Gravity and Strings

4.4 The Cartan–Sciama–Kibble theory 131
The equations of motion are the evident covariantization of the flat-space ones:
(i ∇ −m)ψ = 0, (4.99)
and the spin-angular-momentum tensor S µ ab and spin–energy potential µν a are identical
to the ones calculated in Section 2.4.1. By varying with respect to the Vielbeins, we find
the Vielbein energy–momentum tensor, which has the general form Eq. (4.94) with
Tcan a µ =− i
2 ¯ψγ µ ∇aψ + i
2 ∇a ¯ψγ µ ψ + ea µ Lmatter, (4.100)
giving
Ta µ =− i
2 ¯ψ(γ µ ea ν + γag µν )∇νψ + i
2 ∇ν ¯ψ(γ µ ea ν + γag µν )ψ
+ ea µ Lmatter − i
2 ¯ψγ µ a ∇ψ + i
2 ¯ψ ←
∇ γa µ ψ, (4.101)
which is not symmetric because of the last two terms, which vanish on-shell, as expected.
This is what saves the consistency of the Einstein equation
Ga µ 2 χ
=
2 Ta µ , (4.102)
whose l.h.s. is symmetric in the absence of torsion. This is not too different from the way in
which consistency is achieved in the standard GR theory in which the l.h.s. is divergenceless
(due to the contracted Bianchi identity) and the r.h.s. is divergenceless only when the matter
equations of motion are satisfied.
4.4.2 The coupling to torsion: the CSK theory
Perhaps the simplest generalization of GR one can think of is the use of a (still metriccompatible)
connection with non-vanishing torsion Tµν ρ .Now, the torsion is a new field
whose value we have to determine. The simplest possibility is to consider it a fundamental
field and just include it in a generalized Einstein–Hilbert action and in the covariant derivatives
acting on matter fields (minimal coupling). Then its equation of motion is determined,
as usual, by varying the action with respect to it and imposing the vanishing of the variation.
As we are going to see, the resulting equation of motion is algebraic and simply gives the
torsion as a function of other fields. In fact, in the torsion equation of motion one can see
the matter spin–energy potential µν a as the source for torsion Tµν a . This is essentially the
definition of the Cartan–Sciama–Kibble (CSK) theory (reviewed in [523]; and, in a more
pedagogical form, in [818]; and in the Newman–Penrose formalism in [768]).
Why should we couple intrinsic spin to torsion? The CSK theory is based on Weyl’s
Vielbein formalism in which there are two distinct gauge symmetries: reparametrizations
and local Lorentz transformations in tangent space. Reparametrization invariance leads to
the coupling of the energy–momentum tensor to the metric and, similarly, local Lorentz
invariance leads to the coupling of the spin–energy potential to torsion.
In the CSK theory, torsion is not a propagating new field. Furthermore, there is no way
to couple it to vector gauge potentials without breaking the gauge symmetry, which is