Physically, these represent the paths of (usually ideal) particles with no proper acceleration, their motion satisfying the geodesic equations.
Because the particles are subject to no proper acceleration, the geodesics generally represent the straightest path between two points in a curved spacetime.
The Christoffel symbols are functions of the metric and are given by: where the comma indicates a partial derivative with respect to the coordinates: As the manifold has dimension
As the laws of physics can be written in any coordinate system, it is convenient to choose one that simplifies the geodesic equations.
Mathematically, this means a coordinate chart is chosen in which the geodesic equations have a particularly tractable form.
In this case, many of the heuristic methods of analysing energy diagrams apply, in particular the location of turning points.
Most attacks secretly employ the point symmetry group of the system of geodesic equations.
In general relativity, to obtain timelike geodesics it is often simplest to start from the spacetime metric, after dividing by
This method has the advantage of bypassing a tedious calculation of Christoffel symbols.