# geodesic

### From Mathematics wiki

A **geodesic** on a manifold is a curve for which the arc length is an extremum. Let be a differentiable curve with coordinates . Its arc length is the functional of given by

- ,

where the negative sign is needed if the metric is not positive definite. Using a dot to indicate differentiation with respect to , we can write this as

- .

## Geodesic equation

Consider a variation of the functions , i.e., such that . For to extremize the arc length, we require that

- ,

evaluated at the solution . This could also be written as , but this notation is cumbersome. Then

- .

With the variation performed, we can now use our coordinate freedom to set , or equivalently, . Then

- .

We can integrate by parts:

- ,

but , so the contribution from the endpoints vanishes:

- ,

Using , and the fact that , and relabeling indices

, | |

, | |

, | |

, | |

, |

where we have defined the Christoffel symbol of the second kind . In order for to vanish for an arbitrary variation , by the fundamental lemma of calculus of variations,

,

since is generally not zero. This is known as the geodesic equation, and is satisfied by **geodesic curves** for which is an affine parameter, and is supplemented by our previous condition . We can write it in another way:

.

If we denote the tangent vector by , then this can be written as

.

Written in this form, the geodesic equation makes use of what is known as the affine connecion (specifically, the Levi-Civita connection). Furthermore, the geodesic equation implies that a **geodesic** parallel transports its own tangent vector.

## Reparameterization

Note that above we used the invariance of the integral

under reparameterizations of , i.e,. to choose a parameter for which takes a convenient form. Thus, suppose or . Then the integrals

and

- ,

are equal, and we can extermize either. The second form is especially useful for dealing with **null geodesics**, i.e., geodesics with zero length.

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