PlanetPhysics/Geodesic

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

Given a curved space $S_C$ one can find the geodesic by writing the equation for the length $l_v$ of a curve -- which is defined as a function $f:(R)\to S_C$ from an open interval $(R)$ of $ℝ$ to the manifold $S_C$-- 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 energy as well as the length of the curve.

However, in 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 spacetime. As a more general physical example in general relativity theory, relativistic geodesics describe the motion of 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 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 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 metric geometry is defined as a curve $\mathrm{\Gamma }:I\to M$ from an interval $I\subset ℝ$ to the 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 .