PlanetPhysics/Geodesic

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A geodesic  is generally described as the shortest possible, or topologically allowed, path between two points in a curved space.

Given a curved space ${\mathrm{\S }}_C$ one can find the geodesic by writing the equation for the length $l_v$ of a curve -- which is defined as a [[../Bijective/|function]] $f:(R)\to {\mathrm{\S }}_C$ from an open interval $(R)$ of $ℝ$ to the [[../NoncommutativeGeometry4/|manifold]] ${\mathrm{\S }}_C$-- and then by using the calculus of variations minimizing this length. In physical applications, however, to simplify the calculation one may also require the minimization of [[../CosmologicalConstant/|energy]] as well as the length of the curve.

However, in [[../NonabelianAlgebraicTopology3/|Riemannian geometry]] geodesics are not coinciding with the "shortest length curves" joining two points, even though a close connection may exist between geodesics and the shortest paths; thus, moving around a great circle on a Riemann sphere the `long way round' between two arbitrary, fixed points on a sphere is a geodesic but it is not obviously the shortest length curve between the points (which would be a straight line that is not permitted by the topology of the surface of the Riemann sphere).

The orbits of satellites and planets are all geodesics in curved [[../SR/|spacetime]]. As a more general physical example in general relativity theory, relativistic geodesics describe the [[../CosmologicalConstant/|motion]] of [[../CenterOfGravity/|point particles]] in a spacetime with a curvature determined only by gravity.

Consider such a point particle $z^{\mu }$ that moves along a trajectory or "track" in physical spacetime; also assume that the track is parameterized with the values of $\tau$. Then, the [[../Velocity/|velocity]] [[../Vectors/|vector]] pointing in the direction of motion of the point particle in spacetime can be written as:

$u^{\mu }=\frac{d{z}^{\mu }}{d\tau }.$

If there are no [[../Thrust/|forces]] acting on a point particle, then its velocity is unchanged along the trajectory or `track' and one has the following geodesic equation :

$\frac{d{u}^{\nu }}{d\tau }+{\mathrm{\Gamma }}_{\mu \sigma }^{\nu }{u}^{\mu }{u}^{\sigma }=\frac{d^2{z}^{\nu }}{d{\tau }^2}+{\mathrm{\Gamma }}_{\mu \sigma }^{\nu }\frac{d{z}^{\mu }}{d\tau }\frac{d{z}^{\sigma }}{d\tau }=0.$

More generally, a geodesic in [[../MetricTensor/|metric]] geometry is defined as a a curve $\mathrm{\Gamma }:I\to M$ from an interval $I\subset ℝ$ to the [[../NormInducedByInnerProduct/|metric space]] $M$ for which there exists a constant $v\le 0$ such that for any $t\in I$ there is a neighborhood $J$ of $t\in I$ such that for any $t_1,t_2\in J$ one has that

$d(\mathrm{\Gamma }(t_1),\mathrm{\Gamma }(t_2))=v|t_1-t_2|.$

When the equality $d(\mathrm{\Gamma }(t_1),\mathrm{\Gamma }(t_2))=|t_1-t_2|$ is satisfied for all $t_1,t_2\in I$, the geodesic is called the shortest path or a minimizing geodesic .