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122 9 Invariants and the characterization of geometriesΨ 0 , Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 ✏ ✏✏✏ ✏❄ ❅ ❅I 3 =27J❅2 ❅ ❅❅ ❅✲ yes ❅❅I=J=0 ❅❅❅ ❅yes✲❄TypeIno❄no ❅❅K=N=0 ❅❅❅ ❅❄ noTypeIIyes❄TypeD ❅❅K=L=0❅❅❅ ❅no❄TypeIIIyes❄TypeNFig. 9.1. Flow diagram for determining the Petrov type by the classicalmethodand the algorithm proceeds as before. For Ψ 0 =Ψ 4 = 0, the multiplicitiesof the roots of (9.5) are very simple to determine.If both the invariants I and J vanish, then at least three of the principalnull directions coincide. If not, then from the diagram and the definitions(9.6) it follows that gravitational fields with a repeated principal nulldirection k (Ψ 0 =Ψ 1 = 0) are type D if and only if the remaining tetradcomponents of the Weyl tensor satisfy the condition3Ψ 2 Ψ 4 =2Ψ3 2 . (9.7)An equivalent method for determining the Petrov type is based on theeigenvalue equation (4.4). One can use the invariant criteria for the matrixQ which are listed in Table 4.1, Q being calculated with respect to anarbitrary orthonormal basis {E a }.These algorithms are far from optimal computationally. The first improvementswere those of Fitch (1971), which used ideas similar to thoselater (re-)introduced and extended by others (Hon 1975, Åman et al. 1984,