Solving the geodesic equations

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Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles, their motion satisfying the geodesic equations.

Template:Mathematical methods in general relativity

1 The geodesic equation
2 Heuristics
3 Effective potentials
4 Solution techniques
5 Examples
6 See also
7 References

[edit] The geodesic equation

Main article: Geodesic

On an n-dimensional Riemannian manifold $M$ , the geodesic equation written in a coordinate chart with coordinates $x a$ is:

$\frac{d^2x^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} = 0$

where the coordinates x^a(t) are regarded as the coordinates of a curve γ(t) in $M$ and $\Gamma^{a}_{bc}$ are the Christoffel symbols. The Christoffel symbols are functions of the metric and are given by:

$\Gamma^a_{bc} = \frac{1}{2} g^{ad} \left( g_{cd,b} + g_{bd,c} - g_{bc,d} \right)$

where the comma indicates a partial derivative with respect to the coordinates:

$g_{ab,c} = \frac{\partial {g_{ab}}}{\partial {x^c}}$

As the manifold has dimension $n$ , the geodesic equations are a system of $n$ ordinary differential equations for the $n$ coordinate variables. Thus, allied with initial conditions, the system can, according to the Picard-Lindelöf theorem, be solved.

[edit] Heuristics

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.

[edit] Effective potentials

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[edit] Solution techniques

Solving the geodesic equations means obtaining an exact solution, possibly even the general solution, of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions implicitly, but in many examples does yield the general solution in explicit form.