Gravity and Strings

4.4 The Cartan–Sciama–Kibble theory 127
To obtain the Einstein equations in the presence of matter in this formalism, one has to
use more complicated techniques. 5
4.4 The Cartan–Sciama–Kibble theory
The formalism developed so far can be used to couple matter fields that behave as tensors
under GCTs. In general, the tensorial character of the matter fields under GCTs is determined
from their behavior under Poincaré transformations and the only possible ambiguity
is whether the field is just a tensor or a tensor density. However, this identification does not
work for spinor fields, because it is based on a relation that exists only between the tensor
representations of the Poincaré group and tensor representations of the diffeomorphism
group. Thus, to couple fermions to gravity, we must first find out how to define spinors in a
general curved spacetime.
In a classical paper, 6 [954], Weyl proposed to define spinors in tangent space using an
orthonormal Vielbein basis {e a µ} as fundamental fields instead of the metric and developed
aformalism that is invariant under Lorentz transformations of this Vielbein basis even
if we perform a different Lorentz transformation in (the tangent space associated with)
every spacetime point. Thus, in d spacetime dimensions, the d(d + 1)/2 off-shell degrees
of freedom of the metric (the number of independent components of a d × d symmetric
matrix) are replaced by the same number of off-shell degrees of freedom of the Vielbein
(the number of independent components of a generic d × d matrix minus the d(d − 1)/2
independent local Lorentz transformations). In modern language, 7 this is a gauge theory
of the Lorentz group SO(1, d − 1) and requires the introduction of a Lorentz covariant
derivative Dµ and a Lorentz (spin) connection ωµ ab . Otherwise, the Vielbeins will describe
more degrees of freedom than the metric.
However, if we want to recover GR, we do not want to introduce new fields apart from the
metric (Vielbeins) and thus we have to relate the spin connection to the Vielbeins, destroying
the similarity with a standard Yang–Mills theory in which the connection is the dynamical
field. The natural way to relate connection and Vielbeins is through the first Vielbein
postulate Eqs. (1.83) which connects the spin and the affine connections by Eq. (1.84). This
does not seem to help much, because the affine connection is completely undetermined.
However, metric-compatibility is automatic for spin and affine connections satisfying the
first Vielbein postulate, because, by assumption, the spin connection ωµ ab is antisymmetric
in the indices ab,which implies ∇µηab = 0, which, with the first Vielbein postulate, implies
∇µgρσ = 0. Therefore, the first Vielbein postulate determines the connection in terms of the
Vielbein up to the torsion term. Now if we want to have as fundamental fields the Vielbeins
alone, we need to impose the vanishing of torsion. In that case, the affine connection is the
Levi-Cività connection Ɣ(g) whose components are the Christoffel symbols Eq. (1.44) and
5 See e.g. [682] and references therein. Further generalizations of the Einstein–Hilbert action are also reviewed
there.
6 A guide to the old literature on this formalism and its generalizations to include torsion is [523]. A pedagogical
introduction to this formalism is [818] (see also [817]). A more recent reference is [851].
7 The basic formalism of Yang–Mills gauge theories is developed in Appendix A and, for the Lorentz group
in particular, for the present application, in Section 1.4.