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Tetrads in General Relativity

XII. Einstein Cartan Theory

Einstein-Cartan Action

   In the Einstein-Cartan theory the Cartan curvature tensor (10.6) plays central role in the construction of the action. The first step to derive this action is to bring the Einstein-Hilbert action (11.6) into the form of a directed integral. This may be accomplished by inserting the product of the unit pseudo scalar and its inverse (A.1):

12.3

\[{S_{{\text{EH}}}} = \frac{1}{{2\kappa }}\int {{d^4}x} \sqrt {\left| g \right|} {\operatorname{I} _4} \operatorname{I}^{4}R: = \frac{1}{{2\kappa }}\int {d{x_4}} \operatorname{I} ^{4}R\]

The pseudoscalar volume $d{x_4}: = {d^4}x\sqrt {\left| g \right|} {\operatorname{I} _4}$ is the signed invariant volume measure; see Integration Measure.

   Subsequently, the expression for the Ricci scalar in the integrand may be  reworked into

12.4

\[R = R_{cd}^{cd} = \frac{1}{2}\delta _{cd}^{mn}R_{mn}^{cd} \]

The pre-factor is the generalized Kronecker-delta symbol (B.1) for $p=2$. The first identity (B.4), together with the tetrad form (A.2) of the inverse unit volume and formula (10.5) for the tetrad curvature 2-form, then allows the Einstein-Hilbert action, including the cosmological constant, to be cast in the tetradic Einstein-Cartan form:

12.5

\[{S_{{\text{EC}}}}[{\mathbf{e}},{\boldsymbol{\omega }}] = \frac{1}{{4\kappa }}\int {dx_4}\, {\varepsilon _{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge \left( {{{\mathbf{R}}^{cd}}[{\boldsymbol{\omega }}] + \frac{\Lambda }{6}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right)\]

   Without embellishments, Einstein-Cartan theory is a viable modification of Einstein-Hilbert gravitation, the frame field and the connection having the dynamical interpretation of $\text{T}(4)$ and ${\text{SO}}(1,3)$ gauge potentials. Moreover, the theory is completely consistent with all the experimental tests of gravity so far. For these reasons the Einstein-Cartan formulation is often preferred over the canonical Einstein-Hilbert formulation in explorations of quantum gravity.