Types of Lorentzian Geometries  

In General > s.a. 2D, 3D and 4D geometries; types of metrics [including Zollfrei] and spacetimes [including conformally flat].
* Approach: Use a frame in which gμνs are constant, and use the Riemann tensor and its derivatives to classify.
* Equivalence problem: Solved by E Cartan, in general requires comparison of up to 10th derivatives of Rabcds; However, depending on the Petrov type of the metrics, one may need a smaller number q of derivatives (& Karlhede); For types I, II and III, q ≤ 5; For general type D, q ≤ 6; For vacuum type D, q ≤ 3; For general types N and O, q ≤ 7; For vacuum type N, q ≤ 6.
* Time orientability: If a Lorentzian manifold is not time-orientable, it admits a 2-fold time-orientable covering [@ Markus AM(55)]; The necessary and sufficient condition for the existence of a time-orientable metric on M is χ(M) = 0 (as for existence of a Lorentzian structure).
@ General references: Karlhede GRG(80), GRG(80); Karlhede & Lindström GRG(83); Kreinovich IJTP(91) [NP-hard]; Koutras CQG(92) [q = 4 example]; Siklos CQG(96) [parameters]; Skea CQG(00) [q = 5]; Schmidt gq/01-GR14 [indistinguishable metrics]; Karlhede GRG(06); McNutt et al JMP(17)-a1704 [classification in 5D]; Mergulão & Batista RBEF(20)-a2007 [Cartan-Karlhede algorithm, pedagogical].
@ Comparing metrics: Aguirregabiria et al GRG(01)gq; Llosa & Soler CQG(05)gq/04 [as deformations of constant curvature]; Hall & Lonie JGP(11) [projectively related 4D Lorentzian manifolds and holonomy]; > s.a. riemann tensor [classification].
@ And scalar curvature invariants: Coley et al CQG(04)gq [vanishing invariants]; Hervik & Coley CQG(10)-a1002 [geometries characterised by scalar polynomial curvature invariants]; Coley et al CQG(10)-a1003; Hervik CQG(11)-a1109 [a spacetime not characterised by its invariants is of aligned type II]; Hervik et al JGP(15)-a1410 [metrics with identical polynomial curvature invariants]; > s.a. riemann tensor.
@ Special types of metrics: Haddow & Carot CQG(96) [warped products]; Milson & Pelavas CQG(08)-a0710 [type N]; Sousa et al CQG(08) [3D, solution of equivalence problem]; Coley et al CQG(09)-a0901 [4D, determined by curvature scalars]; Hall & Lonie Sigma(09)-a0906 [projectively equivalent manifolds]; Choi AJM-a1204 [3D, complete, flat, with free fundamental group]; Papadopoulos a1405 [geometries which can not be locally described using curvature scalars solely]; Lake a1912 [with a vanishing second Ricci invariant]; Nozawa & Tomoda CQG-a1910 [3D with 3 Kvfs]; > s.a. petrov types.
@ Geometries of bounded curvature: Klainerman & Rodnianski IM(05)m.AP/03 [vacuum]; Anderson JMP(03) [n+1 dimensions]; de Araujo Costa JGP(04) [bounded sectional curvature Einstein metrics]; Chen & LeFloch CMP(08)m.AP/06 [injectivity radius]; in Punzi et al AP(07)gq/06; Alexander & Bishop CAG-a0804 [Alexandrov curvature bounds]; LeFloch a0812 [injectivity radius]; > s.a. solutions of general relativity.
> Other types: see black-hole phenomenology [effective metric]; light; solutions of general relativity; types of spacetimes.
> Classification: see metrics [characterization]; riemannian geometry [invariants]; James Vickers' page.
> Emergence: see emergent gravity [including analogs of spacetime metrics]; lorentzian geometry.

Homogeneous and Other Symmetric Geometries > s.a. bianchi models; solutions of einstein's equation with symmetries.
* Symmetric spacetimes: Locally symmetric spacetimes are those with vanishing covariant derivatives of the Riemann tensor, Rabcd;m = 0; Second-order symmetric ones have vanishing second covariant derivatives of the Riemann tensor, Rabcd;mn = 0; Contrary to the Riemannian geometry case, there exists proper second-order symmetric Lorentzian spacetimes which are not locally symmetric.
* Semi-symmetric spaces: Conformally semi-symmetric, ∇[mn] Cabcd = 0; Ricci semi-symmetric, ∇[mn] Rab = 0.
@ Homogeneous, constant curvature: Mess pr(90)-a0706 [constant curvature]; Coley et al CQG(06)gq/05; Gilkey 07; Milson & Pelavas IJGMP(09)gq/07 [4D, curvature homogeneous]; Coley et al CQG(09)-a0904 [4D, constant curvature invariants]; Brozos-Vázquez et al DG&A(09) [as realizations of curvature models]; Bowers a1203 [3D, and the Petrov classification]; > s.a. 3D geometries.
@ Symmetric spacetimes: Senovilla CQG(08), Blanco et al JPCS(10)-a1001, JEMS-a1101 [second-order]; Åman JPCS(11)-a1006 [semi-symmetric spaces].

Generalized Lorentzian Geometries > s.a. discrete spacetime; spacetime singularities; types of metrics [degenerate].
@ General references: Mayerhofer PRSE(08)mp/06 [Colombeau, Lorentzian]; Hohmann PRD(13)-a1304 [Cartan geometry on observer space and Finsler spacetimes]; Alexander et al a1909 [Lorentzian length spaces].
@ Continuous, low-regularity metrics: Sorkin & Woolgar CQG(96)gq/95 [causal order]; Kunzinger et al GRG(14)-a1310 [C1,1 metrics, causality]; Steinbauer CQG(14) [Lipschitz metrics have C1 geodesics]; > s.a. causal structures.
@ Disclinations: Holz CQG(88) [2+1 dimensions]; Larsen JGP(92); Li CQG(01); > s.a. defects.
@ Conical singularities: Clarke et al CQG(96)gq [cosmic strings]; Dahia & Romero MPLA(99)gq/98 [curvature]; Fursaev et al PRD(13)-a1306 [squashed cones].
@ Singular metrics, distributional curvature: in Penrose in(72); Taub pr(80); Vickers & Wilson CQG(99)gq/98; Garfinkle CQG(99)gq; Steinbauer mp/01-proc [impulsive gravitational waves]; Pantoja & Rago IJMPD(02)gq/00; LeFloch & Mardare PM(07)-a0712; Traschen CQG(09)-a0809 [codimension-2]; Steinbauer & Vickers CQG(09)-a0811 [lorentzian, distributional in the sense of Colombeau]; Steinbauer NSJM-a0812-conf; Gravanis & Willison JMP(09)-a0901; Stoica IJGMP(14)-a1105, AP(14)-a1205 [dimensional reduction at singularities]; Vickers JGP(12); Stoica PhD(13)-a1301; > s.a. general-relativity solutions with matter; gravitational-wave solutions [impulsive]; regge calculus [piecewise flat]; types of singularities.
@ Observer-dependent geometries: Hohmann a1403-in [rev].
@ Type-changing metrics: Aguirre-Dabán & Lafuente-López DG&A(06); Aguirre et al JGP(07) [transverse Riemann-Lorentz manifolds]; > s.a. modified general relativity [signature change].
@ Statistical spacetime: Calmet & Calmet TCS(04)mp [metric].

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