Types
of Metrics| Types
of Metrics |
Metrics with Symmetries > s.a. axisymmetry; bianchi models; FRW; general relativity solutions with symmetries; spherical.
Degenerate Metrics > s.a. causal
structure; geometrodynamics; models
in canonical general relativity; singularities;
spin structure.
* Example: [@ Yoneda
et al PRD(97)gq] Metric which is flat everywhere
but degenerate at x = t = 0,
ds2 = –[1 – (f'(t) h(x))2] dt2 + [–2 f'(t) h(x) (1–f(t) h'(x))] dt dx + [1 – f(t) h'(x)]2 dx2
((t, x) 
(t, x–fh));
E.g., f =
exp{–t2}, h = x exp{–x2},
–1 < f'(t), h(x) < 1 & –1 < f(t),
h'(x) 
1
(= 1 at 0).
@ General references: Kreisel et al AdP(63);
Crampin PCPS(68); D'Auria & Regge NPB(82)
[instanton]; Koshti & Dadhich
CQG(89);
Bengtsson CQG(91);
Varadarajan CQG(91);
Percacci in(92) [quantum field theory approach]; Gratus & Tucker
JMP(96)gq [2D];
Dray IJMPD(97)gq [tensor
distributions]; Baez CMP(98)
[from 2D BF-theory]; Borde et al CQG(99)gq [causal
continuity]; Deser CQG(06)
[reason for invertibility].
@ And matter: Cabral & Rivelles CQG(00)ht/99 [particle
dynamics]; > s.a.
lagrangian systems; quantum
fields
in curved backgrounds; unified
theories.
Non-Smooth / Generalized > s.a. [riemann
curvature];
hamiltonian and lagrangian
systems;
regge calculus; singularities.
@ C0 metrics: Sorkin & Woolgar
CQG(96)gq/95 [causal
order].
@ Singular and distributional metrics: Holz CQG(88)
[2+1 disclinations];
Larsen JGP(92);
Li CQG(01)
[disclination]; Steinbauer mp/01-in
[impulsive gravitational waves]; Kunzinger & Steinbauer
AAM(02)m.FA/01 [Colombeau];
Mayerhofer mp/06-PhD
[Colombeau, Lorentzian]; LeFloch & Mardare PM(07)-a0712 [distributional curvature,
Lorentzian].
@ On statistical spacetime: Calmet & Calmet TCS(04)mp.
Constant Curvature Manifolds > s.a. sphere.
* Metric: For any signature
and dimension d > 2, in stereographic
coordinates
it is
ds2 = (1 + 
kr2)–2 
ij dxi dxj , with r2:= ij xixj , ij =
diag(
1, 1,
..., 1) .
* Curvature: Rabcd =
(1/n(n–1)) R (gac gdb – gad gcb),
so Rab = (1/n) Rgab;
in 4D Cabcd = 0, and Gab = – Rgab.
* Examples: Minkowski
space (R = 0), de Sitter (R > 0), anti-de Sitter
(R < 0).
@ References: Wolf 87; Dryuma m.DG/05 [3D].
Other Special Types > s.a. 2D
manifolds; 3D manifolds; lie
groups; st
singularities; weyl
tensor.
* Zoll metric: A Riemannian
gab on a compact M, all of whose geodesics are simply periodic,
with period 2
; For example,
the standard
metric on the 2-sphere.
* Zollfrei metric: A Lorentzian
gab on a compact M,
all of whose null geodesics are periodic (a conformally invariant property).
@ Vanishing curvature invariants: Pravda
et
al CQG(02)gq;
Coley PRL(02)ht [and
string theory].
@ Finiteness results: Grove et al BAMS(89).
@ Bounded curvature: Cheeger & Colding JDG(97)
[R bounded
below]; > s.a. lorentzian and riemannian
geometry,
solutions of general relativity.
@ Spacetime metrics: Bona & Coll JMP(94)
[3D, with isometries]; Bueken & Vanhecke CQG(97)
[curvature homogenous]; Gover & Nurowski m.DG/04 [conformally
Einstein]; > s.a. black hole phenomenology [effective
metric], lorentzian geometry [including analogs], light, solutions
of general relativity, types
of spacetimes.
@ Zoll, zollfrei: in Guillemin 89; Nakata JGP(07)
[singular self-dual].
@
Other types: Win gq/96 [diagonal,
efficient
calculation]; Shi & Tam CMP(04)
[quasispherical]; Cuccu & Loi JGP(07)
[balanced metrics on Cn];
Anderson & Herzlich JGP(08)
[with prescribed Ricci curvature].
Information Geometry > s.a. entropy;
information;
probabilities in physics; solutions
of gauge theories.
* Idea: The introduction
of a metric on the space of parameters for
models,
e.g., in statistical mechanics.
* For probability distributions:
If pi is
the probability for the i-th
event, a natural choice is ds2 = 
i
dpi2/pi .
@ General references: Streater in(97); Amari & Nagaoka 00; Janke et
al PhyA(04)cm [and
phase transitions].
@ For probability distributions: Bengtsson qp/05-in
[Fisher-Rao]; > s.a. riemannian geometry.
For States of a Physical System > s.a. distances;
mixed states;
riemannian geometry; thermodynamic
systems.
* Cayley-Fubini-Study metric:
For a small change d
from
a pure quantum state ,
ds2 = [
d|
d
– d| |d]
/ |
;
Can be considered as the infinitesimal version of the distinguishability distance
d(1,2)
:= |1 – 1|2 |2.
@ Fubini-Study metric: Anandan PLA(90)
[physical meaning]; > s.a. types of distances, phase
space.
Generalized Metrics
@ Distributional curvature: Taub pr(80); Clarke et al CQG(96)gq [cosmic
strings]; Vickers & Wilson CQG(99)gq/98;
Garfinkle CQG(99)gq;
Pantoja & Rago IJMPD(02)gq/00.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008