Appendix A The Petrov Classification The Weyl tensor Cabcd, due to its symmetries, can be represented as a 6 x 6 matrix, denoted by CAB. One can then classify the Weyl tensor by using the following simple approach. The Weyl symmetry c~bc = 0 allows us to write cAB as where A and B are 3 x 3 matrices, such that A is symmetric and both A and B are trace free. Above matrix can also be described by the complex 3 x 3 matrix D = A+ i B for which there are the following three possible Jordan forms: Petrov Type I Q 0 0) ( 0 f3 0 0 0 1 Petrov Type II Q 1 0 ) ( 0 Q 0 0 0 -2a Petrov Type III (H!) where the trace free condition on A and B ( and hence on D) is used. Following are subcases of Petrov Type I and II: Petrov Type D Subcase of type I where a= {3. Petrov Type 0 Subcase of type I where a = f3 = 1 = 0 {:::::::} C = 0. Petrov Type N Subcase of type II where a = 0. 193 194 APPENDIX A. THE PETROV CLASSIFICATION An alternative and very useful version of the Petrov classification is due to the work of L. Bel [10], known as the Bel criteria. In this criteria, null eigen bivectors and their principal null directions play an important role. Define the complex self + dual Weyl tensor Cabcd by + * Cabcd= Cabcd + i Cabcd, where Cis* the dual of C. Let a null direction k satisfy the following k[eCa]bc[dkf] kb kc = 0. (A.1) Then the Bel criteria are as follows: (1) Cis Petrov type I if there are exactly 4 distinct null vectors (called its principal null directions) k satisfying (A.1). (2) C is Petrov type II if there are two coincident null directions k satisfying (A.1). (3) C is Petrov type III if there are three coincident null directions k satisfying . + (A.1). Also CIS of type III {::::::} Cabcd ka kc = 0. (4) Cis Petrov type N if there are all four coincident null directions k satisfying + (A.1). Also Cis of type N {::::::} Cabcd kd = 0. 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