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].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16
mar
2012