Petrov, Petrov-Pirani Classification of Spacetimes  

In General > s.a. solutions of general relativity; types of spacetimes.
* Idea: A classification scheme for algebraically special spacetimes, based on properties of the Weyl tensor, or more precisely on the structure of the degenerate spinors of the Weyl spinor ψABCD = oA ιB αC βD in 4-dimensional Lorentzian manifolds.
* Applications: Studies of gravitational radiation.
* Types: Depending on the number of distinct principal null directions,
- I or {1111}, four distinct principal null directions.
- II, one doubly degenerate principal null direction, or a geodesic shear-free null congruence.
- D or {2 2}, two doubly degenerate principal null directions.
- III, one triply degenerate principal null direction.
- N, one four-fold degenerate principal null direction.

Examples > s.a. Robinson-Trautman Spacetimes.
* Type D and black holes: All black-hole solutions are of Petrov type D, as are radially homothetic spacetimes (spherical collapse); The Kerr metric is the only asymptotically flat, stationary, axially symmetric, type-D solution of the vacuum Einstein equation; The most important solutions for numerical relativity are of type I, I-D, and II, and pp-waves are of type N.
@ References: Wu & Bai PRD(08) [Kerr spacetime].

References
@ General: Petrov re GRG(00); Petrov; Kalotas & Eliezer AJP(83)jan [elementary]; Penrose & Rindler 86; Yoon ht/92 [1+1 view]; in Stephani et al 03; Zakhary et al GRG(03) [new algorithm]; & Hauser, & Held; Cherubini et al CQG(05)gq/04 [speciality index and BKL map]; Acevedo & López-Bonilla GRG(05); Pravda JPCS(06)gq/05 [rev].
@ Type I: Edgar IJTP(79); McIntosh & Arianrhod CQG(90) [degeneracy]; Ferrando & Sáez CQG(03)gq, JMP(06)gq/03 [aligned Papapetrou fields]; Wylleman CQG(08)-a0801 [generically asymmetric rotating dust].
@ Type D, classification: Collins et al CQG(91) [vacuum, Karlhede classification]; Collins & d'Inverno CQG(93) [non-vacuum, Karlhede classification]; Ferrando & Sáez CQG(98) [characterization], JMP(04)gq/02 [classification]; Edgar et al CQG(09)-a0812 [vacuum, identities and classification]; Coley & Hervik CQG(09) [vacuum].
@ Type D, other: Penrose GRG(95) [behavior of complex component of Weyl curvature]; Hodgkinson JMP(01) [embedding class two]; Cherubini et al CQG(04)gq [perturbations]; Pravda et al CQG(07)-a0704 [higher dimensions]; De Groote & Van den Bergh a0812 [pure radiation]; Wylleman JPCS(10)-a1204 [perfect fluid, with constant zero-order Riemann invariants].
@ Type III: Hacyan & Plebański IJTP(75) [electrovac]; Nesterov G&C(03) [electrovac with cosmological constant]; Fuster & van Holten PRD(05) [Einstein-Yang-Mills]; Ortaggio et al PRD(10) [Einstein spacetimes, types III and N]; > s.a. huygens principle; Kundt Waves.
@ Type N: Nesterov G&C(03) [electrovac with cosmological constant]; Edgar & Machado CQG(05) [pure radiation]; > lorentzian geometry [classification].
@ Results: Wagh & Govinder gq/02 [radially homothetic is type D].
@ Related topics: Owen PRD(10)-a1004 [degeneracy measures, "approximate" Petrov classes, numerical black-hole mergers]; Gath et al a1506 [and holographic reconstruction of spacetime]; > s.a. Catastrophe Theory; Goldberg-Sachs Theorem.

Other Classifications and Generalizations > s.a. Segre Classification.
@ Riemannian: Hacyan PLA(79); Karlhede CQG(86).
@ Approximate: Ellis & McCarthy AP(87).
@ General tensors and dimensions: De Smet CQG(02) [5D]; Coley et al CQG(04), CQG(08) [higher-dimensional]; Milson et al IJGMP(05)gq/04; Coley & Pelavas GRG(06); Reall a1105-ch [in higher dimensions]; Batista a1204 [4D, all signatures]; Reall et al a1211 [5D algebraically-special vacuum spacetimes]; Batista a1311-PhD [all dimensionalities and signatures].


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