-dual connection is the self-dual part of the curvature of the connection helps to simplify the 3+1 formalism (details of the decomposition into the 3+1 formalism are to be given below).
The results of calculations done here can be found in chapter 3 of notes Ashtekar Variables in Classical Relativity.
[6] The method of proof follows that given in section II of The Ashtekar Hamiltonian for General Relativity.
[7] We need to establish some results for (anti-)self-dual Lorentzian tensors.
From the definition of the Lie bracket and with the use of the basic identity Eq.
we obtain So we have Consider where in the first step we have used the anti-symmetry of the Lie bracket to swap
, as can easily be verified by direct computation: Applying this in conjunction with Eq.
can be written as a sum of its self-dual and anti-sef-dual parts, i.e.
This implies: Altogether we have, which is our main result, already stated above as Eq.
The proof given here follows that given in lectures by Jorge Pullin[8] The Palatini action where the Ricci tensor,
be the projector onto the three surface and define vector fields which are orthogonal to
Hence, we can immediately write Variation of action with respect to the non-dynamical quantities
, that is the time component of the four-connection, the shift function
, it has been rescaled to make the constraint polynomial in the fundamental variables.
To recover the real theory one has to impose what are known as the reality conditions.
These require that the densitized triad be real and that the real part of the Ashtekar connection equals the compatible spin connection.