# Trouble with conflicting results for time dilation for two sets of events [closed]

I was just trying to calculate the time dilation for two sets of events using the Lorentz-Transformation. (I'm trying to understad the special theory of relativity at the moment - I mean what better could you do in this time of isolation?) But in one case I got the result $$\Delta t'=\gamma\Delta t$$ and in the other case the result $$\Delta t'=\frac{\Delta t}{\gamma}$$. And I don't know what's going on. Is this correct and I just need to understand how to interpret it or did I make a mistake?

So here are the two sets of events. Of course we have two observers $$O$$ and $$O'$$ witht heir frames of reference $$S$$ and $$S'$$ and relative to $$O$$ the observer $$O'$$ is moving along the x-axis with the velocity $$v$$.

For the first set of events $$E_1$$ and $$E_2$$ I assumed that for $$O$$ they happen on the same spot, i.e. in $$S$$ we have that $$x_1=x_2$$. I wanted to calculate $$\Delta t'=t_2'-t_1'$$. I just used the Lorentz-Transformation $$t'=\gamma(t-\frac{v}{c^2}x)$$ to substitute $$t_2'$$ and $$t_1'$$ and got the result $$\Delta t'=\gamma\Delta t$$. Straight forward enough I thought.

$$\Delta t'=\gamma(t_2-\frac{v}{c^2}x_1)-\gamma(t_1-\frac{v}{c^2}x_1)$$ (note that $$x_1=x_2$$)

$$\Delta t'=\gamma(t_2-\frac{v}{c^2}x_1-t_1+\frac{v}{c^2}x_1)$$

$$\Delta t'=\gamma(t_2-t_1)$$

$$\Delta t'=\gamma\Delta t$$

Next I took a different sets of events $$E_3$$ and $$E_4$$ for which I assumed that they both happened where the observer $$O'$$ was (at different times). That means for example that $$x_3'=x_4'=0$$ and more importantly that $$x_3=vt_3$$ and that $$x_4=vt_4$$. Again I substituted these fatcs and the Lorentz-transformation. But this time I got the seemingly conflicting result $$\Delta t'=\frac{\Delta t}{\gamma}$$.

$$\Delta t'=\gamma(t_4-\frac{v}{c^2}vt_4)-\gamma(t_3-\frac{v}{c^2}vt_3)$$

$$\Delta t'=\gamma(t_4-\frac{v^2}{c^2}t_4-t_3+\frac{v^2}{c^2}t_3)$$

$$\Delta t'=\gamma(t_4(1-\frac{v^2}{c^2})-t_3(1-\frac{v^2}{c^2}))$$

$$\Delta t'=\frac{(1-\frac{v^2}{c^2})}{\sqrt{1-\frac{v^2}{c^2}}}(t_4-t_3)$$

$$\Delta t'=\frac{(1-\frac{v^2}{c^2})\sqrt{1-\frac{v^2}{c^2}}}{(1-\frac{v^2}{c^2})}(t_4-t_3)$$

$$\Delta t'=\sqrt{1-\frac{v^2}{c^2}}\Delta t$$

$$\Delta t'=\frac{\Delta t}{\gamma}$$

Am I to believe then that $$\Delta t'=\gamma\Delta t=\frac{\Delta t}{\gamma}$$? I think not! But what is going on? I'd appreciate any help I can get.

There is no contradiction, because $$\Delta t$$ and $$\Delta t'$$ mean different things in different situations! In the first case $$\Delta t$$ is the time elapsed in the frame where the events happen at the same place, while in the second case that would be $$\Delta t'$$. And the results are perfectly consistent: in both cases, the time measured by an observer that is moving with respect to the events is $$\gamma$$ times the time measured by an observer for whom the events happen at the same position.