Self-dual variables
The Lie bracket
An important object is the Lie bracket defined by
 
it appears in the curvature tensor (see the last two terms of Eq. 1), it also defines the algebraic structure. We have the results (proved below):
 
and
 
That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write
 
where  contains only the self-dual (anti-self-dual) elements of
 contains only the self-dual (anti-self-dual) elements of 
The Self-dual Palatini action
We define the self-dual part,  , of the connection
, of the connection  as
 as
 
which can be more compactly written
 
Define  as the curvature of the self-dual connection
 as the curvature of the self-dual connection
 
Using Eq. 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:
. 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 resulting Hamiltonian formalism resembles that of a Yang-Mills gauge theory (this does not happen with the 3+1 Palatini formalism which basically collapses down to the usual ADM formalism).
Derivation of main results for self-dual variables
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.
Identities for the totally anti-symmetric tensor
Since  has signature
 has signature  , it follows that
, it follows that
 
to see this consider,
 
With this definition one can obtain the following identities,
 
(the square brackets denote anti-symmetrizing over the indices).
Definition of self-dual tensor
It follows from Eq. 4 that the square of the duality operator is minus the identity,
 
The minus sign here is due to the minus sign in Eq. 4, which is in turn due to the Minkowski signature. Had we used Euclidean signature, i.e.  , instead there would have been a positive sign. We define
, instead there would have been a positive sign. We define  to be self-dual if and only if
 to be self-dual if and only if
 
(with Euclidean signature the self-duality condition would have been  ). Say
). Say  is self-dual, write it as a real and imaginary part,
 is self-dual, write it as a real and imaginary part,
 
Write the self-dual condition in terms of  and
 and  ,
,
 
Equating real parts we read off
 
and so
 
where  is the real part of
 is the real part of  .
.
Important lengthy calculation
The proof of Eq. 2 in straightforward. We start by deriving an initial result. All the other important formula easily follow from it. From the definition of the Lie bracket and with the use of the basic identity Eq. 3 we have
 
That gives the formula
 
Derivation of important results
Now using Eq.5 in conjunction with  we obtain
 we obtain
 
So we have
 
Consider
 
where in the first step we have used the anti-symmetry of the Lie bracket to swap  and
 and  , in the second step we used
, in the second step we used  and in the last step we used the anti-symmetry of the Lie bracket again. So we have
 and in the last step we used the anti-symmetry of the Lie bracket again. So we have
 
Then
 
where we used Eq. 6 going from the first line to the second line. Similarly we have
 
by using Eq 7. Now as  is a projection it satisfies
 is a projection it satisfies  , as can easily be verified by direct computation:
, as can easily be verified by direct computation:
 
Applying this in conjunction with Eq. 8 and Eq. 9 we obtain
 
From Eq. 10 and Eq. 9 we have
 
where we have used that any  can be written as a sum of its self-dual and anti-sef-dual parts, i.e.
 can be written as a sum of its self-dual and anti-sef-dual parts, i.e.  . This implies:
. This implies:
 
Summary of main results
Altogether we have,
 
which is our main result, already stated above as Eq. 2. We also have that any bracket splits as
 
into a part that depends only on self-dual Lorentzian tensors and is itself the self-dual part of  and a part that depends only on anti-self-dual Lorentzian tensors and is the anit-self-dual part of
 and a part that depends only on anti-self-dual Lorentzian tensors and is the anit-self-dual part of 
The proof given here follows that given in lectures by Jorge Pullin  [8] 
The Palatini action 
 
where the Ricci tensor,  , is thought of as constructed purely from the connection
, is thought of as constructed purely from the connection  , not using the frame field. Variation with respect to the tetrad gives Einstein's equations written in terms of the tetrads, but for a Ricci tensor constructed from the connection that has no a priori relationship with the tetrad. Variation with respect to the connection tells us the connection satisfies the usual compatibility condition
, not using the frame field. Variation with respect to the tetrad gives Einstein's equations written in terms of the tetrads, but for a Ricci tensor constructed from the connection that has no a priori relationship with the tetrad. Variation with respect to the connection tells us the connection satisfies the usual compatibility condition
 
This determines the connection in terms of the tetrad and we recover the usual Ricci tensor.
The self-dual action for general relativity is given above.
 
where  is the curvature of the
 is the curvature of the  , the self-dual part of
, the self-dual part of  ,
,
 
It has been shown that  is the self-dual part of
 is the self-dual part of 
Let  be the projector onto the three surface and define vector fields
 be the projector onto the three surface and define vector fields
 
which are orthogonal to  .
.
Writing
 
then we can write
 
where we used  and
 and  .
.
So the action can be written
 
We have  . We now define
. We now define
 
An internal tensor  is self-dual if and only if
 is self-dual if and only if
 
and given the curvature  is self-dual we have
 is self-dual we have
 
Substituting this into the action (Eq. 12) we have,
 
where we denoted  . We pick the gauge
. We pick the gauge  and
 and  (this means
 (this means  ). Writing
). Writing  , which in this gauge
, which in this gauge  . Therefore,
. Therefore,
 
The indices  range over
 range over  and we denote them with lower case letters in a moment. By the self-duality of
 and we denote them with lower case letters in a moment. By the self-duality of  ,
,
 
where we used
 
This implies
 
We replace in the second term in the action  by
 by  . We need
. We need
 
and
 
to obtain
 
The action becomes
 
where we swapped the dummy variables  and
 and  in the second term of the first line. Integrating by parts on the second term,
 in the second term of the first line. Integrating by parts on the second term,
 
where we have thrown away the boundary term and where we used the formula for the covariant derivative on a vector density  :
:
 
The final form of the action we require is
 
There is a term of the form " " thus the quantity
" thus the quantity  is the conjugate momentum to
 is the conjugate momentum to  . Hence, we can immediately write
. 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
, that is the time component of the four-connection, the shift function  , and lapse function
, and lapse function  give the constraints
 give the constraints
 
 
 
Varying with respect to  actually gives the last constraint in Eq. 13 divided by
 actually gives the last constraint in Eq. 13 divided by  , it has been rescaled to make the constraint polynomial in the fundamental variables. The connection
, it has been rescaled to make the constraint polynomial in the fundamental variables. The connection  can be written
 can be written
 
and
 
where we used
 
therefore  . So the connection reads
. So the connection reads
 
This is the so-called chiral spin connection.