Geodesics in Rindler Space: How Do They Differ from Minkowski Space?

[PerpStudent]

How would one determine a geodesic in Rindler space? Why would geodesics not be simply the same as those of Minkowsky space? Is it not analogous to using polar vs. Cartesian coordinates in euclidean space, where a straight line is the same in either case?

 

[Adam Lewis]

How would one determine a geodesic in Rindler space? Why would geodesics not be simply the same as those of Minkowsky space? Is it not analogous to using polar vs. Cartesian coordinates in euclidean space, where a straight line is the same in either case?

Hi Perp Student,

Yes, that is correct. The term "Rindler space" is actually a bit inaccurate since the spacetime being described is just the Minkowski vacuum - "Rindler coordinates" would be better.

Geodesics in Rinder coordinates are indeed the same as those of Minkowski space in the sense that they will appear as straight lines after performing a Rindler->Minkowski coordinate transformation. They will not, however, appear as straight lines in Rindler coordinates. The actual form of the geodesic equations here is quite complicated, but Wikipedia has a decent discussion.

"Rindler observers"; i.e. the worldlines of particles at rest in the Rindler frame, do not correspond to geodesics in Minkowski space. This is because Rindler coordinates describe a coordinate system which is undergoing uniform acceleration - such a coordinate system is not inertial, so the observers at rest in such a system do not undergo geodesic motion.

 

[PerpStudent]

That's very helpful, thank you.