In General > s.a. differential
geometry [completeness]; group
action [preserving geodesics]; types of geodesics [null and other types, special types of spaces].
$ 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 + Γmpq (dxp/dt) (dxq/dt) = 0 .
* As extremal curves: If the connection is torsionless, geodesics
extremize the distance between two points; > s.a. lagrangian dynamics.
* 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.
* Maslov index: For a semi-Riemannian geodesic, a homological invariant which substitutes the notion of geometric index in Riemannian geometry.
@ General references: Busemann 55; in Arnold 89, app1 [concise]; Boccaletti et al GRG(05)gq [Beltrami method + Schwarzschild and Kerr spacetimes].
@ Geodesic circles: Matsyuk a1407-proc [2D, variational description].
@ Other related topics: Rotman Top(07) [length of minimal geodesic net on a closed Riemannian manifold]; Ehrlich et al JMP(09) [non-isolated focal and conjugate points]; Matveev JGP(12)-a1101 [reconstructing a metric from its unparametrized geodesics].
> Space of geodesics: see lines [topology]; symplectic structure.
Geodesic Deviation and Focusing > s.a. Congruence of Lines; regge
calculus; spacetime [interpretation]; torsion.
* Idea: Geodesic deviation is the phenomenon by which the distance between initially parallel geodesics varies, and gives rise to tidal forces.
$ 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 geodesic deviation equation states that the relative acceleration between the geodesics is
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]; Colistete et al CQG(02) [higher-order, in Kerr spacetime]; Koekoek & van Holten PRD(11) [and analytic approximations to geodesics]; Philipp et al a1604 [range of applicability].
@ Jacobi equation: Núñez-Yépez & Salas-Brito PLA(00)mp [variational principle]; Chicone & Mashhoon CQG(02)gq, Perlick GRG(08)-a0710 [generalized]; Philipp et al MfA(15)-a1508-conf [in Schwarzschild spacetime].
@ In f(R) gravity: Guarnizo et al GRG(11)-a1010; de la Cruz-Dombriz et al PRD(14)-a1312; Guarnizo et al GRG(15)-a1402.
@ Other modified theories: Kerner et al PRD(01)gq/00 [Kaluza-Klein theory]; Baffou et al a1509 [f(R,T) gravity].
@ 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]; Kinlaw JMP(11) [refocusing of null geodesics in strongly causal spacetimes].
Physics Applications > s.a. coordinates [geodesic-based]; de
Sitter-Fokker Precession; particle models;
* Main application: Paths of test particles in curved spaces; > s.a. test-particle motion; chaos; general-relativistic chaos.
* Practical realization: 2013, The best one so far is probably the LARES satellite; > s.a. LARES.
* 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]; Müller & Frauendiener CP(11) [in the classroom]; Podolský & Švarc PRD(12) [and geometrical and physical interpretation of an arbitrary spacetime].
@ Dynamics and geodesic motion: Hedlund BAMS(39); Dalvit & Mazzitelli AIP(99)gq, PRD(99)gq [with quantum corrections]; Ghosh CQG(14)-a1303 [corrections from non-commutative geometry, and violation of the equivalence principle]; Drago & Pinamonti JPA(14)-a1402 [influence of quantum matter fluctuations]; Pramanik PRD(14)-a1404 [with gup corrections]; > s.a. conserved quantities; jacobi dynamics; geometrodynamics; motion of gravitating bodies; types of geodesics.
> In gravitation and cosmology: see cosmological models in general relativity; hořava-lifshitz gravity; minkowski spacetime [κ-deformed]; perfect fluids.
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