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3.5 Decomposition of the curvature tensor 37k fixed:V ′ab = V ab ,U ′ ab = U ab − BW ab + B 2 V ab ,W ′ ab = W ab − 2BV ab .(3.42b)A general bivector can be expanded in terms of the basis {Z α , Z α }:Finally, we mention the relationX ab = c α Z α ab + d α Z α ab. (3.43)Z α a[bZ β c] a =0. (3.44)Ref.: For bivectors and their application, see also Debever (1966), Zundand Brown (1971), and Israel (1970). For the connection between bivectorsand the complex 3-space used in Chapter 4 see Cahen et al. (1967).3.5 Decomposition of the curvature tensorThe curvature tensor, with components (2.79) with respect to a basis{e a }, can be uniquely decomposed into parts which are irreducible representationsof the full Lorentz group,R abcd = C abcd + E abcd + G abcd , (3.45)where the following abbreviations have been used:E abcd ≡ 1 2 (g acS bd + g bd S ac − g ad S bc − g bc S ad ), (3.46)G abcd ≡ 112 R(g acg bd − g ad g bc ) ≡ 112 Rg abcd, (3.47)S ab ≡ R ab − 1 4 Rg ab, R ≡ R a a. (3.48)R and S ab respectively denote the trace and the traceless part of the Riccitensor R ab defined by (2.83).The decomposition (3.45) defines Weyl’s conformal tensor C abcd (see§3.7 for the relation of this tensor to conformal transformations). It andthe other parts in the decomposition (3.45) have the same symmetries(3.26) as the Riemann tensor. Moreover, we have the relationsC a bad =0, E a bad = S bd , G a bad = 1 4 g bdR. (3.49)The Weyl tensor is completely traceless, i.e. the contraction with respectto each pair of indices vanishes, and it has ten independent components.A space-time with zero Weyl tensor is said to be conformally flat.