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XI. Field Equations
In the Einstein-Cartan theory of gravity, one usually assumes metric
compatibility of the Lorentz (spin) connection, i.e.
${{\mathbf{\omega }}_{ab}} = - {{\mathbf{\omega }}_{ba}}$. In that
case, also assuming that the variation $\delta $ and the exterior
derivative ${\text{d}}$ commute, one concludes from eq (10.6) that
the variation of the curvature 2-form is the exterior
coderivative of the variation of the connection: $\delta
{{\mathbf{R}}^{ab}} = {\text{D}}\delta {{\mathbf{\omega }}^{ab}}$.
Hence
\[{\delta _{\mathbf{\omega }}}
{S_{{\text{EC}}}}[{\mathbf{e}},{\mathbf{\omega }}] =
\frac{1}{{4\kappa }}\int {d{x_4}{\text{ }}} {\varepsilon
_{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge
{\text{D}}\delta {{\mathbf{\omega }}^{cd}}\]
An integration by parts can be performed with the help of the
Leibniz rule
\[\begin{gathered}{\varepsilon
_{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge
{\text{D}}\delta {{\mathbf{\omega }}^{cd}} =
\hfill\\{\text{D(}}{\varepsilon _{abcd}}{{\mathbf{e}}^a} \wedge
{{\mathbf{e}}^b} \wedge \delta {{\mathbf{\omega }}^{cd}}) -
2{\varepsilon _{abcd}}{{\mathbf{T}}^a} \wedge {{\mathbf{e}}^b}
\wedge \delta {{\mathbf{\omega }}^{cd}}\end{gathered}\]
where ${{\mathbf{T}}^a}$ is the torsion 1-form as defined in
(4.24). Since in (11.26) the exterior coderivative at the right-hand
side acts on a scalar, this ${\text{D}}$ may be replaced just by the
exterior derivative $\text{d}$. Ignoring the resulting boundary term
in (11.25), one obtains the torsion field equation of
the (vacuum) Einstein-Cartan theory
\[{\varepsilon
_{abcd}}{{\mathbf{T}}^a} \wedge {{\mathbf{e}}^b} = 0\]
When the tetrad is assumed to be invertible, one is led to the
equation ${{\mathbf{T}}^a} = {\text{D}}{{\mathbf{e}}^a} = 0$. Hence,
the vanishing torsion condition is achieved dynamically in
Einstein-Cartan gravity, Thus, even when a vanishing torsion is not
imposed as a constraint, the theory defined by the
Einstein-Hilbert-Palatini-Cartan action is equivalent to standard
GRT, at least in the vacuum case.
This equivalence is broken by adding matter to the system. This is
so, because spinning matter couples to the connection so that the
right hand side of the torsion constraint (11.27) acquires a source.
Dynamical torsion is thus sourced by spinning matter.
However, even then the torsion equation is an algebraic equation (as
opposed to a differential equation). This means that the torsion is
completely determined by the matter content, and does not have
dynamical degrees of freedom on its own. For this reason, one says
that in the Einstein-Cartan theory torsion is non-propagating
; It cannot propagate through empty space like, say, gravitational
waves.