# Geodesics in a rotating coordinate system

1. Jun 22, 2012

### snoopies622

In a uniformly rotating coordinate system the trajectories of freely moving objects are influenced by an apparent centrifugal and Coriolis force. Is there a coordinate system or metric (or both) in which these trajectories are geodesics instead?

2. Jun 22, 2012

### DrGreg

The property of being a geodesic doesn't depend on the coordinate system. The trajectories of freely moving objects are always geodesics, whatever coordinate system you use.

In non-rotating Cartesian coordinates $x = vt$ is a geodesic, which in rotating Cartesian coordinates might become $X \cos \omega T + Y \sin \omega T = VT$. In these coordinates $X = VT$ would not be a geodesic.

I'm not sure if that answers your question.

3. Jun 22, 2012

### pervect

Staff Emeritus
The trajectories of freely moving objects will be geodesics in any coordinate system. The condition for a path being a geodesic is that there are no "real" forces influencing the path. In an inertial frame, there are no real and no apparent forces on a geodesic trajectory. In a non-inertial frame, such as your rotating frame, there may be apparent forces on a geodesic trajectory, but there are still no "real" forces.

4. Jun 23, 2012

### snoopies622

Hmm.. Thank you both. Must think about this some more.

5. Jun 23, 2012

### A.T.

I think the metric you are looking for is given in the last paragraph of chapter 2 in this:
http://www.projects.science.uu.nl/igg/dieks/rotation.pdf [Broken]

Last edited by a moderator: May 6, 2017