Lorentz Transformations Acceleration: A simple problem

[Albertgauss]

Hi all,

I came up with the following problem myself and am trying to solve myself. I haven't seen it in any txtbook, grad or undergrad.

Suppose you have the ground frame (Earth).
Earth sees ship1 start at t=0, v=vo1, at x=xo1.
Earth sees ship2 start at t=0, v=vo2, at x=xo2

All velocities are near c. All objects travel along the x-axis only. Let ship2 start ahead of ship1, that is xo2 > xo1.

Earth sees ship2 begin to accelerate at a constant acceleration A at t=0.

If ship1 sends a radio signal (light speed) to ship2, at what position and time will all frames----- Earth, ship1, and ship 2--- say the signal arrives at ship2?

I can easily get x,t v,a for ship 2 as the Earth measures them. That's pretty common, but I can't find any lorentz-transformations for the same set of variables in the frame of ship1 or ship2. I'm willing to do the hard work, but can someone guide me to a resource or some help with how to get X,t for the different frames for such a situation?

 

[jfy4]

Ah yes, The great Albert Guass, one of the greatest mathematicians and physicits of the 1900s :)

I think MTW treats accelerated frames in special relativity. You might find some answers there.

 

[Passionflower]

Since you are asking for an approach I would first calculate everything in coordinate time, which would be simplest if you take the ground frame.

So you first need to convert the proper accelerations from the rockets into coordinate accelerations which are not constant but decreasing with coordinate time. Then you need to setup the equations for the two spaceships in terms of location and coordinate time. Then pick a point where you want a light signal to leave, with that you can setup the equation for the light path and use this to solve where this light meets the other rocket.

Once you got that you need to convert the coordinate times into proper times.

So the fist thing I would ask you is do you know how to express an equation for a rocket with constant proper acceleration in terms of the ground frame?