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3.5 Decomposition of the curvature tensor 39In these definitions, C ∗ abcd /2 may be substituted for C abcd. The variousterms in (3.58) admit the following physical interpretation (Szekeres1965): the Ψ 4 -term represents a transverse wave in the k-direction, theΨ 3 -term a longitudinal wave component, and the Ψ 2 -term a ‘Coulomb’component; the Ψ 0 - and Ψ 1 -terms represent transverse and longitudinalwave components in the l-direction.With the aid of (3.42) we find the transformation laws of Ψ 0 ,...,Ψ 4under the null rotations (3.14), (3.15):l fixed: Ψ ′ 4 =Ψ 4, Ψ ′ 3 =Ψ 3 + EΨ 4 ,Ψ ′ 2 =Ψ 2 +2EΨ 3 + E 2 Ψ 4 ,Ψ ′ 1 =Ψ 1 +3EΨ 2 +3E 2 Ψ 3 + E 3 Ψ 4 ,(3.60)Ψ ′ 0 =Ψ 0 +4EΨ 1 +6E 2 Ψ 2 +4E 3 Ψ 3 + E 4 Ψ 4 .k fixed: Ψ ′ 0 =Ψ 0, Ψ ′ 1 =Ψ 1 + BΨ 0 ,Ψ ′ 2 =Ψ 2 +2BΨ 1 + B 2 Ψ 0 ,Ψ ′ 3 =Ψ 3 +3BΨ 2 +3B 2 Ψ 1 + B 3 Ψ 0 ,(3.61)Ψ ′ 4 =Ψ 4 +4BΨ 3 +6B 2 Ψ 2 +4B 3 Ψ 1 + B 4 Ψ 0 .Generalizing (3.37), (3.38), we can express Cabcd ∗ in terms of the complextensor−Q ab ≡ Cabcdu ∗ b u d ≡ E ac +iB ac , u c u c = −1, (3.62)according to the formula− 1 2 C∗ abcd =4u [a Q b][d u c] + g a[c Q d]b − g b[c Q d]a+iε abef u e u [c Q d] f +iε cdef u e u [a Q b] f . (3.63)E ac and B ac respectively denote the ‘electric’ and ‘magnetic’ parts of theWeyl tensor for the given four-velocity u a (Matte 1953). The componentsQ ab satisfy the relationsQ a a =0, Q ab = Q ba , Q ab u b =0, (3.64)and can be considered as a symmetric complex (3×3) matrix Q with zerotrace. Using (3.40), (3.58) and (3.62), and expressing the 3×3 matrix withrespect to the orthonormal basis given by (3.12),⎛Ψ 2 − 1 2 (Ψ 10 +Ψ 4 )2 i(Ψ ⎞4 − Ψ 0 ) Ψ 1 − Ψ 3Q = ⎜ 1⎝ 2 i(Ψ 4 − Ψ 0 )Ψ 2 + 1 2 (Ψ 0 +Ψ 4 )i(Ψ 1 +Ψ 3 ) ⎟⎠ . (3.65)Ψ 1 − Ψ 3 i(Ψ 1 +Ψ 3 ) −2Ψ 2