Lorentzian Geometry  

In General > s.a. differential geometry; metric types and matching; [riemann tensor].
* Idea: A Lorentzian metric is one with signature (–, +, ..., +) [Remark: This gives normal signs to spatial components, while the opposite signature gives pm pm = m2 for a relativistic particle].
$ Lorentzian structure: A reduction of the bundle of frames F(M) to the Lorentz group, as a subgroup of GL(n,R).
* Conditions: The nasc for a manifold M is that M be non-compact, or that the Euler number (M) = 0.
* Time orientability: If a Lorentzian manifold is not time-orientable, it admits a 2-fold time-orientable covering [@ Markus AM(55)]; The nasc for existence of a time-orientable metric is still (M) = 0.
* Result: Given a metric g with scalar curvature R, there is another g' with R' = 0 iff for all C0infty,

–(1/8)R d < ||2 d    (sufficient condition: R L^{3/2} < some known c) .

@ References: Steenrod 51; O'Neill 83; Bugajska JMP(89) [open spin 4-manifolds]; Beem et al 96; Hall 04.

Specific Concepts and Results > s.a. causality; holonomy; minkowski space; spectral geometry; types of metrics and spacetimes.
* Flat deformation theorem: Given a semi-Riemannian analytic metric g on a manifold, there always exists a two-form F, a scalar function c, and an arbitrarily prescribed scalar constraint depending on the point x of the manifold and on F and c, say (c,F,x) = 0, such that the deformed metric = c g –  F2 is semi-Riemannian and flat; It implies that every (Lorentzian analytic) metric g can be written in the extended Kerr-Schild form ab:= a gab – 2 b k(a lb), where is flat and ka, la are two null covectors such that ka l a = –1.
@ Homogeneous, constant curvature: Mess pr(90)-a0706 [constant curvature]; Coley et al CQG(04)gq [vanishing invariants]; Coley et al CQG(06)gq/05; 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]; > s.a. 3D geometries.
@ Comparing metrics: Aguirregabiria et al GRG(01)gq; Llosa & Soler CQG(05)gq/04 [as deformations of constant curvature].
@ Related topics: Kim BAusMS(90); Tod CQG(92) [diagonalizability]; Pezzaglia & Adams gq/97-in [(–,+,+,+) vs (+,–,–,–)]; Gerhardt GRG(03)m.DG/02 [volume estimates]; Milson et al IJGMP(05)gq/04 [alignment]; Sánchez DG&A(06) [compact, causality]; Llosa & Carot CQG(09)-a0809 [flat deformation theorem and symmetries]; Kim JGP(09) [comparison of volumes between level hypersurfaces]; Pugliese et al a0910-in [deformations].
> Related topics: see Extremal Surface, Hypersurface; Osserman Manifolds; Pythagorean Theorem; Splitting Theorem; world function.

Isomorphism and Classification > s.a. 3D geometries; petrov classification; riemann tensor.
* Approach: Use frame in which gmn's 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 Rabcd's; 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.
@ 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-in [indistinguishable metrics]; Karlhede GRG(06).
@ 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]; > s.a. petrov types.

Other Structures and Related Topics > s.a. affine connection; fluids; jacobi metric; riemann tensor.
@ Metrics from volumes and gauge symmetries: Wilczek PRL(98)ht.
@ Relation with Riemannian: Iliev JGP(00)gq/98.
@ 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].
@ Effective metrics and analog gravity: Klidis & Spyrou CQG(00) [in astrophysics]; Barceló et al CQG(01)gq [field modes in non-trivial background]; De Lorenci & Klippert PRD(02) [electromagnetism in non-linear media]; Novello & Perez Bergliaffa AIP-gq/03 [flowing dielectric]; Barceló et al NJP(04)gq [causal structure], LRR(05)gq [rev]; Liberati et al PRL(06), CQG(06)gq/05 [quantum gravity analog from BECs]; Weinfurtner et al JPA-gq/05-in [analog of Klein-Gordon field in curved spacetime]; Milgrom PRD(06) [particles and mass]; Unruh & Schützhold ed-07; Weinfurtner et al PRD(07)gq [boson gas, signature change]; Visser & Weinfurtner PoS-a0712 [rev]; > s.a. black-hole analogs; de sitter space; finsler geometry; FRW models [condensed matter analogs]; gravity theories [emergent]; Lorentz-Fitzgerald Contraction; optics [optical geometry]; sound [acoustic geometry]; spacetime; types of quantum field theories.

Space of Lorentzian Geometries > s.a. distance between geometries; spacetime singularities; solutions of general relativity.
* Ck open topology: (aka Whitney fine or uniform convergence topology) A neighborhood basis for g is

Bf(g):= {g' | for all p M g – g'(p) < f(p), ..., kg – kg'(p) < f(p)} ,

where f : M → R is continuous and strictly positive. Intuitively, for C0, the light cones are close; For C1, the geodesic systems are close; For C2, the curvature tensors are also close; This is an extremely fine topology.
* Wk compact-open topology: A neighborhood subbasis is

B_{U,}(g):= {g' | (g – g')|U W^k < } ,

where U is an open set of compact closure in M and a positive constant.
* Partial order: For each A M, g <A g' iff for all p A, all non-spacelike vectors wrt g are non-spacelike wrt g' (the light cones of g are narrower).
@ General references: Geroch JMP(70), in(70); Hawking GRG(71); in Hawking & Ellis 73; Lerner CMP(73); in Beem et al 96.
@ Structures: Beem 81 [Lorentzian distance function]; Bombelli JMP(00)gq [pseudodistance]; Noldus CQG(02) [topology]; García-Parrado & Senovilla mp/02-in, CQG(03)gq/02 [causal equivalence]; Noldus CQG(04)gq/03, CQG(04)gq/03 [distance]; Bombelli & Noldus CQG(04)gq.
@ Bounded curvature: Klainerman & Rodnianski 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].
@ Limits of spacetimes: Geroch CMP(69); Paiva et al CQG(93)gq.

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