Relativity, reference frames and Lorentz transformations

[Petar Mali]

Homework Statement

The system $$S'$$ moves in relation to the system $$S$$ with velocity $$\upsilon$$ along the -$$x$$- axis. At the time when the beginnings of the coordinate system are in the same point, clocks in both system shows $$t=t'=0$$. Which coordinates will have a reference point during the motion in every of these systems, which has the property that in some next moment clocks in systems $$S$$, $$S'$$ shows the same time $$t=t'$$. Determine the law of motion of motion of this point.

Homework Equations

Lorentz transformation

$$x'=\frac{x-\upsilon t}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}$$

$$y'=y$$

$$z'=z$$

$$t'=\frac{t-\frac{\upsilon}{c^2}x}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}$$

The Attempt at a Solution

I tried like this

$$t'=\frac{t-\frac{\upsilon}{c^2}x}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}$$

$$t=\frac{t'+\frac{\upsilon}{c^2}x'}{\sqrt{1-\frac{{\upsilon}^2}{c^2}}}$$

$$t=t'$$

$$t-\frac{\upsilon}{c^2}x=t'+\frac{\upsilon}{c^2}x'$$

$$t-t'=\frac{\upsilon}{c^2}(x+x')$$

$$0=\frac{\upsilon}{c^2}(x+x')$$

and get

$$x=\frac{\upsilon t}{\sqrt{1-\frac{\upsilon^2}{c^2}}+1}$$

Is this correct?

 

[Petar Mali]

Any answer?