We have two frames of reference: frame $F$ and frame $F'$ such that $F'$ is moving at velocity $v$ in the positive $x$ direction of $F$. I get the overall idea of special relativity (time dilation, length contraction etc.) but I still have problems with working on the equations. Lets say our velocity is $0.7c$ of $F'$. In $F'$ is a flash lamp and in distance $d'$ (in $x'$ direction) is a detector installed. The time between emission and detection of the light pulse for the observer in $F'$ is $t' = 1.5 *{10}^{-8}s$.
I calculated the distance $d'$, which is approx $4.5m$ and the timeinterval between emission and detection of the light-pulse for the observer in the resting reference frame: $t = 2.1 * {10}^{-8}s$ with $t = \gamma * t'$.
Now I have to to calculate the spatial distance $x$ between the emission and the detection for the observer in $F$. I tried 2 separate ways. First with the space-time invariant $$(c*t')^2 - (d')^2 = (c*t)^2-(x)^2$$ and got $6.3m$ iirc. The second way was using the Lorentz Transformation, the obvious solution I guess, but I'm new to special relativity so forgive me. That's the point where I got confused with the equations.
"The Lorentz transform for the $x$ coordinate is given by: $$x'=\gamma (x-vt)$$ Everything on the RHS of this equation is measured in the frame $F$ and every thing on the LHS is measured in frame $F'$."
That was posted in another thread. What I don't get is the reference of the variables. So my $x'$ is my $d'$ of course. My distance in the moving frame. My $x$ is what I'm searching for. The distance the resting observer is seeing (?). My velocity is the velocity of the moving frame (so $0.7c$, right?) and now my time.. I thought I should use the time measured in the resting frame, so $t = 2.1 * {10}^{-8}s$. But that doesn't give me the same solution as the the space-time invariant. If I take $t'$ however, it works. But why? Did I mix something up? If $t'$ is the right time to use, why don't write the Lorentz Transformation differently?
Sorry but it really confused me - everywhere it is somewhat ambiguous and not absolutely clear what the variables are referring to. An example would help me a lot!