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5.2 The energy-momentum tensor 61T ab . The Bianchi identities (2.81) imply the important relationAs well as vacuum fields (empty spaces)κ 0 T ab ;b =(R ab − 1 2 Rgab ) ;b =0. (5.5)R ab =0, (5.6)with zero T ab and Λ, we shall consider solutions of the field equations(5.4) for the following physically relevant energy-momentum tensors:(i) electromagnetic field (Maxwell field):T ab = F ac F b c − 1 4 g abF cd F cd = 1 2 (F acF b c + ˜F ac ˜F b c )= 1 2 F ∗ca F ∗ bc,F ∗ ab ≡ F ab +i˜F ab , ˜F ab ≡ 1 2 ε abcdF cd , F ∗ab ;b =0,(ii) pure radiation field (null dust):(iii) perfect fluid:(5.7)T ab =Φ 2 k a k b , k a k a =0, (5.8)T ab =(µ + p)u a u b + pg ab , u a u a = −1. (5.9)In the perfect fluid case we normally assume µ + p ̸= 0, µ > 0. Inthe particular case where T ab = 0 and Λ ̸= 0, or where T ab is of perfectfluid type (5.9) but with µ + p = 0, we shall say the Ricci tensor isof Λ-term type. Thus the perfect fluid solutions formally include Λ-termcases, the Einstein spaces R ab =Λg ab . They also include the combinationof a perfect fluid and a Λ-term. The latter can be incorporated in thefluid quantities by substituting (p − Λ/κ 0 ) for p and (µ +Λ/κ 0 ) for µ;of course this substitution may violate the condition µ>0. Note thatno invariant direction for u a is determined by a Λ-term; the kinematicquantities defined in §6.2 have no invariant meaning in this case. In theother non-vacuum cases the cosmological constant Λ is usually set equalto zero; occasionally solutions including Λ are listed.In general we do not consider superpositions of these energy-momentumtensors.By virtue of the field equations (5.4), T ab has the same algebraic typeas R ab . We shall now determine these types for the energy-momentumtensors (5.7)–(5.9).(i) The complex self-dual electromagnetic field tensor F ∗ ab can be expandedin terms of the basis (U, V , W ) (see §3.4) as12 F ∗ ab =Φ 0 U ab +Φ 1 W ab +Φ 2 V ab , (5.10)