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An orthonormal tetrad, or vierbein, is a local basis $e_I{}^a$ of tangent vectors together with inverse coframe $e_a{}^I$, satisfying $g_{ab}{e}_I{}^a{e}_J{}^b={\eta }_{IJ}$ and $g_{ab}=e_a{}^I{e}_b{}^J{\eta }_{IJ}$. Capital indices label the locally orthonormal Lorentz frame; lower-case indices label spacetime coordinates. Changing the frame by a local Lorentz transformation leaves the metric unchanged. A spinor description uses a compatible local spin frame and the torsion-free spin connection.

Let fixed flat matrices satisfy $\{{\Gamma }_I,{\Gamma }_J\}=2{\eta }_{IJ}1$. The curved spacetime gamma matrices are ${\gamma }_a=e_a{}^I{\Gamma }_I$ and ${\gamma }^a=e_I{}^a{\Gamma }^I$. ThereforeThe identity matrix is four by four for a four-dimensional Dirac spinor. The combined tensor/spin covariant derivative makes ${\nabla }_a{\gamma }_b=0$, the Clifford version of the tetrad postulate.

Fix the curvature slots by $R_{abcd}=g_{ae}{R}^e{}_{bcd}$, with $[{\nabla }_c,{\nabla }_d]v^a=R^a{}_{bcd}{v}^b$, and $R=g^{ac}{g}^{bd}{R}_{abcd}$. The torsion-free spin connection then has $[{\nabla }_a,{\nabla }_b]\psi =\frac{1}{4}{R}_{abcd}{\gamma }^c{\gamma }^d\psi$. These choices fix the signs in the requested identities.

Write ${\gamma }^{ab}={\gamma }^{[a}{\gamma }^{b]}$ and similarly for four indices. Clifford anticommutation givesOn contraction with the Riemann curvature tensor, the four-form term vanishes by the first Bianchi identity. The first and last two-form terms vanish by pair antisymmetry; the remaining two-form terms are symmetric Ricci contractions against antisymmetric gamma matrices and vanish too. The scalar terms are zero, minus R and minus R respectively. Thus the Riemann tensor Clifford contraction isThis algebraic result is independent of Lorentz signature when the contraction convention is held fixed.

Apply the Dirac operator twice, using covariant constancy of the gamma matrices. Define the rough wave operator by $\square =g^{ab}{\nabla }_a{\nabla }_b$, with the full connection acting on the intermediate covector-spinor derivative. Splitting the product into symmetric and antisymmetric parts givesConsequently the Lichnerowicz spinor-square formula and the assumed massless Dirac equation imply $\boxed{(\square -R/4)\psi =0}$. Torsion would modify the commutator and introduce extra terms, so the torsion-free connection is part of this argument.

A vierbein is a local orthonormal coframe, consisting of four linearly independent covectors $e^{\mu }{}_a$ such thatHere $a,b$ label spacetime coordinates, while $\mu ,\nu$ label a Lorentz frame with $\eta =\mathrm{diag}(-1,1,1,1)$. Its inverse $e_{\mu }{}^a$ obeys $e_{\mu }{}^a{e}^{\mu }{}_b={\delta }_b^a$ and $e_{\mu }{}^a{e}^{\nu }{}_a={\delta }_{\mu }^{\nu }$. The vierbein converts tensor indices between the coordinate and orthonormal descriptions.

The metric is unchanged under an arbitrary local Lorentz transformation $e^{\mu }{}_a\mapsto {\Lambda }^{\mu }{}_{\nu }(x)e^{\nu }{}_a$, since ${\Lambda }^T\eta \Lambda =\eta$. This is local frame redundancy, with corresponding transformations of fields and connections; it is distinct from coordinate diffeomorphism covariance. A generic curved spacetime does not have global Lorentz isometries. Rather, local tangent frames have Lorentz freedom, and the field equations can be written covariantly under their changes. On flat spacetime, global Lorentz transformations are also spacetime isometries.

A spinor field is a section of the spinor bundle associated to a spin structure. Local components transform in a Spinor representation of the Lorentz group, more precisely a representation of its spin double cover. For the proper orthochronous group in four dimensions, that cover is ${\mathrm{Spin}}^{+}(1,3)\simeq SL(2,\mathbb{C})$. Thus a spinor field transforms as $ϵ\mapsto S(\Lambda )ϵ$ under frame changes, rather than as a coordinate-index tensor. The double-valued lift of a frame rotation requires a consistent spin structure globally; local spinor calculations alone do not choose that global structure. One can use a Dirac spinor for the gamma-matrix calculation below.

Write ${\omega }_a{}^{\mu }{}_{\nu }$ for the Lorentz-frame spin connection. Expanding the tetrad postulate givesMultiply by the inverse vierbein to solve for the connection:With the torsion-free Levi-Civita connection, metric compatibility makes ${\omega }_{a\mu \nu }={\eta }_{\mu \rho }{\omega }_a{}^{\rho }{}_{\nu }$ antisymmetric in $\mu ,\nu$. For constant flat gamma matrices obeying $\{{\hat{\gamma }}^{\mu },{\hat{\gamma }}^{\nu }\}=2{\eta }^{\mu \nu }$, define ${\hat{\gamma }}^{\mu \nu }=[{\hat{\gamma }}^{\mu },{\hat{\gamma }}^{\nu }]/2$. The spinor covariant derivative isUnder a local Lorentz transformation, the matrix connection acquires the inhomogeneous term needed to make ${\nabla }_aϵ$ transform as a spinor. The derivative of the vierbein in the boxed formula is therefore essential.

For the conformally flat metric, choose $e^{\mu }{}_a=\Omega {\delta }^{\mu }{}_a$ and $e_{\mu }{}^a={\Omega }^{-1}{\delta }_{\mu }^a$. Let ${\ell }_a={\partial }_a\log \Omega$ and raise these coordinate indices temporarily with the flat metric $\eta$. Directly evaluating the Christoffel symbols givesSince ${\partial }_a{e}^{\mu }{}_b=\Omega {\ell }_a{\delta }_b^{\mu }$, the connection formula yieldsThis is the spin connection of a conformally flat metric in the specified conformal frame. It is frame dependent, unlike an invariant curvature tensor.

Set $L={\hat{\gamma }}^c{\ell }_c$. Antisymmetry of $\omega$ makes its action on spinorsFor $ϵ={\Omega }^{1/2}{ϵ}_0$ with ${\partial }_a{ϵ}_0=0$, we have ${\partial }_aϵ={\ell }_aϵ/2$. The Clifford algebra gives $\{{\hat{\gamma }}_a,L\}=2{\ell }_a$, soThe curved gamma matrices are ${\gamma }_a=\Omega {\hat{\gamma }}_a$ and ${\gamma }^a={\Omega }^{-1}{\hat{\gamma }}^a$. ConsequentlyThus the rescaled field is a twistor spinor. Multiplying this last equation by ${\gamma }_b$, symmetrizing in $a,b$ and using $\{{\gamma }_a,{\gamma }_b\}=2g_{ab}$ provesThis is the conformal rescaling of a parallel spinor to a twistor spinor. It does not claim that the rescaled spinor remains parallel in the conformal metric; the nonzero connection and the derivative of ${\Omega }^{1/2}$ combine precisely into its gamma-trace part.

For $g_{ab}={\Omega }^2{\eta }_{ab}$, use the vierbein $e^{\mu }{}_a=\Omega {\delta }^{\mu }{}_a$ and constant flat gamma matrices. The tetrad postulate and ${\Gamma }^c{}_{ab}={\delta }_a^c{\partial }_b\log \Omega +{\delta }_b^c{\partial }_a\log \Omega -{\eta }_{ab}{\eta }^{cd}{\partial }_d\log \Omega$ give the displayed lowered spin connection. With $L={\hat{\gamma }}^c{\partial }_c\log \Omega$, its spinor action is ${\nabla }_a={\partial }_a+[{\hat{\gamma }}_a,L]/4$. The Lorentz index lowering matters in indefinite signature.

The tetrad postulate expresses compatibility of the spacetime and Lorentz-frame connections on a vierbein. For the inverse coframe $e_{\nu }{}^b$, solving gives ${\omega }_a{}^{\mu }{}_{\nu }=e_{\nu }{}^b({\Gamma }^c{}_{ab}{e}^{\mu }{}_c-{\partial }_a{e}^{\mu }{}_b)$. Metric compatibility makes the lowered connection antisymmetric in its Lorentz indices. Lifting this connection through the Spinor representation of the Lorentz group defines the spinor covariant derivative.