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I'm having trouble with the concept of a co-moving frame, specifically in the context of constant proper-acceleration (hyperbolic motion). I feel like I don't "get it"; and perhaps this is related to some deeper misunderstandings I may have regarding the definitions of certain quantities in SR.

I would really appreciate it if someone could explain this in very plain words, specifically the following points:

These seem like enough for a start. Thanks a lot for the help!

- When speaking of trajectories in space-time as seen in a certain frame. What exactly do mean by this when relating to the co-moving frame? It is my intuitive understanding that in this frame the particle/observer lays at the origin for all values of tau, there can be no trajectory which isn't identically zero. What are we talking about when saying that in this frame the trajectory is a hyperbola?

- How is the acceleration defined in a co-moving frame, if by definition in this frame the velocity is constant? It would seem to me that the derivative of the velocity wrt proper time should vanish in this frame.
- Does a body have momentum in it's co-moving frame? That would seem to contradict that fact that the velocity is null, as 4-momentum and 4-velocity are always simply related. On the other hand, if we have proper acceleration, then we have proper force, and the momentum should be changing. How do we settle the definitions of momentum and velocity in this case?
- I've seen the relation stating that the 4-velocity and 4-acceleration are always perpendicular. To my understanding, the 4 velocity is just the "velocity of the world line", in other words the velocity of the trajectory wrt to the path parameter - correct? If the world line is a straight line in space time (for example), how can the acceleration be perpendicular to the velocity? That would suggest a change in the path direction. Or am I confusing the definitions here?

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# The co-moving frame - acceleration, momentum and velocity

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