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40 3 Some topics in Riemannian geometryThe matrix Q determines ten real numbers corresponding to the ten independentcomponents of the Weyl tensor.3.6 SpinorsSpinor formalism provides a very compact and elegant framework for numerouscalculations in general relativity, e.g. algebraic classification of theWeyl tensor (Chapter 4) and the Newman-Penrose technique (Chapter 7).It can be shown that the (connected) group SL(2, C) of linear transformationsin two complex dimensions, with determinant of modulus 1, hasa two-to-one homomorphism onto the group L ↑ +. The space on whichSL(2, C) acts is called spinor space, and its elements are (one-index)spinors with components ϕ A . Spinor indices like A obviously range over1 and 2, or, commonly, 0 and 1. Every proper Lorentz transformation definesan element of SL(2, C) up to overall sign. Since the defining propertyof L ↑ + (within all linear transformations in four dimensions) is that it is the(connected) group preserving the Minkowski metric, and since SL(2, C)is defined (within all linear transformations of two complex dimensions)as the (connected) group that preserves determinants, we expect that thedeterminant-forming 2-form in spin space, with components( ) 01ε AB = = ε−1 0AB , (3.66)will play the role of the metric. Spinor indices are raised and loweredaccording to the ruleϕ A = ε AB ϕ B ⇔ ϕ A = ϕ B ε BA . (3.67)Note that ϕ A ε AB ̸= ε BA ϕ A . The scalar product of two spinors (withcomponents ϕ A and ψ A ) is then defined byε AB ϕ A ψ B = ϕ A ψ A = −ϕ A ψ A . (3.68)If ϕ B transforms under S A B ∈ SL(2, C), the complex conjugate spinorϕḂ must, for consistency, transform under the complex conjugate SȦḂ ,and similarly ϕ A transforms under the inverse of S A B. Dotted indices areused to indicate that the complex conjugate transformations are to beapplied. The order of dotted and undotted indices is clearly irrelevant.One can obviously build multi-index spinors, in just the same way thattensors are developed from vectors.It is now natural to seek a correspondence between the vectors v ofMinkowski space and spinors. To do so we shall need not one-index