# Solving the Equation of Geodesic Deviation

1. Mar 20, 2010

### Altabeh

Hello

Anybody know how we can solve the equations of geodesic deviation in a given spacetime whether approximately or exactly?

More interestingly, all solutions are known analytically (at least in a finite region) in terms of the spacetime's world function (also known as Synge's function). This is a two-point scalar $\sigma(x,x') = \sigma(x',x)$ that returns one-half of the geodesic distance between its arguments. See Dixon, Proc. R. Soc. A 314, 499 (1970) or http://arxiv.org/abs/0807.1150" [Broken]. Standard results in bitensor perturbation theory can be used to turn this solution into a power series near the initial point, if desired. Part of the exact solution also looks very nice in Riemann normal coordinates (without approximation).