Geodesics |

**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

d^{2}*x*^{m}/d*t*^{2} +
Γ^{m}_{pq}
(d*x*^{p}/d*t*) (d*x*^{q}/d*t*) =
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]; Boonserm et al a1710 [coordinate velocity and acceleration of near-horizon geodesics]; Sämann & Steinbauer a1710 [metrics of low regularity].

> __Space of geodesics__:
see lines [topology]; symplectic structure.

**Geodesic Deviation and Focusing ** > s.a. Congruence of Lines; regge
; 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 *X*^{a}:= ∂*x*^{a}/∂*s* is
the connecting vector between nearby
geodesics in a one-parameter family, and *T*^{ a}:=
∂*x*^{a}/∂*t* the
tangent vector, the geodesic deviation equation states that the relative acceleration between the geodesics is

*a*^{a}:= *D*^{2}*X*^{a}/*Dt*^{2}
= *R*^{a}_{bcd}* 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.

@

@

**Physics Applications** > s.a. coordinates [geodesic-based]; de
Sitter-Fokker Precession; particle models;
Sachs-Wolfe Effect.

* __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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