Lorentzian Geometry| 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 IJMPD(16)-a1601
[orientability, use of tetrads as variables, etc];
Kulkarni a1911-BS [rev].
@ 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, including acoustic].
@ Related topics:
Bugajska JMP(89) [open spin 4-manifolds].
> Related topics:
see affine connection; fluids; jacobi
metric; lines [space of timelike lines]; types of distances
[Lorentzian length spaces].
> 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 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) .
@ 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];
Hintz & Uhlmann IMRN(18)-a1705
[reconstruction of Lorentzian manifolds from boundary light observation sets].
@ Related topics: Impera JGP(12) [Hessian and Laplacian comparison theorems];
Robinson a2104 [spinorial coordinates, complex metrics].
@ 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 p ∈ M ||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' | ||(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-proc,
CQG(03)gq/02 [causal equivalence];
Noldus CQG(04)gq/03,
CQG(04)gq/03 [distance];
Bombelli & Noldus CQG(04)gq;
Fletcher JMP(18)-a2005 [topology].
@ Limits of spacetimes: Geroch CMP(69);
Bampi & Cianci IJTP(80);
Paiva et al CQG(93)gq;
Sormani a1006-fs [convergence].
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 10 apr 2021