Types of Lorentzian Geometries| 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, ∇[m
∇n]
Cabcd
= 0; Ricci semi-symmetric, ∇[m
∇n]
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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