# Spin connection
A **spin connection** is a connection on the bundle of orthonormal frames of a manifold equipped with a metric, written as a one-form \( \omega_{\mu}{}^{a}{}_{b} \) carrying one spacetime index and two Lorentz (frame) indices. In the principal-bundle formulation of general relativity, a principal connection can be pulled back to the base manifold in a canonical way and restricted to its antisymmetric component, giving the spin connection[1](https://www.mdpi.com/2218-1997/5/10/206). Its defining job is to make covariant derivatives well defined for quantities that carry Lorentz or spinor indices, which the ordinary Christoffel connection cannot handle[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf).
| Key fact | Detail |
|---|---|
| Index structure | One spacetime (Greek) index and two Lorentz (Latin) indices, \( \omega_{\mu}{}^{a}{}_{b} \); each Latin index of a tensor in an orthonormal basis acquires a factor of the spin connection[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) |
| Metric compatibility | Equivalent to antisymmetry in the Latin indices, \( \omega_{\mu ab} = -\omega_{\mu ba} \)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) |
| Tetrad postulate | \( \nabla_{\mu} e^{a}{}_{\nu} = 0 \), relating \( \omega \), the vielbein, and the Christoffel symbols; it holds without assuming metric compatibility or torsion-freeness[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) |
| Structure equations | \( T^{a} = de^{a} + \omega^{a}{}_{b} \wedge e^{b} \) and \( R^{a}{}_{b} = d\omega^{a}{}_{b} + \omega^{a}{}_{c} \wedge \omega^{c}{}_{b} \)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) |
| Transformation law | Under local Lorentz transformations it transforms inhomogeneously, like a gauge field, not as a tensor[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) |
| Spinor coupling | \( D_{\mu}\psi = \partial_{\mu}\psi + \tfrac{1}{4}\,\omega_{\mu ab}\,\gamma^{ab}\psi \), with \( \gamma^{ab} = \tfrac{1}{2}[\gamma^{a},\gamma^{b}] \)[3](https://physics.stackexchange.com/questions/865236/showing-d-mu-psi-transforms-as-a-spinor-with-the-spin-connection-supergravi) |
| Einstein–Cartan split | \( \tilde{\omega} = \omega + K \), a torsion-free part plus a contortion one-form \( K \) related to spacetime torsion[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5866282/) |
## Definition and role
In a coordinate basis, covariant differentiation of tensors uses the [Christoffel symbols](https://www.edgepedia.org/christoffel-symbols) \( \Gamma^{\lambda}{}_{\mu\nu} \). In an orthonormal (tetrad) basis, the same role is played by the spin connection: each Latin index gets a factor of \( \omega_{\mu}{}^{a}{}_{b} \), so that for a mixed tensor
\[ \nabla_{\mu} X^{a}{}_{b} = \partial_{\mu} X^{a}{}_{b} + \omega_{\mu}{}^{a}{}_{c} X^{c}{}_{b} - \omega_{\mu}{}^{c}{}_{b} X^{a}{}_{c}. \]
The name comes from the fact that this connection can be used to take covariant derivatives of spinors, which is impossible with the conventional Christoffel connection alone[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). Beyond a metric, defining a curved Dirac field also calls for an orientation with a time orientation, a spin structure, a tetrad, a suitably matched spin connection, and a well-defined adjoint; the tetrad translates spacetime indices into Clifford indices, and the spin connection renders local Lorentz transformations mere gauge redundancy[5](https://qft.org/curved-spacetime/fields-local-algebras-curved-backgrounds/spinors-tetrads-and-spin-connections/).
## Relation to the tetrad and Christoffel symbols
The vielbein \( e^{a}{}_{\mu} \) (tetrad in four dimensions) relates the metric to the flat metric, \( g_{\mu\nu} = e^{a}{}_{\mu} e^{b}{}_{\nu} \eta_{ab} \), which is why the vielbeins are sometimes called the square root of the metric[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). Demanding that a tensor be independent of the basis in which it is written yields the relation
\[ \Gamma^{\nu}{}_{\mu\lambda} = e^{\nu}{}_{a}\,\partial_{\mu} e^{a}{}_{\lambda} + e^{\nu}{}_{a}\, e^{b}{}_{\lambda}\, \omega_{\mu}{}^{a}{}_{b}, \]
or equivalently \( \omega_{\mu}{}^{a}{}_{b} = e^{a}{}_{\nu} e^{\lambda}{}_{b} \Gamma^{\nu}{}_{\mu\lambda} - e^{\lambda}{}_{b}\, \partial_{\mu} e^{a}{}_{\lambda} \)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). This can be written as the vanishing of the covariant derivative of the vielbein, \( \nabla_{\mu} e^{a}{}_{\nu} = 0 \), the tetrad postulate; it is always true and requires no assumption that the connection be metric compatible or torsion free[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf).
When the spin connection is required to be compatible with the tetrad in this way, it is not fully determined by the tetrad and its derivatives alone; imposing vanishing torsion as well determines the torsion-free spin connection from the tetrad and its derivatives, and an explicit formula using only the tetrad can be given[6](https://www.tpi.uni-jena.de/~floerchinger/assets/pdfs/SymmetriesFloerchinger.pdf) • [12](https://link.springer.com/article/10.1140/epjc/s10052-019-6852-y). Metric compatibility, \( \nabla g = 0 \), evaluated in the orthonormal basis where the metric components are the constant \( \eta_{ab} \), reduces to \( \omega_{\mu ab} = -\omega_{\mu ba} \): metric compatibility is equivalent to antisymmetry of the spin connection in its Latin indices[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). Together with the torsion-free condition \( \omega^{a}{}_{b} \wedge e^{b} = -de^{a} \), these two conditions fix the spin connection in terms of the vielbeins[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf).
## Transformation law and gauge character
The spin connection does not obey the tensor transformation law. Under general coordinate transformations its one lower Greek index transforms as a one-form, but under local Lorentz transformations it transforms inhomogeneously,
\[ \omega'_{\mu}{}^{a}{}_{b} = \Lambda^{a}{}_{c}\, \omega_{\mu}{}^{c}{}_{d}\, \Lambda_{b}{}^{d} + \Lambda^{a}{}_{c}\, \partial_{\mu} \Lambda_{b}{}^{c}, \]
which is exactly the transformation rule of a non-abelian gauge field for the local [Lorentz group](https://www.edgepedia.org/lorentz-group)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf) • [7](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2). A local [Lorentz transformation](https://www.edgepedia.org/lorentz-transformation) therefore acts on the tetrad and the spin connection simultaneously[7](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2). This gauge character is what makes the covariant derivative of a spinor, \( D_{\mu}\psi = \partial_{\mu}\psi + \tfrac{1}{4}\,\omega_{\mu ab}\,\gamma^{ab}\psi \), transform as a spinor[3](https://physics.stackexchange.com/questions/865236/showing-d-mu-psi-transforms-as-a-spinor-with-the-spin-connection-supergravi). In the [Palatini formulation](https://www.edgepedia.org/palatini-formulation), general relativity can be developed as a genuine connection theory in which the gravitational field is represented by a Lorentz-valued spin connection[8](https://arxiv.org/abs/gr-qc/0412032).
## Curvature and torsion
The torsion and curvature of the connection take compact form-language expressions, sometimes called the Cartan (or Maurer–Cartan) structure equations:
\[ T^{a} = de^{a} + \omega^{a}{}_{b} \wedge e^{b}, \qquad R^{a}{}_{b} = d\omega^{a}{}_{b} + \omega^{a}{}_{c} \wedge \omega^{c}{}_{b}. \]
They are equivalent to the usual component definitions; converting the torsion back to a coordinate basis recovers \( T^{\lambda}{}_{\mu\nu} = \Gamma^{\lambda}{}_{\mu\nu} - \Gamma^{\lambda}{}_{\nu\mu} \)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). The associated Bianchi identities read \( dT^{a} + \omega^{a}{}_{b} \wedge T^{b} = R^{a}{}_{b} \wedge e^{b} \) and \( dR^{a}{}_{b} + \omega^{a}{}_{c} \wedge R^{c}{}_{b} - R^{a}{}_{c} \wedge \omega^{c}{}_{b} = 0 \)[2](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf). In Riemann–Cartan geometry, where torsion is allowed, the spin connection remains metric compatible (antisymmetric in its Latin indices) but is no longer torsion free[9](https://www.mdpi.com/2218-1997/9/12/516).
## As an independent variable: Palatini and Einstein–Cartan
Although the torsion-free, metric-compatible spin connection is determined by the tetrad, it is often useful to treat it as an independent field, at least at intermediate stages of a calculation[6](https://www.tpi.uni-jena.de/~floerchinger/assets/pdfs/SymmetriesFloerchinger.pdf). In the Palatini (first-order) formalism the tetrad and the spin connection \( \omega_{\mu}^{ab} \), required only to be antisymmetric in its Lorentz indices, are varied independently; in vacuum general relativity the equation of motion with respect to \( \omega^{ab} \) is nothing but the torsion-free condition[10](https://www-tap.scphys.kyoto-u.ac.jp/jgrg/proc/JGRG1.pdf). Varying an action with respect to the spin connection at fixed tetrad defines the spin current \( S_{\mu}{}^{AB} \), and the interplay between the two variations reproduces the Belinfante–Rosenfeld form of the energy-momentum tensor[6](https://www.tpi.uni-jena.de/~floerchinger/assets/pdfs/SymmetriesFloerchinger.pdf).
**Einstein–Cartan theory.** When matter with intrinsic spin is present, the independent variation instead links the antisymmetric part of the connection to the spin density. [Einstein–Cartan theory](https://www.edgepedia.org/einstein-cartan-theory) is a geometric extension of general relativity in which the spin density of massive objects sources torsion, with fermionic fields such as protons, neutrons, and leptons providing natural sources[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5866282/). The full spin-connection one-form splits into a torsion-free part (the usual \( \omega \), related to the Christoffel symbol) and a contortion one-form \( K \) related to spacetime torsion[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5866282/). Two features follow from the simplest Lagrangian: the torsion field equations are purely algebraic equalities linking torsion to the spin of gravitating matter, so torsion does not propagate; and because the same scalar-curvature Lagrangian with the single Newton–Einstein gravitational constant governs both fields, the spin-torsion interaction is proportional to that constant and is therefore extremely weak[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC8919128/). With fermions, classically only the totally antisymmetric component of torsion is nonvanishing, and the on-shell spin connection acquires a term proportional to the axial fermion current, \( \omega_{\mu}^{ab} = \mathring{\omega}_{\mu}^{ab} + \tfrac{\kappa^{2}}{4}\,\epsilon^{ab}{}_{cd}\, e^{c}{}_{\mu}\, j^{5d} \)[9](https://www.mdpi.com/2218-1997/9/12/516).
## Teleparallel interpretation
In teleparallel gravity the same letter \( \omega \) plays a different physical role: the spin connection there represents inertial effects only, whereas in general relativity the [Levi-Civita connection](https://www.edgepedia.org/levi-civita-connection) mixes gravitational and inertial effects and the two cannot be separated[7](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2). There exists a class of frames, called proper frames, in which the teleparallel spin connection vanishes; in any other frame it takes a nonzero form fixed by the local Lorentz transformation relating the frames[7](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2). Choosing the spin connection associated with a given tetrad removes the inertial contributions from the action locally at each point, a procedure the authors describe as a renormalization that yields a purely gravitational action and a physically sensible energy-momentum density[7](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2).
## References
1. [On the Mathematics of Coframe Formalism and Einstein–Cartan Theory—A Brief Review, Universe (MDPI)](https://www.mdpi.com/2218-1997/5/10/206)
2. [Lecture Notes on General Relativity, S. Carroll (gr-qc/9712019)](https://www.physics.smu.edu/scalise/P3368sp17/CarrollGR.pdf)
3. [Showing \( D_\mu \Psi \) transforms as a spinor with the spin connection, Physics Stack Exchange](https://physics.stackexchange.com/questions/865236/showing-d-mu-psi-transforms-as-a-spinor-with-the-spin-connection-supergravi)
4. [Does space-time torsion determine the minimum mass of gravitating particles? (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5866282/)
5. [Spinors, Tetrads, and Spin Connections, QFT.org](https://qft.org/curved-spacetime/fields-local-algebras-curved-backgrounds/spinors-tetrads-and-spin-connections/)
6. [Lectures on Symmetries, S. Floerchinger, University of Heidelberg](https://www.tpi.uni-jena.de/~floerchinger/assets/pdfs/SymmetriesFloerchinger.pdf)
7. [Spin connection and renormalization of teleparallel action, European Physical Journal C](https://link.springer.com/article/10.1140/epjc/s10052-015-3749-2)
8. [General Relativity as a Genuine Connection Theory (arXiv:gr-qc/0412032)](https://arxiv.org/abs/gr-qc/0412032)
9. [Torsion at Different Scales: From Materials to the Universe, Universe (MDPI)](https://www.mdpi.com/2218-1997/9/12/516)
10. [Formulations of general relativity, JGRG proceedings (Kyoto)](https://www-tap.scphys.kyoto-u.ac.jp/jgrg/proc/JGRG1.pdf)
11. [100 years of mathematical cosmology: Models, theories, and problems, Part A (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC8919128/)
12. [Client Challenge](https://link.springer.com/article/10.1140/epjc/s10052-019-6852-y)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Mathematical structure of curved spacetime › Tetrads, frames, and spin structures*
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