Self-dual Palatini action

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.[1][2][3] These proofs were given in terms of spinors. 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 edit

The Palatini action for general relativity has as its independent variables the tetrad   and a spin connection  . 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   We define the `curvature' by


The Ricci scalar of this curvature is given by  . The Palatini action for general relativity reads


where  . Variation with respect to the spin connection   implies that the spin connection is determined by the compatibility condition   and hence becomes the usual covariant derivative  . Hence the connection becomes a function of the tetrads and the curvature   is replaced by the curvature   of  . Then   is the actual Ricci scalar  . Variation with respect to the tetrad gives Einsteins equation


Self-dual variables edit

(Anti-)self-dual parts of a tensor edit

We will need what is called the totally antisymmetry tensor or Levi-Civita symbol,  , which is equal to either +1 or −1 depending on whether   is either an even or odd permutation of  , respectively, and zero if any two indices take the same value. The internal indices of   are raised with the Minkowski metric  .

Now, given any anti-symmetric tensor  , we define its dual as


The self-dual part of any tensor   is defined as


with the anti-self-dual part defined as


(the appearance of the imaginary unit   is related to the Minkowski signature as we will see below).

Tensor decomposition edit

Now given any anti-symmetric tensor  , we can decompose it as


where   and   are the self-dual and anti-self-dual parts of   respectively. Define the projector onto (anti-)self-dual part of any tensor as


The meaning of these projectors can be made explicit. Let us concentrate of  ,




The Lie bracket edit

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):




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  

The Self-dual Palatini action edit

We define the self-dual part,  , of the connection   as


which can be more compactly written


Define   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:


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 edit

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 edit

Since   has signature  , 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 edit

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   to be self-dual if and only if


(with Euclidean signature the self-duality condition would have been  ). Say   is self-dual, write it as a real and imaginary part,


Write the self-dual condition in terms of   and  ,


Equating real parts we read off


and so


where   is the real part of  .

Important lengthy calculation edit

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 edit

Now using Eq.5 in conjunction with   we obtain


So we have




where in the first step we have used the anti-symmetry of the Lie bracket to swap   and  , in the second step we used   and in the last step we used the anti-symmetry of the Lie bracket again. So we have




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  , 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.  . This implies:


Summary of main results edit

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  

Derivation of Ashtekar's Formalism from the Self-dual Action edit

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  , 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  , the self-dual part of  ,


It has been shown that   is the self-dual part of  

Let   be the projector onto the three surface and define vector fields


which are orthogonal to  .



then we can write


where we used   and  .

So the action can be written


We have  . We now define


An internal tensor   is self-dual if and only if


and given the curvature   is self-dual we have


Substituting this into the action (Eq. 12) we have,


where we denoted  . We pick the gauge   and   (this means  ). Writing  , which in this gauge  . Therefore,


The indices   range over   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  . We need




to obtain


The action becomes


where we swapped the dummy variables   and   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   is the conjugate momentum 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  , and lapse function   give the constraints


Varying with respect to   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   can be written




where we used


therefore  . So the connection reads


This is the so-called chiral spin connection.

Reality conditions edit

Because Ashtekar's variables are complex it results in complex general relativity. 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.

More to be said on this, later.

See also edit

References edit

  1. ^ Samuel, Joseph (1987). "A lagrangian basis for ashtekar's reformulation of canonical gravity". Pramana. Springer Science and Business Media LLC. 28 (4): L429–L432. Bibcode:1987Prama..28L.429S. doi:10.1007/bf02847105. ISSN 0304-4289. S2CID 120704976.
  2. ^ Jacobson, Ted; Smolin, Lee (1987). "The left-handed spin connection as a variable for canonical gravity". Physics Letters B. Elsevier BV. 196 (1): 39–42. Bibcode:1987PhLB..196...39J. doi:10.1016/0370-2693(87)91672-8. ISSN 0370-2693.
  3. ^ Jacobson, T; Smolin, L (1988-04-01). "Covariant action for Ashtekar's form of canonical gravity". Classical and Quantum Gravity. IOP Publishing. 5 (4): 583–594. Bibcode:1988CQGra...5..583J. doi:10.1088/0264-9381/5/4/006. ISSN 0264-9381. S2CID 250866876.
  4. ^ Goldberg, J. N. (1988-04-15). "Triad approach to the Hamiltonian of general relativity". Physical Review D. American Physical Society (APS). 37 (8): 2116–2120. Bibcode:1988PhRvD..37.2116G. doi:10.1103/physrevd.37.2116. ISSN 0556-2821. PMID 9958915.
  5. ^ Henneaux, M.; Nelson, J. E.; Schomblond, C. (1989-01-15). "Derivation of Ashtekar variables from tetrad gravity". Physical Review D. American Physical Society (APS). 39 (2): 434–437. Bibcode:1989PhRvD..39..434H. doi:10.1103/physrevd.39.434. ISSN 0556-2821. PMID 9959655.
  6. ^ Ashtekar Variables in Classical General Relativity, Domenico Giulini, Springer Lecture Notes in Physics 434 (1994), 81-112, arXiv:gr-qc/9312032
  7. ^ The Ashtekar Hamiltonian for General Relativity by Ceddric Beny
  8. ^ Knot theory and quantum gravity in loop space: a primer by Jorge Pullin; AIP Conf.Proc.317:141-190,1994, arXiv:hep-th/9301028