Special relativity, a non-inertial perspective

[pepediaz]

Homework Statement

Titan is 1.2 billion km from Earth, where is the observer O.
Observer O ' moves to u = 0.866c with respect to O.
Another non-inertial observer Õ with constant quadceleration A = (0, a, 0, 0) seen by O
it begins its trajectory being stationary on Earth (that is, it moves at four speeds
initial v = (c, 0, 0, 0) measured by the observer O). To the acceleration a stated in my attempt (here it isn't written properly):

a = \frac{2c^{2}\beta _u^{2}}{(1-\beta _u^{2})\Delta x}

find how long it takes to get to Titan according to O (u is the one associated with speed u
of the observer O 'with respect to O). And according to O '? What does Õ 's clock measure when it arrives at Titan?

Relevant Equations

Uniform Accelerated Movement equations, Lorentz transformations.

I have tried calculating it as a Uniform Accelerated Movement problem:

, where t is the time for the observer at Earth, O.
For calculating t'' (the proper time for the accelerated spaceship observer), it is just using a Lorentz transformation?
It seems easy, but as is stated that the frame is non-inertial, maybe I'm forgetting something.

 

[jbsteele]

No, you are not forgetting anything. The calculation of the proper time for the accelerated spaceship observer, t'', is indeed done using a Lorentz transformation, as you have suggested. The Lorentz transformation is used to transform the coordinates from one inertial frame (the observer at Earth, O) to another inertial frame (the accelerated spaceship observer). Once you have transformed the coordinates, then you can calculate the proper time for the accelerated spaceship observer, t'', by simply integrating the velocity of the spaceship with respect to time. This will give you the total distance traveled by the spaceship over the given time period, and thus the proper time, t''.