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5Classification of the Ricci tensor and theenergy-momentum tensor5.1 The algebraic types of the Ricci tensorIn §3.5 we decomposed the curvature tensor into irreducible parts. Theinvariant classification of the Weyl tensor was treated in Chapter 4. Nowwe consider the algebraic classification of the remaining part, the tracelessRicci tensor S ab .In a Riemannian space, every second-order symmetric tensor defines alinear mapping which takes a vector v into another vector w. To classifyS ab , it is natural to examine the eigenvalue equationS a bv b = ̂λv a . (5.1)Because a term proportional to g a b merely shifts all eigenvalues by thesame amount, we may as well consider the eigenvalue equation for theRicci tensor R a bR a bv b = λv a ; λ = ̂λ + 1 4R. (5.2)In a positive definite metric, a real symmetric matrix can always bediagonalized by a real orthogonal transformation. However, the Lorentzmetric g ab leads to a more complicated algebraic structure; the elementarydivisors can be non-simple, and the eigenvalues can be complex. Theeigenvalue equation (5.2) determines the orders m 1 ,...,m k of the elementarydivisors belonging to the various eigenvalues. The Segre notation(1884), which also appears in Weiler (1874), gives just these orders, andround brackets indicate that the corresponding eigenvalues coincide. Iftwo eigenvalues are complex conjugates they are symbolized by Z and Z.The Plebański notation (1964) indicates whether the space spanned bythe eigenvectors belonging to a certain real eigenvalue is timelike (T ),null (N) or spacelike (S). The multiplicity of the eigenvalue is written infront of this symbol. Finally the orders of the corresponding factors in57