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].

@ __Unparametrized geodesics__:
Matveev JGP(12)-a1101 [metric reconstruction];
Gover et al a1806 [conserved quantities and integrability].

@ __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];
Boonserm et al Univ(18)-a1710 [coordinate velocity and acceleration of near-horizon geodesics];
Sämann & Steinbauer a1710 [metrics of low regularity];
Dimakis et al a1901 [integrability from non-local conserved charges].

> __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
*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 in(19)-a1604 [range of applicability];
Puetzfeld & Obukhov en(19)-a1901 [in relativistic geodesy].

@ __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];
Cariglia et al EPJC(18)-a1805 [integrability].

@ __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; perfect fluids;
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);
Mashhoon FTP-a1806 [gravity gradiometry];
Mannheim a2105 [light rays do not always follow null geodesics];
> s.a. conserved quantities; jacobi dynamics;
geometrodynamics; motion of gravitating bodies;
types of geodesics.

@ __Quantum corrections__: Dalvit & Mazzitelli AIP(99)gq,
PRD(99)gq;
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 [gup corrections];
Vieira et al PRD(18)-a1805 [and spacetime geometry fluctuations];
Pipa et al JCAP(19)-a1801 [entanglement-induced deviations].

> __In gravitation and cosmology__:
see cosmological models in general relativity;
hořava-lifshitz gravity; minkowski
spacetime [*κ*-deformed]; world function.

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