Lorentzian Geometry  

In General > s.a. differential geometry; riemann tensor / metric types and matching; types of lorentzian geometries [including scalar invariants, generalizations].
* 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, \(\mathbb R\)).
* 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.
@ General references: Iliev JGP(00)gq/98 [relation with Riemannian geometry]; Müller & Sánchez JDMV-math/06; Chen 11 [submanifolds, δ-invariants and applications]; news ea(11)apr [visualization through tendex lines and vortex lines]; Gilkey et al IJGMP(13) [with boundary, universal curvature identities].
@ Global aspects: Nawarajan & Visser a1601 [physical considerations on orientability, the use of tetrads as variables, and other issues].
@ Emergent geometry: Wilczek PRL(98)ht [metrics from volumes and gauge symmetries]; Brown a0911 [metric as spacetime property or emergent field]; Mukohyama & Uzan PRD(13)-a1301, Kehayias et al PRD(14)-a1403 [Lorentzian signature as emergent from Riemannian one and classical fields]; Majid EPJwc(14)-a1401; Cirilo-Lombardo & Prudêncio IJGMP(14) [from supergeometries]; > s.a. emergent gravity [analog models of spacetime metrics].
@ Related topics: Bugajska JMP(89) [open spin 4-manifolds].
> Related topics: see affine connection; fluids; jacobi metric; lines [space of timelike lines].
> Online resources: see Technische Universität Berlin group [2012].

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 gε 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) dμ < |∇φ|2 dμ    (sufficient condition: || R ||L3/2 < some known c) .

@ General 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?]; Kim JMP(11) [covering spaces and homotopy classes of causal curves]; Impera JGP(12) [Hessian and Laplacian comparison theorems].
@ Extensions: Chruściel JDG(10) [conformal boundary extensions]; Low CQG(12) [maximal extensions and Zorn's lemma]; > s.a. spacetime boundaries and completions.
> Related topics: see distance; Extremal Surface; geodesics; Hypersurface; Osserman Manifolds; Pythagorean Theorem; Splitting Theorem; world function.

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 pM ||g g'||(p) < f(p), ..., ||∂kg – ∂kg'||(p) < f(p)} ,

where f : M → \(\mathbb R\) is continuous and strictly positive and the norms are calculated using some positive-definite (inverse) metric; 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; For example, on a non-compact M a sequence g(n) of metrics cannot converge unless there exists a compact subset in M such that for sufficiently large n all metrics coincide outside it [@ in Golubitsky & Guillemin 73].
* Wk compact-open topology: A neighborhood subbasis is

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

where U is an open set of compact closure in M and δ a positive constant.
* Partial order: For each AM, g <A g' iff for all pA, 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-proc, 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-fs [convergence].

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