First-Order Palatini Gravity

First-order Palatini gravity treats the oriented coframe $e^I$ and Lorentz connection $\omega^{IJ}$ as independent variables. The connection variation then imposes zero torsion, and only after solving that equation does the coframe equation reduce to Einstein’s equation. This classical equivalence is conditional on an invertible tetrad, suitable boundary data, and matter that does not independently source the connection.

Required background. Frames, Coframes, and Anholonomy supplies coframes and Lorentz gauge, The Gibbons-Hawking-York Boundary Term supplies the comparison variational problem, and Presymplectic Potentials and Currents supplies the interpretation of the surface term in the first variation.

Helpful background. The Einstein Equation supplies the metric equation recovered on shell, Hypersurface Normals and Tangential Projectors supplies boundary normals, and Formulations of General Relativity situates first-order gravity among equivalent formulations.

Independent coframe and connection

Take $\eta_{IJ}=\operatorname{diag}(-1,1,1,1)$, $\epsilon_{0123}=+1$, and $\kappa=8\pi G$. Define

$$F^{IJ}=d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ}, \qquad T^I=De^I.$$

The Palatini action with cosmological constant is

$$S_{\rm P}[e,\omega] =\frac1{4\kappa}\int_{\mathcal M} \epsilon_{IJKL}e^I\wedge e^J\wedge \left(F^{KL}[\omega] -\frac{\Lambda}{6}e^K\wedge e^L\right) +S_{\partial\mathcal M}.$$

No Levi–Civita relation is assumed in this definition. The 16 coframe components and 24 connection components are varied separately, subject to their boundary conditions. The action, connection equation, and symplectic potential are derived in Corichi, Rubalcava-García, and Vukašinac 2016, §IV.A, pp. 22–24, PDF.

Varying the Lorentz connection

The curvature variation is covariant:

$$\delta F^{IJ}=D(\delta\omega^{IJ}).$$

Integrating by parts gives the connection contribution

$$\begin{aligned} \delta_\omega S_{\rm P} ={}&-\frac1{4\kappa}\int_{\mathcal M} D\!\left(\epsilon_{IJKL}e^I\wedge e^J\right) \wedge\delta\omega^{KL}\\ &+\frac1{4\kappa}\int_{\partial\mathcal M} \epsilon_{IJKL}e^I\wedge e^J\wedge\delta\omega^{KL}. \end{aligned}$$

The surface term must be canceled by boundary conditions or by $S_{\partial\mathcal M}$. The bulk equation is

$$\epsilon_{IJKL}e^I\wedge T^J=0.$$

The internal dual is invertible, so this is equivalent to $e^I\wedge T^J-e^J\wedge T^I=0$. To see why an invertible coframe forces zero torsion, let $\iota_I$ denote contraction with its dual vector and put $t=\sum_J\iota_JT^J$. Contract the preceding equation with $\sum_J\iota_J$. The identity $\sum_J e^J\wedge\iota_J\alpha=p\alpha$ for a $p$-form gives, in four dimensions,

$$T^I-e^I\wedge t-2T^I=0, \qquad T^I=-e^I\wedge t.$$

Contract once more and sum over $I$: $t=-3t$, so $t=0$ and every $T^I$ vanishes. Therefore

$$T^I=0, \qquad \omega^{IJ}=\omega^{IJ}(e).$$

This inference fails for a degenerate coframe because contractions with the inverse tetrad are then unavailable.

Varying the coframe

Since the connection is held fixed in the coframe variation, the bulk equation is

$$\epsilon_{IJKL}e^J\wedge \left(F^{KL}[\omega] -\frac{\Lambda}{3}e^K\wedge e^L\right)=0.$$

On the invertible, torsion-free branch, substitute $\omega=\omega(e)$. Multiplication by the inverse tetrad then gives

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=0.$$

The order of reasoning matters. Varying $e$ and $\omega$ independently and then solving the connection equation agrees with varying the second-order tetrad action only under the stated hypotheses. It does not license setting $T^I=0$ before variation in a theory whose matter action depends on $\omega^{IJ}$.

Where equivalence can fail

For matter with spin current

$$\tau_{IJ}\equiv\frac{\delta S_{\rm m}}{\delta\omega^{IJ}},$$

the connection equation is sourced. A minimally coupled Dirac field produces algebraic contorsion; eliminating it yields a local fermion interaction in addition to the torsion-free Einstein–Dirac action. Scalar and Maxwell matter coupled only through the metric do not source $\omega$.

Degenerate tetrads can support connection components or solution branches absent from metric general relativity. Boundaries also matter: two actions can have identical bulk equations but different allowed variations, symplectic potentials, and charges. Finally, “Palatini” sometimes denotes a metric and independent affine connection rather than a tetrad and Lorentz connection. That metric-affine theory has a different gauge structure and should not be identified with the present action without a separate derivation.

Flat solution check

With the Cartesian Minkowski coframe, $\omega^{IJ}=0$, and $\Lambda=0$, both $T^I$ and $F^{IJ}$ vanish. The connection and coframe equations are satisfied independently. A Lorentz-gauge transform produces nonzero pure-gauge $\omega'$ while preserving both equations.

Exercise: derive the connection equation

Starting from the curvature term in $S_{\rm P}$, carry out the connection variation and show explicitly why the bulk equation is $\epsilon_{IJKL}e^I\wedge T^J=0$.

Solution

Using $\delta F^{KL}=D\delta\omega^{KL}$,

$$\delta_\omega S_{\rm P} =\frac1{4\kappa}\int \epsilon_{IJKL}e^I\wedge e^J\wedge D\delta\omega^{KL}.$$

Covariant integration by parts gives a boundary term and

$$-\frac1{4\kappa}\int D\!\left(\epsilon_{IJKL}e^I\wedge e^J\right) \wedge\delta\omega^{KL}.$$

Because $D\epsilon_{IJKL}=0$ and one-forms anticommute,

$$D(e^I\wedge e^J) =T^I\wedge e^J-e^I\wedge T^J =2T^{[I}\wedge e^{J]}.$$

Antisymmetry of $\epsilon_{IJKL}$ therefore reduces the Euler–Lagrange coefficient to a nonzero multiple of $\epsilon_{IJKL}e^I\wedge T^J$. Its vanishing is the connection equation; invertibility then implies $T^I=0$.

References

• Corichi, Alejandro, Irais Rubalcava-García, and Tatjana Vukašinac. “Actions, Topological Terms and Boundaries in First Order Gravity: A Review.” International Journal of Modern Physics D 25, no. 4 (2016): 1630011. DOI. Open PDF.