Geodesics  

In General > s.a. differential geometry [completeness]; group action [actions preserving geodesics].
$ Def: A geodesic is a curve in a manifold whose tangent vector X is parallel to itself along the curve, or

X X = f X ,   for some function f .

* Affine parametrization: It is always possible to parametrize a geodesic such that X X = 0; This is equivalent to

d2xm/dt2 + mpq (dxp/dt) (dxq/dt) = 0 .

* As extremal curves: If the connection is torsionless, geodesics extremize the distance between two points.
* Properties: The geodesics of a manifold depend on the metric, but are insensitive to its torsion, as can be seen from above; For an affinely parametrized geodesic, the norm of the tangent vector is constant along it.
@ General references: Busemann 55; in Arnold 89, app1 [concise]; Boccaletti et al GRG(05)gq [Beltrami method + Schwarzschild and Kerr].
@ Conformal geodesics: Friedrich & Schmidt PRS(87); Friedrich CMP(03)gq/02.
@ Quantum corrections: Dalvit & Mazzitelli gq/99-in, PRD(99)gq.

Geodesic Deviation and Focusing > s.a. regge calculus; torsion.
$ Def: If Xa:= xa/s is the connecting vector between nearby geodesics in a one-parameter family, and T a:= xa/t the tangent vector, the relative acceleration

aa:= D2Xa/Dt2 = Rabcd T b T c X d .

* Jacobi equation: The linearized equation for a particle (or a field theory) around an extremal of the action; Used for the linearized geodesic flow, it gives the geodesic deviation equation; A Jacobi field is a solution of the Jacobi equation, i.e., a linearized solution of the equations of motion.
@ General references: Barraco et al GRG(90); Roberts GRG(96)gq/99 [from action, quantization]; Kerner et al PRD(01)gq/00 [in Kaluza-Klein]; Colistete et al CQG(02) [higher-order, in Kerr].
@ Jacobi equation: Núñez-Yépez & Salas-Brito PLA(00)mp [variational principle]; Chicone & Mashhoon CQG(02)gq, Perlick GRG(08)-a0710 [generalized].
@ Geodesic focusing: in Hawking in(73) [simple intro]; O'Sullivan MA(74); Rosquist IJTP(83); Visser PRD(93) [Lorentzian spacetime]; Ehlers & Newman JMP(00)gq/99 [caustics].

Related Notions > see Congruence and Congruence Expansion; Totally Geodesic [mapping, submanifold].
* Maslov index: For a semi-Riemannian geodesic, a homological invariant which substitutes the notion of geometric index in Riemannian geometry.
@ Geodesic flow as dynamical system: Hedlund BAMS(39); > s.a. conserved quantities.
@ Other references: Rotman Top(07) [length of minimal geodesic net on a closed Riemannian manifold]; Bolsinov et al a0806 [Fubini theorem, for pseudo-Riemannian metrics].
> Space of geodesics: see symplectic structure.

Null Geodesics > s.a. coordinates [based on a null geodesic].
* Set of null geodesics: If (M, g) is a globally hyperbolic (d+1)-dimensional spacetime, this set is naturally a smooth (2d–1)-dimensional contact manifold.
* Sky: The sky of an event is the subset of defined by all null geodesics through that event, and is an embedded Legendrian submanifold of diffeomorphic to a Sd–1.
@ General references: Low JMP(89); Urbantke JMP(89); Frittelli & Newman gq/98-in [time of arrival].
@ Types of spacetimes: Frittelli et al CQG(98) [asymptotically flat, conjugate points]; Stuckey AJP(93), Cadez & Kostic PRD(05)gq/04 [in Schwarzschild spacetime]; Hall & Lonie JMP(08) [FRW spacetimes].
@ Properties of skies: Natário CQG(02)gq/01 [skies]; Natário & Tod PLMS(04)gq/02.

Types of Spaces > s.a. connections [non-commutative]; solutions with symmetries [static]; graph theory; regge calculus.
@ 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].
@ Riemannian: Berger 65; Bolsinov & Jovanovic in(04)mp/03 [integrability].
@ Lorentzian: Larsen JDG(96) [sic]; Candela & Sánchez m.DG/06 [rev].
@ Bianchi metrics: Nilsson et al GRG(00)gq/99 = GRG(00) [dynamical system]; Valent & Yahia CQG(07) [integrable flows]; > s.a. bianchi models.
@ Black hole spacetimes: Marck CQG(96)gq/95 [Schwarzschild-Kerr]; Leiva et al MPLA(09)-a0808 [Schwarzschild metrics in rainbow gravity]; Slany et al IJMPA(09) [Kerr-de Sitter]; > s.a. kerr spacetimes.
@ Other metrics: Moraes PLA(96) [dislocation]; Barrow & Levin PLA(97) [k = –1 FRW models]; Podolsky & Belán CQG(04) [Kundt spacetimes]; Herrera FPL(05) [quash-spherical]; D'Afonseca et al CQG(05)gq [Weyl-Bach ring solution]; Kiosak & Matveev CMP(09)-a0806 [geodesically rigid]; Brink PRD(08)-a0807 [stationary axisymmetric vacuum]; > s.a. FRW metrics; Lemaître-Tolman-Bondi, Lewis Metric, test-particle motion.

Physics Applications > s.a. de Sitter-Fokker Precession; jacobi dynamics; particle models; perfect fluids; Sachs-Wolfe Effect.
* Main application: Paths of test particles in curved spaces; > s.a. chaos, general relativistic chaos.
* Multiple imaging: The local geometry equivalent of "gravitational lensing," weaker than focusing.
@ References: Kuusk & Paal TTTU(92)-a0803 [geodesic multiplication, Akivis algebras]; Perlick CQG(96) [multiple imaging]; Adamo et al LRR(09)-a0906 [asymptotically shear-free null geodesic congruences].
> In gravitation and cosmology: see geometrodynamics; cosmological models in general relativity [dynamics as geodesic motion]; motion of gravitating bodies.

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