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This is in response to the following in another thread:
I'm assuming that Fredrik refers to using the plane of simultaneity of the co-moving inertial observer as a means of assigning coordinates to events in an accelerating frame. This works for Rindler coordinates in the case of uniform acceleration in flat spacetime, but I see no reason why it shouldn't work for arbitrary accelerating observers.
You will have to impose some restrictions on the smoothness of the observer's worldline. ([itex]C^\infty[/itex] would certainly be good enough, in my guess, and probably much worse would do.) And you are guaranteed to get a one-one correspondence between coordinates and events only locally near the observer. (I would guess within a distance of [itex]c^2/a[/itex], the "radius of curvature" of the worldline, where a is the instantaneous proper acceleration; this radius is pretty large for most physically realistic accelerations, e.g. about one light year for 1g. But even beyond that radius, coordinates are still defined, just no longer one-one. That being said, there may nevertheless be some events lying behind an event horizon with no coordinates.) But apart from that I don't see why it shouldn't work, and why you shouldn't regard this as "natural" -- or to be more specific, I don't see why uniform acceleration should be singled out as being different. (I'm using my geometric intuition here, I have no proof.)
I answer it here to avoid diverting that other thread's course.Fredrik said:The problem is that there's no natural way to extend the accelerating coordinate system to points that aren't close on the rocket's world line so that it assigns a time coordinate to events on Earth as well. (There is a choice that can be considered natural when the acceleration is constant, but not when the acceleration is arbitrary).
I'm assuming that Fredrik refers to using the plane of simultaneity of the co-moving inertial observer as a means of assigning coordinates to events in an accelerating frame. This works for Rindler coordinates in the case of uniform acceleration in flat spacetime, but I see no reason why it shouldn't work for arbitrary accelerating observers.
You will have to impose some restrictions on the smoothness of the observer's worldline. ([itex]C^\infty[/itex] would certainly be good enough, in my guess, and probably much worse would do.) And you are guaranteed to get a one-one correspondence between coordinates and events only locally near the observer. (I would guess within a distance of [itex]c^2/a[/itex], the "radius of curvature" of the worldline, where a is the instantaneous proper acceleration; this radius is pretty large for most physically realistic accelerations, e.g. about one light year for 1g. But even beyond that radius, coordinates are still defined, just no longer one-one. That being said, there may nevertheless be some events lying behind an event horizon with no coordinates.) But apart from that I don't see why it shouldn't work, and why you shouldn't regard this as "natural" -- or to be more specific, I don't see why uniform acceleration should be singled out as being different. (I'm using my geometric intuition here, I have no proof.)