Two Spinor-valued formulations of Einstein-Cartan Equations

The field equations of a Metric theory can be derived from a first order Palatini formalism where an orthonormal frame  is used and in which the constraint that the dynamical connection is a metric compatible connection  is put in by hand. The field equations read:

where the components of the energy-momentum,  and spin tensors,  are given in terms of the three forms ,  with . 

Here  is an orthonormal “dual” (to tetrad) frame where indices  are Lorentzian time-space indices of a freely falling frame.

Kinematical differential Multivector framework

As Euler-Lagrange equations of a Lagrangian field theory, these Einstein-Cartan equations lend themselves to many alternative kinematical descriptions. Indeed they may be considered as a conservation law for a certain Sparling 3-form defined on the bundle of orthonormal frames over spacetime, . They are most elegantly presented as two spinor-valued differential 3-form equations, 
so that  where  and  are known source quantities.

For our purposes the main kinematical objects of a Variational principle are the 1-forms . These are the co-frame, verbein objects that are Einstein’s freely falling elevator.  This dynamical variable,  is a Hermitian matrix-valued one-form from which the (real Lorentzian) metric is given as 

For Real General Relativity the soldering functor is required to be real (, .

In constructing the 4-form Palatini Lagrangian we can make use of higher order multivector constructions based on the co-frame such as 

The internal indices  associated to the spin structure over space-time only acquires the interpretation as spinor indices through the dynamical soldering form, . A priori there is no relation between the tangent space and the internal space of the vector bundle  associated to the spinor structure over space time, M. Rather if a (non compact) spacetime manifold,  admits a global null tetrad it has a spinor-structure defined on it. A spinor structure,  is a principal fibre bundle with structure group , the gauge group for spinor dyads. The (real) space-time manifold carries a  spin (trivial vector) bundle, B associated to  and its conjugate on it. The tensor product of these two bundles can be identified with the complexified tangent bundle. Each fibre,  of B consists of a 2-complex dimensional vector space equipped with a symplectic metric, .

Cartan’s Structure Equations

With  and  (complex conjugate) -valued connection one-forms and the torsion two form denoted as , the first Cartan structure equation reads

That is,
 where  denotes the exterior covariant derivative with respect to the -valued connection(s).

The internal `symplectic metric’,  is given as fixed so that the internal connection is then traceless  due to 

Defining the basis of anti-self dual two-forms as

The second Cartan structure equations take the complex form

where the curvature two-form, , has been decomposed into spinor fields of dimension 5,9,1 and 3 respectively , corresponding to the anti-self dual part of the Weyl conformal spinor, , the spinor representation of the trace-free part of the Ricci tensor,  and the Ricci scalar , – all with respect to the curvature of the  connection and  arising from the presence of non-zero torsion.

Here we have used the basis of  anti-self dual two-forms , defined in terms of the co-frame dynamical variable for mere ease of exposition. As we will see in a later post these objects can be treated as the bona fide, fully chiral dynamical “graviton” field object in tis own right.

Semi-Chiral Lagrangian formulation Equations

Two Euler-Lagrange equations arise from Lagrangians of the general form
 with  hermitian and  representing some matter fields.

The internal  connection is not associated to the tangent bundle and is thus not a linear connection but a spinor connection. The variation of the Lagrangian with respect to  will determine this connection in terms of the co-frame so that the bundle  can then be considered soldered to . The co-frame variation evaluated at the particular value of the connection just determined gives equations for the co-frames only. There exists a unique Levi Civita connection,  (with curvature ) so the connection,  can be decomposed according to


where  is the contorsion one form, irreducibly written in terms of totally symmetric and`axial’ parts as

The Einstein-Matter equations owing to the triviality of the Bianchi Identity

have the simpler form,

It is possible to solve for  and replace any  in  by .

Semi-Chiral Lagrangian for Fermion fields

The Lagrangian for fermion matter has a dependence on the connection so admits torsion contributions but nevertheless can be written as the sum of a semi-chiral complex Lagrangian for vacuum General Relativity, , a complex (semi)chiral fermion matter Lagrangian,  and a term,  that ensures the standard Einstein-Weyl form of the field equations,

The  are the left (resp. right)-handed zero forms.

The theory uses only the anti-self dual connection,  (which does not act on tensors, so for example

but is complete and it turns out, (by varying ), that the real source current,  supports only the axial part of the torsion of ,

Because ultimately the real theory is of interest (where  and  is hermitian) it proves useful to extend  to . Although it is argued that the spin  field variables can be taken to be either Grassman [or complex]-valued, in fact the use of complex spin  fields leads to a non-standard energy-momentum tensor which includes quartic spin  fields.