Special relativity inertial reference frames.

[C.E]

1. Suppose the space time co-ordinates of two events in inertial frame S are as follows:
Event 1:
x1=x0, t1=x0/c y1=z1=0
Event 2:
x2=2x0, t2= x0/2c y2=z2=0
Show that there exists an inertial frame s' in which these events occur at the same time (i.e t1'=t2') and find the value of time for which these events occur in this reference frame.


2. You may assume that: (Δx)^2+c^2Δt^2=(Δx’)^2 + (Δt’)^2


3. I think I have a solution but annoyingly have not used the above asssumption which I want to do as it is given in the question. Anyway I did the following:

Firstly I set t1'=t2' from this it follows that γ(x0/c - ux0/c^2)=γ(x0/2c-2ux0/c^2) (by the lorentz x co-ordinate transformations) rearranging gives that u=-0.5c. Indeed u=-0.5c seems to work giving t1'=t2'=√3x0/c. However, I am unhappy with my answer in that firstly I seem to have assumed it rather than proved it and secondly I did not use the above assumption. Any guidance would be very welcome.(By the way this is not assesed work merely revision so feel free to give as much help as you deem appropriate).

 

[D H]

Finding a solution is certainly a valid approach to solving a problem of the nature "show that a solution exists ...".

 

[C.E]

So, do you think what I have done is correct?

 

[C.E]

I am finding gamma correctly but as it depends only on u^2 and not on the sign of u, u=0.5c or u=-0.5c (by taking positive or negative square roots in finding u from gamma), in selecting 0.5c I clearly chose the wrong option. Anyway how could I know that I should have selected u=-0.5c and not 0.5c? Do you just have to try both and see what works?