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XII. Einstein-Cartan Theory
In the Einstein-Cartan theory of gravity, one usually assumes metric
compatibility of the Lorentz (spin) connection, i.e.
${{\boldsymbol{\omega }}_{ab}} = - {{\boldsymbol{\omega }}_{ba}}$.
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: $\delta
{{\mathbf{R}}^{ab}} = {\text{D}}\delta {{\boldsymbol{\omega
}}^{ab}}$. Hence
\[{\delta _{\boldsymbol{\omega
}}} {S_{{\text{EC}}}}[{\mathbf{e}},{\boldsymbol{\omega }}] =
\frac{1}{{4\kappa }}\int {d{x_4}{\text{ }}} {\varepsilon
_{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge
{\text{D}}\delta {{\boldsymbol{\omega }}^{cd}}\]
An integration by parts may be performed with the help of the
Leibniz rule
\[\begin{gathered}{\varepsilon
_{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge
{\text{D}}\delta {{\boldsymbol{\omega }}^{cd}} =
\hfill\\{\text{D(}}{\varepsilon _{abcd}}{{\mathbf{e}}^a} \wedge
{{\mathbf{e}}^b} \wedge \delta {{\boldsymbol{\omega }}^{cd}}) -
2{\varepsilon _{abcd}}{{\mathbf{T}}^a} \wedge {{\mathbf{e}}^b}
\wedge \delta {{\boldsymbol{\omega }}^{cd}}\end{gathered}\]
where ${{\mathbf{T}}^a}$ is the torsion 1-form as defined in
(4.26). Since in (12.10) the exterior coderivative at the right-hand
side acts on a scalar, this ${\text{D}}$ may be replaced by just the
exterior derivative $\text{d}$. Ignoring the resulting boundary
term, one obtains the torsion field equation of the (vacuum)
Einstein-Cartan theory:
\[{{\mathbf{S}}_{ab}} : = -
{\varepsilon _{abcd}}{{\mathbf{T}}^c} \wedge {{\mathbf{e}}^d}=0\]
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 (4.27) of GRT is achieved dynamically
in Einstein-Cartan gravity. Even when a vanishing torsion is not
imposed as a constraint, the theory defined by the Einstein-Cartan
action is equivalent to standard GRT, at least in the vacuum case.
Modified Torsion Tensor
- The torsion equation in tensor components may be obtained by
expanding the torsion 2-form ${{\mathbf{T}}^{c}}$ in (12.11) and
then by constructing the dual; see Equation
of Motion.
- Next one needs (B.1) and (B.4) for $p=3$ to find:
\[{*
\mathbf{S}}_{ab} = \left( {T_{ab}^c + 2\delta
_{[a}^cT_{b]d}^d} \right){{\mathbf{e}}_c} : =
S_{ab}^c{{\mathbf{e}}_c} = 0\]
The tensor $S_{ab}^c$ defined here is called the modified
torsion tensor.