In General > s.a. differential
geometry;
metric types and matching; [riemann
tensor].
* Idea: A Lorentzian
metric is one with signature (–, +, ..., +) [Remark: 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.
@ 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 gq/07 [4D
curvature homogeneous]; > 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].
> 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]; > s.a. petrov
types.
Other Structures and Related Topics > s.a.
affine connection;
fluid;
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
gq/03-in
[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 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 a0712-in
[rev]; > s.a.
black hole analogs, de
sitter
space, finsler geometry, frw
models [condensed matter
analogs], Lorentz-Fitzgerald Contraction, optics [optical
geometry], sound [acoustic
geometry].
Space of Lorentzian Geometries > s.a. distance;
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 a0804-CAG [Alexandrov curvature bounds].
@ Limits of spacetimes: Geroch CMP(69);
Paiva et al CQG(93)gq.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
13 jul 2008