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snoopies622

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- TL;DR Summary
- Given one inertial frame and one frame accelerating with constant proper acceleration, how to transform the coordinates from one reference frame to the other?

I'm not perfectly clear about how Rindler coordinates work. Here's what i do understand:

Suppose I'm in an inertial reference frame and i define the location of the events around me with (x,t), (ignoring the y and z directions here) and a spaceship approaches me from afar and then flies away according to the right half of the hyperbola defined by

[itex] x^2 - c^2t^2 = r^2 [/itex]

Then the person on the spaceship will feel a constant acceleration in the positive x direction [itex] \alpha=c^2/r [/itex]. If [itex] \tau [/itex] is the time according to the clock on the spaceship, then I can re-define the hyperbola above with

x = r cosh ( [itex] \tau \alpha [/itex] /c )

ct = r sinh ( [itex] \tau \alpha [/itex] /c )

If an event takes place on this hyperbola, then given the x and t that I observe, i can conclude that the clock on the spaceship at that event reads

[itex]\tau = c/ \alpha [/itex] inverse cosh (x/r)

or [itex] \tau = c/ \alpha [/itex] inverse sinh (ct/r)

and that the X coordinate of the event as defined by the spaceship observer must be zero.

What i don't know is a general transformation formula. That is, given the x,t coordinates of an event NOT on the hyperbola, how do I translate them into X,T as observed by the person on the spaceship?

Thanks.

Suppose I'm in an inertial reference frame and i define the location of the events around me with (x,t), (ignoring the y and z directions here) and a spaceship approaches me from afar and then flies away according to the right half of the hyperbola defined by

[itex] x^2 - c^2t^2 = r^2 [/itex]

Then the person on the spaceship will feel a constant acceleration in the positive x direction [itex] \alpha=c^2/r [/itex]. If [itex] \tau [/itex] is the time according to the clock on the spaceship, then I can re-define the hyperbola above with

x = r cosh ( [itex] \tau \alpha [/itex] /c )

ct = r sinh ( [itex] \tau \alpha [/itex] /c )

If an event takes place on this hyperbola, then given the x and t that I observe, i can conclude that the clock on the spaceship at that event reads

[itex]\tau = c/ \alpha [/itex] inverse cosh (x/r)

or [itex] \tau = c/ \alpha [/itex] inverse sinh (ct/r)

and that the X coordinate of the event as defined by the spaceship observer must be zero.

What i don't know is a general transformation formula. That is, given the x,t coordinates of an event NOT on the hyperbola, how do I translate them into X,T as observed by the person on the spaceship?

Thanks.

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