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60 5 Classification of the Ricci tensorTable 5.2.Invariance groupInvariance groups of the Ricci tensor typesRicci tensor typesNone [111, 1], [11,ZZ], [11, 2], [1, 3]Spatial rotations (3.16) [(11)1, 1], [(11),ZZ], [(11), 2]Boosts (3.17) [11(1, 1)]Boosts (3.17) and rotations (3.16) [(11)(1, 1)]SO(3) rotations [(111), 1]SO(2, 1): three-dimensional Lorentz group [1(11, 1)]One-parameter group of null rotations [1(1, 2)], [(1, 3)]Null rotations (3.15) and rotations (3.16) [(11, 2)]Full Lorentz group [(111, 1)], R a b =0Here (x, y, z, u) is an orthonormal tetrad, and (x, y, k, l) a real (or half-)null tetrad. In each case the associated orthonormal tetrad is called theRicci principal tetrad.The Ricci tensor types are called degenerate when there is more thanone elementary divisor with the same eigenvalue; in the Segre notationthese degeneracies are indicated by round brackets (in Table 5.1).If the Ricci tensor is non-degenerate and the elementary divisors aresimple, the type is said to be algebraically general. Otherwise it is calledalgebraically special. These ideas are analogous to those for Petrov types(Chapter 4). As we see in the next section, the physically most importanttypes are algebraically special.The Ricci principal tetrads of the non-degenerate types (where theeigenvalues of different elementary divisors are distinct) are uniquely determined(cp. §4.2), but in other cases some freedom is allowed. We listthe possibilities in Table 5.2.Ref.: For other approaches to the classification of symmetric tensors seeCrade and Hall (1982), Penrose and Rindler (1986) and Chapter 9.5.2 The energy-momentum tensorThe Einstein field equations (1.1)R ab − 1 2 Rg ab +Λg ab = κ 0 T ab (5.4)(κ 0 being Einstein’s gravitational constant and Λ the cosmological constant)connect the Ricci tensor R ab with the energy-momentum tensor