Lorentzian Geometry  

In General > s.a. differential geometry; metric types and matching; types of lorentzian geometries [including scalar invariants]; [riemann tensor].
* Idea: A Lorentzian metric is one with signature (–, +, ..., +).
* Remark: This signature convention gives normal signs to spatial components, while the opposite ones 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 necessary and sufficient condition for the existence of a Lorentzian structure on a manifold M is that M be non-compact, or that the Euler number χ(M) = 0.
@ Books: Steenrod 51; O'Neill 83; Beem et al 96; Hall 04.
@ Related topics: Bugajska JMP(89) [open spin 4-manifolds]; Brown a0911 [metric as spacetime property or emergent field].

Specific Concepts and Results > s.a. causality; holonomy; minkowski space; spectral geometry; simplex (Lorentzian geometry case).
* 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 Ψ(c, F, x) = 0, such that the deformed metric η = c g – ε F2 is semi-Riemannian and flat; It implies that every (Lorentzian analytic) metric g can be written in the extended Kerr-Schild form ηab:= a gab – 2 b k(a lb), where η is flat and ka, la are two null covectors such that ka l a = –1.
* Result: Given a metric g with scalar curvature R, there is another g' with R' = 0 iff for all φ ∈ C0∞,

–(1/8) ∫ Rφ dμ < ∫ |∇φ|2 dμ    (sufficient condition: || R ||L3/2 < some known c) .

@ References: Kim BAusMS(90); Tod CQG(92) [diagonalizability]; Pezzaglia & Adams gq/97-conf [(–,+,+,+) 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), JGP(11) [volume comparison between hypersurfaces]; Pugliese et al a0910-proc [deformations]; de Siqueira a1006 [every n-dimensional pseudo-Riemannian manifold is conformal to one of constant curvature?]; Impera JGP(12) [Hessian and Laplacian comparison theorems].
> Related topics: see Extremal Surface; Hypersurface; Osserman Manifolds; Pythagorean Theorem; Splitting Theorem; world function.

Other Structures and Related Topics > s.a. affine connection; fluids; jacobi metric; lines [space of timelike lines]; riemann tensor.
@ General references: Iliev JGP(00)gq/98 [relation with Riemannian geometry]; Chen 11 [submanifolds, δ-Invariants and applications]; news ea(11)apr [visualization through tendex lines and vortex lines].
@ Metrics from volumes and gauge symmetries: Wilczek PRL(98)ht.
@ 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(03)gq [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(06)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]; Visser & Molina-Paris NJP(10)-a1001 [rev]; Cacciatori et al NJP(10) [refractive index perturbations]; Thompson & Frauendiener PRD(10)-a1010 [dielectric, general metrics]; Barceló et al LRR(11); Smolyaninov & Hung JOSA-a1104 [modeling the flow of time]; Chaline et al a1203-ln [surface waves and dispersive horizons]; > s.a. black-hole analogs; de sitter space; emergent gravity; finsler geometry; FRW models [condensed matter analogs]; Lorentz-Fitzgerald Contraction; optics [optical geometry]; quantum field theory effects in curved spacetimes; sound [acoustic geometry]; spacetime; types of quantum field theories.
@ Extensions: Chruściel JDG(10) [conformal boundary extensions]; > s.a. spacetime boundaries and completions.

Space of Lorentzian Geometries > s.a. distance between geometries; spacetime singularities; solutions of general relativity.
* Ck open topology: (a.k.a. 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

BU, δ(g):= {g' | ||(g – g')|U ||Wk < δ} ,

where U is an open set of compact closure in M and δ a positive constant.
* Partial order: For each A ⊂ M, g <A g' iff for all p ∈ A, all non-spacelike vectors with respect to g are non-spacelike with respect to 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)-a1104 [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.
@ Limits of spacetimes: Geroch CMP(69); Bampi & Cianci IJTP(80); Paiva et al CQG(93)gq; Sormani a1006-in [convergence].

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