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Tetrads in General Relativity

XI. Field Equations

Equation of Motion

   The field equations in the Einstein-Cartan theory are obtained from the tetradic action (11.22) by performing independent stationary variations with respect to the frame field and the Lorentz connection. The variation with respect to the frame field yields:

11.23

\[{\delta _{\mathbf{e}}}{S_{{\text{EC}}}} [{\mathbf{e}},{\mathbf{\omega }}] = \frac{1}{{2\kappa }}\int {d{x_4}}\, {\varepsilon _{abcd}}\delta {{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge \left( {{{\mathbf{R}}^{cd}} + \frac{\Lambda }{3}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right)\]

The condition that the action vanishes for arbitrary variations $\delta {{\mathbf{e}}^a}$ gives the equation of motion

11.24

\[\frac{1}{2}{\varepsilon_{abcd}}{{\mathbf{e}}^b} \wedge \left( {{{\mathbf{R}}^{cd}} + \frac{\Lambda }{3}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right) = 0\]

which is the Einstein (vacuum) field equation in the tetrad formalism, including the cosmological constant.

Proof

  1. Expand the curvature 2-form ${{\mathbf{R}}^{cd}}$ to create the 3-form equation
    \[\frac{1}{4}{\varepsilon _{abcd}}{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^m} \wedge {{\mathbf{e}}^n}\left( {R_{mn}^{cd} + \frac{\Lambda }{3}\delta _{mn}^{cd}} \right) = 0\]
  2. Multiply from the left with the pseudoscalar ${\operatorname{I} _4}$. Then use (11.13) and (11.19) for $p=3$ to get:
    \[ - \frac{1}{4}\delta _{acd}^{kmn}\left( {R_{mn}^{cd} + \frac{\Lambda }{3}\delta _{mn}^{cd}} \right){{\mathbf{e}}_k} = 0\]
  3. Expand the generalized Kronecker delta's with (11.16) to recover the tensor form of the Einstein (vacuum) field equations:
    \[{G_{ab}} - \Lambda {\eta _{ab}}= {R_{ab}} - \frac{1}{2}(R + 2\Lambda ){\eta _{ab}} = 0\]
    with the Ricci tensor and -scalar as defined in Ricci Tensor.

   The multiplication with  ${\operatorname{I} _4}$ above is the inverse of the multiplication with $\operatorname{I} _4^{ - 1}$ in (11.20). Both are duality operations equivalent, up to a sign, to the Hodge Star  $( * )$ operation in the theory of differential forms.