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XI. Field Equations
Alternatively, the Einstein-Palatini action (11.7) may be formulated
in terms of vierbein- and spin-connection fields
$\left\{ {{e^\mu }_a,{\Sigma _\mu }{{^a}_b}} \right\}$ as basic
dynamical variables. From their definitions (2.6,7) and (4.21), it
is seen that they are 1-form fields on ${{\cal M}_4}$ taking values,
respectively, in ${V_4}$ and in the Lie algebra of the Lorentz group
${\rm{SO}}(1,3)$.
By identifying the Ricci scalar as the double contraction of the
2-form gauge curvature tensor (6.3) induced by the spin
connection:
\[R = {\eta ^{ab}}{\eta
^{cd}}R_{cadb}^{} = R_{ab}^{ab}: = {e^\mu }_a{e^\nu }_bR_{\mu \nu
}^{ab}[\Sigma ]\]
the action gets the vierbein Einstein-Palatini form
\[{S_{{\rm{EP}}}}[e,\Sigma ] =
\frac{1}{{2\kappa }}\int {{d^4}x\,{\rm{e}} \left( {{e^\mu }_a{e^\nu
}_bR{{_{\mu \nu }^{ab}}_{}}[\Sigma ] + 2\Lambda } \right)} \]
with ${\rm{e}}: = \det [{e_\mu }^a] = \sqrt {\left| g \right|} $,
which is now a function of the vierbein; no conditions are imposed
on the spin-connection.
Varying around the metric compatible and torsionless (Levi-Civita)
spin-connection, one finds the analogue of (11.8)
\[{\Sigma _\mu }{^a}{_b} =
{\left. {{\Sigma _\mu }{{^a}_b}} \right|_{{\rm{LC}}}} + {A_\mu
}\delta _b^a\]
For the spin-connection to become the Levi-Civita
spin-connection, one must set the gauge parameter ${A_\mu }$
to zero. Equivalently, one may impose ${{\Sigma _\mu }{^a}}{_a} =
0$, which is the metricity condition, since metric
compatibility is equivalent to the anti-symmetry of the spin
connection in its Latin indices; see Tetrad
Postulate.
The vierbein Palatini action (11.10), or rather its extension to the
Einstein-Cartan theory, is essential in the formulation of a
generally covariant fermionic action which couples fermions to
gravity. Since fermions are sources of torsion, this requires a
theory in which torsion is allowed, but metricity
preserved. In this respect, the Einstein-Cartan
theory is a natural extension of GRT when fermions are
involved.