Two-component spinor calculus ($v^a=v^{A{A}^{\prime }},g_{ab}={ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}$)

The complexified tangent space of a four-dimensional Lorentzian manifold factors locally into an unprimed and a primed two-dimensional spinor representation. Their alternating forms ${ϵ}_{AB}$ and ${ϵ}_{A^{\prime }{B}^{\prime }}$ encode the metric tensor and raise or lower spinor indices. A convention must specify the order of contraction with these alternating forms. An antisymmetric spacetime tensor decomposes into symmetric unprimed and primed spinors, $F_{A{A}^{\prime }B{B}^{\prime }}={\varphi }_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}+{\tilde{\varphi }}_{A^{\prime }{B}^{\prime }}{ϵ}_{AB}$. This splits the six-dimensional space of differential two-forms into the two three-dimensional eigenspaces of the complexified Hodge star operator.

A two-component two-component spinor is an element of a two-dimensional complex spin space with invariant antisymmetric form ${ϵ}_{AB}$. In complexified four-dimensional geometry there are independent unprimed and primed spin spaces, and vectors are represented by $x^{A{A}^{\prime }}$. Raising an index uses the antisymmetric form, so ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$. This identity makes the incidence-plane directions ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$ totally null.

The totally symmetric unprimed four-index spinor represents one chiral part of the Weyl tensor. The other part is represented by a primed spinor, conjugate to the unprimed one on a Lorentzian real slice. The trace-free symmetry separates conformal curvature from the Ricci tensor and scalar-curvature terms.

The spinor covariant derivative commutator is a differential two-form in its derivative indices, so the two-component spinor calculus decomposition defines symmetric curvature operators ${\Delta }_{AB}$ and ${\Delta }_{A^{\prime }{B}^{\prime }}$. With ${ϵ}^{AB}{ϵ}_{AB}=2$, the unprimed projection is ${\Delta }_{AB}={ϵ}^{A^{\prime }{B}^{\prime }}[{\nabla }_{A{A}^{\prime }},{\nabla }_{B{B}^{\prime }}]/2$, and the primed projection is analogous. For a torsion-free connection these are algebraic curvature derivations: they annihilate scalar functions and obey the Leibniz rule on tensor products.

Metric compatibility makes the curvature action on a two-component spinor symplectic, so its lowered curvature spinor is symmetric in each pair. The pair-exchange symmetry of the Riemann curvature tensor makes $X_{ABCD}=X_{CDAB}$. This six-component tensor decomposes into the five-component totally symmetric Weyl curvature spinor ${\Psi }_{ABCD}=X_{(ABCD)}$ and one scalar term. In the displayed normalization, with alternating forms satisfying ${ϵ}^{AB}{ϵ}_{AB}=2$, its scalar is $\Lambda =X_{ABCD}{ϵ}^{AC}{ϵ}^{BD}/3$. This coefficient follows by contracting the displayed scalar term; different normalizations of that term give different scalar conventions.

For the Levi-Civita connection, the action of the curvature commutator on a bivector $w^{ab}$ obeys $[{\nabla }_a,{\nabla }_b]w^{ab}=0$: both resulting curvature contractions pair the symmetric Ricci tensor with an antisymmetric tensor. For a bivector of one chirality, $w^{A{A}^{\prime }B{B}^{\prime }}={\varphi }^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$, this is $2{\Delta }_{AB}{\varphi }^{AB}=0$. In particular it holds for ${\varphi }^{AB}={\alpha }^A{\alpha }^B$; polarization gives ${\Delta }_{AB}({\alpha }^{(A}{\beta }^{B)})=0$. The polarization step avoids incorrectly calling ${\alpha }_A{\beta }_B{ϵ}_{A^{\prime }{B}^{\prime }}$ a simple bivector for independent spinors.