Einstein–Cartan Gravity and Spin

Einstein–Cartan gravity promotes the Lorentz connection to an independent variable but keeps it metric compatible. Matter spin then sources torsion just as stress–energy sources curvature. In the minimal theory the connection equation is algebraic, so torsion carries no wave equation: it can be eliminated point by point, leaving general relativity plus contact interactions inside spinning matter.

Required background. Frames, Tetrads, and Spinors supplies tetrads and spin connections; Differential Forms and Integration supplies curvature and torsion forms; and Gravitational Actions and Boundary Terms supplies first-order variation and its boundary assumptions.

Helpful background. Einstein Equations gives the torsion-free limit, and Formulations of General Relativity places the first-order variables among other formulations.

Tetrad and Lorentz-connection variables

Let $e^a=e^a{}_{\mu}dx^\mu$ be an invertible coframe and let $\omega^{ab}=-\omega^{ba}$ be an independent Lorentz connection. Metric compatibility is built in through

$$g_{\mu\nu}=\eta_{ab}e^a{}_{\mu}e^b{}_{\nu}, \qquad \eta_{ab}=\operatorname{diag}(-1,1,1,1).$$

The curvature and torsion two-forms are

$$R^{ab}=d\omega^{ab}+\omega^a{}_c\wedge\omega^{cb}, \qquad T^a=De^a=de^a+\omega^a{}_b\wedge e^b.$$

For an oriented four-dimensional spacetime, the minimal gravitational action is

$$S_{\mathrm{EC}} =\frac{1}{32\pi G} \int \epsilon_{abcd}\,e^a\wedge e^b\wedge R^{cd} +S_{\mathrm m}[e,\omega,\psi].$$

The coframe equation is an Einstein equation built from the full connection. The connection equation is the decisive one. Define the matter spin three-form by

$$\delta_\omega S_{\mathrm m} =-\int\delta\omega^{ab}\wedge\tau_{ab}.$$

After integrating $D(\delta\omega^{ab})$ by parts and fixing the connection variation at the boundary, one obtains

$$\epsilon_{abcd}\,T^c\wedge e^d =16\pi G\,\tau_{ab}.$$

No derivative of $T^a$ occurs. For an invertible coframe this linear algebraic equation determines the 24 torsion components from the 24 components of the spin current. The component form, its explicit solution, and the reduction to an effective Einstein equation are given in Trautman 2006, pp. 189–195, Eqs. (22)–(32), PDF.

Why torsion is auxiliary

Write the Lorentz connection as $\omega=\mathring\omega+K$, where $\mathring\omega$ is the torsion-free spin connection and $K$ is contorsion. The curvature splits as

$$R^{ab}(\omega) =\mathring R^{ab} +\mathring D K^{ab} +K^a{}_c\wedge K^{cb}.$$

In the Einstein–Cartan action, the $\mathring D K$ term is a boundary term after using $\mathring D e^a=0$. The bulk action therefore contains $K$ quadratically but never with a kinetic term. Schematically,

$$S[e,K,\psi] =S_{\mathrm{GR}}[e]+\int e\left( \frac{1}{2\kappa}K\mathcal M K+K\cdot s \right)+S_{\mathrm m}[e,\mathring\omega,\psi], \qquad \kappa=8\pi G.$$

Varying gives $K=-\kappa\mathcal M^{-1}s$; substituting it back produces a local term quadratic in spin. This is the precise sense in which torsion does not propagate. It is not a statement that every theory containing torsion is nonpropagating. Derivative terms such as $DT\,DT$, suitable curvature-squared terms, or additional independent connection kinetic terms can turn affine components into dynamical fields.

Dirac matter and the induced contact term

For a minimally coupled Dirac field with the Hermitian Dirac kinetic action, only the totally antisymmetric torsion component is sourced. To specify the axial-current sign completely, use the source’s Clifford convention $\{\gamma^a,\gamma^b\}=-2\eta^{ab}$, $\gamma^5=-i\gamma^0\gamma^1\gamma^2\gamma^3$, and $\bar\psi=\psi^\dagger\gamma^0$, together with the site’s $\epsilon_{0123}=+1$ in an oriented orthonormal frame. This Clifford sign is an explicit local convention; a convention with $\{\gamma^a,\gamma^b\}=2\eta^{ab}$ requires translating the Dirac action and adjoint as well. Define

$$\widehat T^a=\epsilon^{abcd}T_{bcd}, \qquad J_5^a=\bar\psi\gamma^5\gamma^a\psi.$$

With this normalization, the torsion-dependent part of the Lagrangian can be arranged as

$$\mathcal L_{\widehat T} =e\left( \frac{1}{48\kappa}\widehat T_a\widehat T^a -\frac18\widehat T_aJ_5^a \right).$$

Its equation is $\widehat T^a=3\kappa J_5^a$. Eliminating $\widehat T^a$ yields

$$\mathcal L_{\mathrm{eff}} =\mathcal L_{\mathrm{GR+Dirac}} -\frac{3\kappa e}{16}J_{5a}J_5^a.$$

The displayed sign belongs to the displayed Clifford, axial-current, and gravitational-action conventions. Reversing the definition of $J_5$ alone reverses its linear coupling but leaves $J_5^2$ unchanged. The axial coupling and the distinction between minimal and nonminimal fermion couplings are derived in Rigouzzo and Zell 2023, conventions on p. 5 and §§2.1–2.2, Eqs. (2.19) and (2.29)–(2.32), PDF.

The interaction is suppressed by $G$. Since $[\kappa]=-2$ and $[J_5]=3$ in mass units, $\kappa J_5^2$ has the required Lagrangian dimension four. At ordinary densities this is extremely small, but the algebraic structure—not its phenomenological size—is the durable result.

Limits and physical interpretation

If the spin current vanishes, the connection equation gives $T^a=0$ and $\omega=\mathring\omega$ up to boundary and topology qualifications. Vacuum solutions therefore coincide with those of general relativity. For scalar matter, a minimal covariant derivative contains no connection and hence no intrinsic spin source. Gauge-field strength should be defined by the exterior derivative so that internal gauge symmetry is not spoiled by torsion.

The separation between canonical stress, spin, and the symmetric effective stress tensor follows from local Lorentz invariance. After torsion is eliminated, the coframe equation can be written with the Levi-Civita Einstein tensor and a symmetric stress tensor containing the spin-squared correction. Existence of this correction does not establish singularity avoidance, cosmological viability, or an observable effect. Current empirical likelihoods belong to Tests of Gravity; quantum-gravity completions and their ultraviolet claims remain outside this classical derivation.

Common pitfalls

Independent does not mean propagating. A field can be varied independently and still have an algebraic equation. Propagation requires a nondegenerate kinetic operator after constraints are accounted for.

Macroscopic rotation is not automatically intrinsic spin. The Cartan source is the variational spin current of matter, not an arbitrary orbital angular-momentum density.

Exercises

Complete the square in $\mathcal L_{\widehat T}$ and verify both the torsion solution and the coefficient of the four-fermion interaction.

Solution

Factor $e/(48\kappa)$ and write $\widehat T^2-6\kappa\widehat T\cdot J_5=(\widehat T-3\kappa J_5)^2-9\kappa^2J_5^2$. The stationary point is therefore $\widehat T^a=3\kappa J_5^a$. The square vanishes on that solution, while the remainder is $e(48\kappa)^{-1}(-9\kappa^2J_5^2)=-3\kappa eJ_5^2/16$. Its mass dimension is four, independently checking the normalization.

References

• Rigouzzo, C., and S. Zell, “Coupling Metric-Affine Gravity to the Standard Model and Dark Matter Fermions,” Physical Review D 108 (2023) 124067. DOI. Open PDF
• Trautman, A., “Einstein–Cartan Theory,” in J.-P. Françoise, G. L. Naber, and S. T. Tsou, eds., Encyclopedia of Mathematical Physics, vol. 2 (Elsevier, 2006), pp. 189–195. Stable record. Open PDF