Types of Metrics| Types of Metrics |
Metrics with Symmetries > s.a. axisymmetry; general relativity solutions with symmetries; killing fields; spherical symmetry.
Degenerate Metrics > s.a. extended signatures;
geometrodynamics; models in canonical general relativity;
quantum gravity and geometry; spin structure.
* Example:
[@ Yoneda et al PRD(97)gq]
A metric which is flat everywhere but degenerate at x = t = 0 is
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) \(\mapsto\) (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];
Deser CQG(06) [reason for invertibility].
@ Geometry: Borde et al CQG(99)gq [causal continuity];
Stoica IJGMP(11)-a1105;
> s.a. causal structure; newton-cartan
theory [connections]; singularities.
@ And matter: Cabral & Rivelles CQG(00)ht/99 [particle dynamics];
> s.a. lagrangian systems; quantum fields
in curved backgrounds; unified theories.
Constant-Curvature Manifolds
> s.a. riemann tensor; sphere.
* Metric: For any signature
and dimension d > 2, in stereographic coordinates it is
ds2 = (1 + \(1\over4\)kr2)−2 ηij dxi dxj , with r2:= ηij xi xj , ηij = diag(±1, ±1, ..., ±1) .
* Curvature: The Riemann tensor
is of the form Rabcd
= \(1\over n(n-1)\)R (gac
gdb
− gad
gcb),
so Rab
= \(1\over n\)Rgab;
in 4D Cabcd = 0,
and Gab
= −\(1\over4\)Rgab.
* Examples: The n-sphere
(R > 0); For the Lorentzian case, Minkowski space (R =
0), de Sitter (R > 0), anti-de Sitter (R < 0).
@ References: Wolf 87;
Dryuma TMP(06)m.DG/05 [3D].
Other Special Types
> s.a. 2D manifolds; 3D manifolds;
lie groups; types of lorentzian
geometries; weyl tensor.
* Zoll metric: A Riemannian
gab on a compact M,
all of whose geodesics are simply periodic, with period 2π; 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).
* Neutral metric: A pseudo-Riemannian
metric of signature (n, n).
@ 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. riemannian geometry;
types of lorentzian geometries.
@ 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 \(\mathbb C\)n];
Anderson & Herzlich JGP(08) [with prescribed Ricci curvature];
Kalafat et al JGP(13) [self-dual, on non-simply-connected 4-manifolds];
Santalla et al NJP(15)-a1407 [random geometry];
Georgiou & Guilfoyle a1605 [neutral 4-manifolds with null boundary];
> s.a. riemann tensor [in terms of curvature invariants].
@ Singular metrics, distributional curvature:
Kunzinger & Steinbauer AAM(02)m.FA/01 [Colombeau];
in Dray in(17)-a1701;
> s.a. coordinates [discontinuous transformations];
gravitational-wave solutions [impulsive]; hamiltonian
and lagrangian systems.
Information Geometry > s.a. entropy;
information; 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;
Naudts & Anthonis LNCS-a1506 [extension to non-statistical systems];
Goddard a1802 [treatise].
@ For probability distributions: Bengtsson AIP(06)qp/05 [Fisher-Rao metric];
> s.a. probabilities in physics; riemannian
geometry / actions for gravity.
@ And phase transitions: Janke et al PhyA(04)cm;
Kumar et al PRE(12)-a1210 [geodesics, classical and quantum second-order phase transitions];
Maity et al PRE(15)-a1503 [and the renormalization group].
@ Quantum:
Álvarez-Jiménez & Vergara IJQI(19)-a1904 [from generating functions];
Erdmenger et al a2001 [in quantum field theory, examples];
> s.a. coherent states; origin of quantum theory.
For States of a Physical System
> s.a. mixed states; riemannian geometry;
thermodynamics [geometry of state space]; types
of distances.
* Fubini-Study metric: A complex tensor
whose real part is the Riemannian metric that measures the 'quantum distance', and
whose imaginary part is the Berry curvature; > s.a. Encyclopedia of Mathematics
page;
Wikipedia page.
* Cayley-Fubini-Study metric:
For a small change dψ from a pure quantum state ψ,
ds2 = \([\langle{\rm d}\psi \mid {\rm d}\psi\rangle - \langle{\rm d}\psi \mid \psi\rangle\,\langle\psi \mid {\rm d}\psi\rangle] \,/\, \langle\psi \mid \psi\rangle\) ;
It can be considered as the infinitesimal version of the distinguishability distance
d(ψ1,ψ2)
:= |1 − \(\langle\psi_1|\psi_2\rangle\) |2.
@ Fubini-Study metric:
Anandan PLA(90) [physical meaning];
Cheng a1012 [pedagogical];
Álvarez-Jiménez & Vergara a1605 [and gauge invariance];
> s.a. types of distances; phase space.
@ Other metrics on the space of quantum states: Man'ko et al JPA(17)-a1612 [from relative entropy].
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