Frames, Tetrads, Spinors, and Clifford Geometry

Frames translate invariant geometry into local components without confusing a basis choice with a coordinate system. Orthonormal tetrads expose Lorentzian measurements; connection forms record how frames change; Clifford multiplication and a spin lift make spinor fields possible. The central lesson is that connection coefficients may change inhomogeneously while torsion and curvature retain covariant meaning.

This chapter develops the geometric language only. It does not derive Newman–Penrose field equations, choose gravitational dynamics, quantize fermions, or survey representation theory. Its local formulas are paired with the global condition that determines whether spinors can be defined consistently across all patches.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian inner products, signature, and index operations; Affine Connections and Covariant Derivatives supplies the connection framework expressed here in moving frames.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps, bilinear forms, and changes of basis. Exterior Algebra and Differential Forms supplies wedge products, while The Exterior Derivative and the Poincare Lemma supplies $d$. Killing Fields and Isometry Groups supplies the spacetime-isometry side of the comparison with internal frame freedom. Tensor and Geometry Bridge offers a shorter review of the notation.

From coordinates to moving frames

A coordinate basis is tied to commuting coordinate directions. A general frame $e_a$ is merely a smooth basis of each tangent space and may satisfy

$$[e_a,e_b]=C^c{}_{ab}e_c.$$

The dual coframe detects the same anholonomy through

$$d\theta^c=-\frac12C^c{}_{ab}\theta^a\wedge\theta^b.$$

When the frame is orthonormal, the metric has constant internal components $\eta_{ab}$ even though the coframe varies. The Levi-Civita connection is then carried by one-forms $\omega^a{}_b$, torsion by $T^a$, and curvature by $\Omega^a{}_b$. Carroll presents this passage from coordinate tensors to tetrads and spin connections in Carroll 1997, §3, pp. 88–95, eqs. (3.114)–(3.143).

The distinction between a nonzero connection and nonzero curvature is especially important. A rotating orthonormal frame in flat space has nonzero $\omega$, while $\Omega$ vanishes. A suitable frame can remove the connection at one point, but nonzero curvature prevents its removal on an open region.

Preparation and shortest repairs

Readers need Lorentzian inner products, changes of basis, differential forms, and covariant derivatives. Review Pseudo-Riemannian Metrics and Signature for signature, orthogonality, and index operations. Review Affine Connections and Covariant Derivatives for the general connection framework. Exterior Algebra and Differential Forms develops forms and wedge products; The Exterior Derivative and the Poincare Lemma separately develops $d$.

The spin pages assume only the algebraic idea of a representation plus the distinction between local data and bundle gluing. They introduce the Clifford sign, the double cover, the obstruction class, and the spinor connection explicitly.

Route through the six pages

Begin with Frames, Coframes, and Anholonomy. It defines dual local bases and proves the equivalence between frame commutators and coframe exterior derivatives. Its polar coframe provides the recurring example.

Orthonormal Tetrads and Measured Components reconstructs the metric from a coframe and resolves tensors in an observer-adapted Lorentz basis. A boost calculation distinguishes frame-dependent components from invariant norms.

Connection Forms and Cartan Structure Equations derives

$$T^a=d\theta^a+\omega^a{}_b\wedge\theta^b, \qquad \Omega^a{}_b=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b.$$

It compares a flat polar frame with a round sphere, keeping anholonomy, connection, and curvature separate. Local Lorentz Freedom and Frame Gauge then derives the inhomogeneous law for $\omega$ and the homogeneous law for $\Omega$.

Clifford Algebras and Spin Structures states $\{\gamma^a,\gamma^b\}=2\eta^{ab}$, constructs the spin double cover, and separates local Clifford modules from a global spin lift. Wernli develops the algebraic cover in Wernli 2019, §§1.2–1.3, pp. 12–21 and the obstruction and classification in Wernli 2019, §§2.3.1–2.3.3, pp. 60–70, with Lorentzian spacetime conditions compared against Geroch 1968, §I, pp. 1739–1742.

Finish with Spin Connections and Spinor Curvature. It lifts $\omega$ to $\Gamma=\tfrac14\omega_{ab}\gamma^{ab}$ and shows that the commutator of spinor derivatives is the Riemann curvature in the spin representation.

Two reader routes

For tensor calculations and local measurements, read the first four pages in order. At every stage reconstruct the metric, check duality, and distinguish the coordinate transformation from the internal Lorentz transformation. This route is sufficient preparation for later hypersurface frames or curvature decompositions without introducing spinors.

For spin geometry, read frames and tetrads, then Cartan’s equations, Clifford algebras, and spin connections. The local chain is

$$\theta^a\longrightarrow\omega^a{}_b \longrightarrow\Gamma \longrightarrow F.$$

The global chain is different: orient and time-orient the frame bundle, test $w_2(TM)$, choose a spin lift, and then form its associated spinor bundle. A successful local gamma-matrix calculation cannot replace that global test.

Recurring checks and common distinctions

The polar coframe $\theta^1=dr$, $\theta^2=r\,d\phi$ is anholonomic. Its Levi-Civita connection is nonzero, but its curvature vanishes on $r>0$. On a round sphere, a similar-looking coframe gives curvature $1/a^2$. This pair prevents the mistaken identifications “anholonomy equals torsion” and “connection equals curvature.”

Under $\theta'=\Lambda\theta$, the metric is invariant, the connection gains $-d\Lambda\Lambda^{-1}$, and curvature transforms by conjugation. In the spin lift, $\psi'=S\psi$ and $\Gamma$ gains the corresponding $-dS\,S^{-1}$. Every sign can be checked by composing two transformations.

Finally, the Clifford bundle and spinor bundle are not the same object. Pointwise Clifford multiplication follows from the metric. A global spinor bundle in the desired representation requires a spin structure, and different spin structures can share identical local formulas.

Boundaries and onward connection

Tetrad components can represent readings by a specified local observer, but comparing separated readings requires transport and a path. A frame gauge is not a spacetime isometry, and fixing it does not remove curvature. Spin-connection geometry does not supply a Dirac equation until a field, action, and dynamical setting are chosen.

The next chapter, Congruences and Kinematical Decomposition, uses observer-adapted and screen frames to split deformation into expansion, shear, and vorticity. More specialized null-tetrad field equations remain outside this chapter.

Chapter diagnostic

In cylindrical Minkowski coordinates, use $\theta^0=dt$, $\theta^1=dr$, $\theta^2=r\,d\phi$, and $\theta^3=dz$. Find the nonzero anholonomy coefficient, the torsion-free connection pair, the curvature, and the spin connection. Explain why none of the nonzero intermediate quantities contradicts flatness.

Solution

The dual angular frame is $e_2=r^{-1}\partial_\phi$, so

$$[e_1,e_2]=-\frac1r e_2, \qquad C^2{}_{12}=-\frac1r.$$

Cartan’s first equation with zero torsion gives $\omega^1{}_2=-d\phi$ and $\omega^2{}_1=d\phi$. Cartan’s second equation gives $\Omega^1{}_2=d(-d\phi)=0$ on $r>0$. With $\gamma^{12}=\tfrac12[\gamma^1,\gamma^2]$ and both antisymmetric index orders included,

$$\Gamma=-\frac12d\phi\,\gamma^{12}, \qquad F=d\Gamma+\Gamma\wedge\Gamma=0.$$

The anholonomy records a noncoordinate orthonormal frame, and the connection records its rotation. Curvature is the invariant obstruction and vanishes. The exclusion of $r=0$ marks the polar patch’s failure, not a curvature singularity.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019 (1997), arXiv.
• Geroch, Robert. “Spinor Structure of Space-Times in General Relativity. I.” Journal of Mathematical Physics 9, 1739–1744 (1968), DOI.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766 (2019), arXiv.