Petrov, Petrov-Pirani Classification of Spacetimes| 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 II: Aksteiner et al a2101 [geometry].
@ 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 GRG(13)-a1204 [4D, all signatures];
Reall et al CQG(13)-a1211 [5D algebraically-special vacuum spacetimes];
Batista a1311-PhD [all dimensionalities and signatures].
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 23 jan 2021