Geodesics| 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
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].
@ 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
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 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.
@ Other modified theories: Kerner et al PRD(01)gq/00 [Kaluza-Klein theory];
Baffou et al ChJP(17)-a1509 [f(R,T) gravity];
Puetzfeld & Obukhov PRD(18)-a1804 [in Riemann-Cartan spacetime].
@ 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; 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.
main page
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 18 may 2021