Spin Connections, Torsion, and Compatibility

The spin connection differentiates internal Lorentz indices, while torsion measures whether the coframe closes under that covariant derivative. For an invertible coframe, Lorentz compatibility plus zero torsion selects the Levi–Civita spin connection uniquely. In a first-order action, however, the connection begins as an independent field and zero torsion must follow from an equation of motion rather than be inserted as a definition.

Required background. Frames, Coframes, and Anholonomy supplies coframes and local Lorentz freedom, The Gibbons-Hawking-York Boundary Term supplies the role of connection variations at boundaries, and Presymplectic Potentials and Currents supplies the distinction between field equations and boundary potentials.

Helpful background. The Einstein Equation supplies the target torsion-free dynamics, Hypersurface Normals and Tangential Projectors supplies induced connections, and Formulations of General Relativity locates connection variables in the wider formulation map.

Lorentz-covariant differentiation

For an internal vector-valued $p$-form $v^I$, define

$$Dv^I=dv^I+\omega^I{}_J\wedge v^J.$$

Compatibility with $\eta_{IJ}=\operatorname{diag}(-1,1,1,1)$ is $D\eta_{IJ}=0$, equivalently

$$\omega^{IJ}=-\omega^{JI}.$$

Under a proper local Lorentz transformation,

$$e' =\Lambda e, \qquad \omega'=\Lambda\omega\Lambda^{-1}-d\Lambda\,\Lambda^{-1}.$$

The two Cartan field strengths are

$$T^I=De^I=de^I+\omega^I{}_J\wedge e^J,$$ $$F^{IJ}=d\omega^{IJ}+\omega^I{}_K\wedge\omega^{KJ}.$$

They transform homogeneously: $T'^I=\Lambda^I{}_JT^J$ and $F'=\Lambda F\Lambda^{-1}$. Their Bianchi identities follow from applying $D$ once more:

$$DT^I=F^I{}_J\wedge e^J, \qquad DF^{IJ}=0.$$

Thus torsion is a tensorial field strength, not merely a failure to choose coordinates conveniently.

The torsion-free compatible connection

Let $e_I{}^\mu$ be the inverse coframe. A Lorentz connection and a spacetime connection are related by the tetrad postulate

$$\partial_\mu e^I{}_{\nu} -\Gamma^\rho{}_{\mu\nu}e^I{}_{\rho} +\omega_\mu{}^I{}_J e^J{}_{\nu}=0.$$

If $T^I=0$, the associated $\Gamma^\rho{}_{\mu\nu}$ is torsion free. Since $D\eta=0$ also implies $\nabla_\rho g_{\mu\nu}=0$, it is the Levi–Civita connection. Conversely, inserting the Levi–Civita symbols in the tetrad postulate gives

$$\omega_\mu{}^{IJ}(e) =e^{I\nu} \left( \partial_\mu e^J{}_{\nu} -\Gamma^\rho{}_{\mu\nu}(g)e^J{}_{\rho} \right).$$

Antisymmetry follows from differentiating $\eta^{IJ}=g^{\mu\nu}e^I{}_{\mu}e^J{}_{\nu}$. Hence an invertible coframe determines one and only one torsion-free Lorentz connection. The connection and torsion conventions used in first-order gravity are reviewed in Corichi, Rubalcava-García, and Vukašinac 2016, §IV.A, pp. 22–24, PDF.

The word “compatible” alone is insufficient. Lorentz compatibility fixes the antisymmetry of $\omega^{IJ}$ but allows torsion. Zero torsion supplies the remaining condition that selects $\omega(e)$.

Independent connection and contorsion

Any other Lorentz connection can be written

$$\omega^{IJ}=\omega^{IJ}(e)+C^{IJ}, \qquad C^{IJ}=-C^{JI},$$

where $C^{IJ}$ is the contorsion one-form. Because $D^{(e)}e^I=0$,

$$T^I=C^I{}_J\wedge e^J.$$

For an invertible tetrad this relation can be inverted algebraically between the 24 torsion and 24 contorsion components. Pure Palatini gravity sets them to zero. Matter with an independent spin current, notably a minimally coupled Dirac field, instead sources an algebraic contorsion. That situation is not classically identical to imposing the Levi–Civita connection before varying.

Degenerate coframes are another boundary of the uniqueness statement: without $e_I{}^\mu$, the tetrad postulate cannot be solved in the displayed way and the zero-torsion equation need not determine every connection component.

Anholonomic flat-space check

On flat spacetime in cylindrical coordinates choose

$$e^0=dt, \qquad e^1=dr, \qquad e^2=r\,d\theta, \qquad e^3=dz.$$

Although $de^2=dr\wedge d\theta$ is nonzero, flatness does not imply a vanishing spin connection in this frame. The torsion-free solution is

$$\omega^2{}_1=d\theta, \qquad \omega^1{}_2=-d\theta,$$

with all other components zero on the cylindrical patch. Its curvature still vanishes away from the coordinate axis.

Common pitfalls

Equating anholonomy with torsion. The $de^I$ term and the connection term in $T^I$ can cancel, as they do for the cylindrical coframe.

Imposing zero torsion too early. In a first-order matter theory, varying the independent connection can produce a nonzero spin-sourced torsion.

Exercise: verify the cylindrical connection

For the cylindrical coframe above, verify $T^I=0$ and $F^{IJ}=0$ for the stated connection.

Solution

Only the $I=2$ torsion equation has two potentially nonzero terms:

$$\begin{aligned} T^2 &=de^2+\omega^2{}_1\wedge e^1\\ &=dr\wedge d\theta+d\theta\wedge dr=0. \end{aligned}$$

For $I=1$,

$$T^1=\omega^1{}_2\wedge e^2 =-d\theta\wedge r\,d\theta=0,$$

and $T^0=T^3=0$. The only possible curvature component is

$$F^1{}_2=d(-d\theta) +\omega^1{}_K\wedge\omega^K{}_2=0$$

on the patch, because there is no second nonzero connection component for the quadratic term. The frame is anholonomic but the Levi–Civita geometry is both torsion free and flat.

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.