Solving Geodesic Equations with Killing Vectors: Is There a General Solution?

[AleksanderPhy]

Hello I am concered about way of solving geodesic equation. Is there a general solution to geodesic equation? How to calculate the Cristoffel symbol at the right side of the equation?
Thanks for helping me out!

 

[Orodruin]

Is there a general solution to geodesic equation?

No, the solution depends on the manifold and connection.

How to calculate the Cristoffel symbol at the right side of the equation?

This depends on your connection. Since this is the relativity forum and not the differential geometry forum, I assume that you are working with the Levi-Civita connection from a metric tensor. You can then compute the Christoffel symbols directly from the expression in terms of the metric components and their derivatives, or from the geodesic equations resulting from finding the extrema of the integral
$$
S[\gamma] = \int g_{\mu\nu} \dot x^\mu \dot x^\nu d\tau,
$$
where $\tau$ is an affine curve parameter that will be identified with the proper time in the case of a time-like geodesic and $x^\mu(\tau)$ are the coordinates of the curve $\gamma$ in some coordinate system.

 

[pervect]

Hello I am concered about way of solving geodesic equation. Is there a general solution to geodesic equation? How to calculate the Cristoffel symbol at the right side of the equation?
Thanks for helping me out!

When you have certain symmetries known in the metric, called Killing vectors, writing the solution of the geodesic equations is a lot easier. It's not a formal definition, but if a coordinate system exists where the metric coefficients are not a function of time, the system has a time-like Killing vector that represents a conserved energy. More generally, if a metric exists where none of the metric coefficients are not a function of one of the coordinates, that coordinate represents a symmetry, and the existence of that symmetry implies that there is a conserved energy-momentum (linear or angular, depending on the nature of the coordinate) which is constant along the geodesic.