Conformal rescaling of a parallel spinor to a twistor spinor ($ϵ={\Omega }^{1/2}{ϵ}_0$)

In the conformal frame for $g={\Omega }^2\eta$, identify the spinor fibres using the chosen vierbein. If ${ϵ}_0$ is a flat constant spinor field, the spin connection of a conformally flat metric gives ${\nabla }_aϵ={\hat{\gamma }}_a{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Contracting with curved ${\gamma }^a={\Omega }^{-1}{\hat{\gamma }}^a$ yields ${\gamma }^a{\nabla }_aϵ=n{\Omega }^{-1}{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Thus ${\nabla }_aϵ={\gamma }_a{\gamma }^b{\nabla }_bϵ/n$: the rescaled parallel spinor is a twistor spinor.