A geodesic is generally described as the shortest possible, or topologically allowed, path between two points in a curved space.
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).
Consider such a point particle
that moves along a trajectory or “track” in physical spacetime; also assume that the track is parameterized with the values of
. Then, the velocity
vector
pointing in the direction of motion of the point particle in spacetime can be written as:
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:
When the equality
is satisfied for all
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