Types of Geodesics| Types of Geodesics |
In General
> see geodesics [quantum corrections]; projective structures.
* Homogeneous geodesics:
In any homogeneous (Riemannian or pseudo-Riemannian) manifold, there is
at least one homogeneous geodesic through each point.
* Compact Lorentzian manifolds:
All lightlike geodesics are periodic, while there are closed and non-closed
spacelike and timelike geodesics.
@ Homogeneous geodesics:
Dusek JGP(10).
@ Conformal geodesics:
Friedrich & Schmidt PRS(87);
Friedrich CMP(03)gq/02;
Tod JGP(12) [examples].
@ Riemannian manifolds: Berger 65;
Bolsinov & Jovanović in(04)mp/03 [integrability];
LaGatta & Wehr CMP(14)-a1206,
a1206 [random perturbations of Euclidean space].
@ Lorentzian manifolds:
Larsen JDG(96) [sic];
Candela & Sánchez in(08)m.DG/06 [rev];
Bolsinov et al JLMS(09)-a0806 [Fubini theorem, for pseudo-Riemannian metrics];
Del Barco et al JGP(14) [compact Lorentzian manifolds];
> s.a. finsler geometry.
@ 2D: Knieper & Weiss JDG(94) [S2, positive topological entropy];
Lévay JPA(00),
JPA(00) [negative curvature Riemann surfaces];
Rowland EJP(06) [simple];
Ying & Candès JCP(06) [computation];
Müller PLA(12)
[on closed surfaces, numerical, and chaotic regions].
@ 3D:
Bose et al CG(12) [3D polyhedral surfaces].
> Related topics: see Congruence
and Congruence Expansion; Totally Geodesic
[mapping, submanifold].
Null Geodesics > s.a. coordinates [based on a null geodesic];
quantum spacetime; spacetime subsets [and complex world-lines].
* Set of null geodesics:
If \((M,g)\) is a globally hyperbolic (d+1)-dimensional spacetime, this
set \(\cal N\) is naturally a smooth \((2d-1)\)-dimensional contact manifold.
* Sky: The sky of an event is the
subset of \(\cal N\) defined by all null geodesics through that event, and is an
embedded Legendrian submanifold of \(\cal N\) diffeomorphic to a S\(^{d-1}\).
@ General references: Low JMP(89);
Urbantke JMP(89);
Frittelli & Newman gq/98-fs [time of arrival].
@ Types of spacetimes: Frittelli et al CQG(98) [asymptotically flat, conjugate points];
Hall & Lonie JMP(08) [FLRW spacetimes];
Décanini et al PRD(10)-a1002 [circular, static spherically symmetric black holes];
Gibbons & Vyska CQG(12)-a1110 [spherically symmetric, and Weierstraß elliptic functions];
Bikwa et al MNRAS(12)-a1112 [FLRW spacetimes];
Kraniotis GRG(14)-a1401 [Kerr-Newman (-de Sitter) black holes];
Paganini et al a1611 [Kerr];
Charbulák & Stuchlík EPJC(17)-a1702 [Kerr-de Sitter];
Gal'tsov & Kobialko PRD(19)-a1901 [Kerr and Kerr-Newman].
@ Null geodesic congruences: Adamo et al LRR(09)-a0906,
LRR(12) [asymptotically shear-free];
Adamo & Newman CQG(11) [generalized good cut equation];
Newman GRG(20) [in Minkowski spacetime].
@ Properties of skies: Natário CQG(02)gq/01 [skies];
Natário & Tod PLMS(04)gq/02;
Bautista et al CQG(15)-a1411 [partial order on the space of skies, and the Malament-Hawking theorem].
Special Types of Spacetimes
> s.a. connections [non-commutative]; graph theory;
regge calculus; types of lorentzian metrics.
* Stationary axisymmetric spacetimes:
The motion of a test particle in a stationary axisymmetric gravitational field is generally
non-integrable unless, in addition to the energy and angular momentum about the symmetry axis,
an extra non-trivial constant of motion exists, as in the Kerr spacetime.
@ Stationary axisymmetric spacetimes: Brink PRD(08)-a0807 [vacuum];
Markakis MNRAS(14)-a1202 [constants of the motion];
Pineda et al a1409-proc;
> s.a. solutions with symmetries [static].
@ Other black-hole spacetimes: Marck CQG(96)gq/95 [Schwarzschild-Kerr];
Cardoso et al PRD(09)-a0812 [stability, Lyapunov exponents, and quasinormal modes];
Slany et al IJMPA(09) [Kerr-de Sitter];
Radosz et al MPLA(11) [black holes in Hořava-Lifshitz gravity];
Chakraborty & Chakraborty CJP(11)-a1109 [spherically symmetric, test-particle motion and light bending];
Yang & Wang A&A(14)-a1311 [Kerr-Newman spacetime, numerical code ynogkm for timelike geodesics];
Pradhan CQG(15)-a1402 [Kerr-Newman-Taub-Nut spacetime];
Hackmann a1506-proc,
Hackmann & Lämmerzahl AIP(14)-a1506 [Plebański-Demiański family of solutions];
Semerák & Suková in(15)-a1509 [around deformed black holes];
Salazar & Zannias PRD(17)-a1705 [Kerr-de Sitter];
Liu et al CQG(17)-a1706 [Kerr-Newman spacetime, neutral particles];
Eskin RVMP(19)-a1807 [Reissner-Nordström spacetime];
> s.a. spinning particles.
@ Bianchi metrics: Nilsson et al GRG(00)gq/99
= GRG(00) [dynamical system];
Valent & Yahia CQG(07) [integrable flows];
> s.a. bianchi models.
@ Cosmological metrics: Barrow & Levin PLA(97) [k = −1 FLRW models];
Podolský & Belán CQG(04) [Kundt spacetimes];
Pereira & Sampson GRG(12)-a1110 [de Sitter spacetime, generalized geodesics];
Sokołowski & Golda IJMPD(16)-a1602 [in AdS spacetime].
@ Other metrics: Moraes PLA(96) [dislocation];
Herrera FPL(05) [quash-spherical];
D'Afonseca et al CQG(05)gq [Weyl-Bach ring solution];
Kiosak & Matveev CMP(09)-a0806 [geodesically rigid];
Andrade & de Berredo-Peixoto G&C(13)-a1203 [spherically symmetric dislocation];
Lim PRD(14)-a1405 [vacuum C-metric];
Podolský et al CQG(15)-a1409 [non-expanding impulsive gravitational waves];
Bambhaniya et al PRD(19)-a1908 [with naked singularities].
@ Generalized settings: Ulhoa et al GRG(14)-a1312 [non-commutative spacetime];
Arrighi & Dowek a1507 [discrete spacetime];
> s.a. finsler geometry [Randers spaces]; geodesics.
Black-hole metrics:
see chaotic motion; particles
in kerr spacetimes; particles in schwarzschild spacetimes;
reissner-nordström spacetimes.
Other metrics: see C-metric;
FLRW metrics; gödel solution;
gowdy spacetime; Lemaître-Tolman-Bondi
Solution; Lewis Metric; multipole moments.
main page
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