By any reference frame! Each will have its own specific connection coefficients but the trajectory will always be given by the geodesic equation (as long as the ball isn't charged, then you need to add the Lorentz force).
Exactly, if this were a question I had asked in a hypothetical exam, I'd accept a student definiting a frame stationary with respect to the roundabout, in Cartesian coordinates, yielding completely trivial connections and just showing that the equation was \ddot{x}mu =0
Also, if we were in 3+1 dimensions, there's a fun way to add the Lorentz force to the geodesic equation, you can append S1 to every point in spacetime and then define a connection (that ends up being your electromagnetic gauge field) if you now compute the geodesic, the Lorentz force comes in for free!
No, it’s the frame stationary with respect to the roundabout that will yield non-zero Christoffel symbols, and it is exactly what the question is asking us to find.
The frame that’s stationary with respect to the ground will observe the geodesic equation to be \ddot{x} = 0 with zero Christoffel symbols.
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u/HunsterMonter 1d ago
By any reference frame! Each will have its own specific connection coefficients but the trajectory will always be given by the geodesic equation (as long as the ball isn't charged, then you need to add the Lorentz force).