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In my notes it says, when explaining an immediate consequence of the Lorentz Transformation: The significance of the parameter $v$ is seen as follows. Consider the origin of spatial coordinates $(x', y', z')' = (0, 0, 0)'$ in frame $F'$. From the Lorentz transformation This corresponds to $ \gamma(v)(x - vt) = 0, y = 0, z = 0 $ or $\mathbf x = \mathbf v t$, which is the worldline of an object moving uniformly with respect to $F$ with velocity $\mathbf v = (v, 0, 0)$.

Do this mean that they have said where $P$ is w.r.t. $F'$ (origin of $F'$) and they are now trying to find $P$ w.r.t. $F$? It looks as if they are, and if that is the case, are they doing it this way: Do the reverse Lorentz transformation i.e. from frame $F'$ to frame $F$. Or is it: Find out what the worldline of the origin of $F'$ would have to be in frame $F$ in order to give the Lorentz transformation $(x', y', z')' = (0, 0, 0)'$. Does it matter which way we do it? I know it doesn't, but at this stage, it is hard to explain/see why.

Moreover, how can this result be true, when the origin of frame $F'$ is moving with velocity $v$, and the distance between $F$ and $F'$ must therefore be $\gamma(v)$ Also, how comes we are allowed to define such a $v$ existing. $v$, after all, is distance over time, and the "coordinates" or (loosely) distances $|x-x'|$ depend on $v$ itself???

I am guessing that this is a common trap, and I am also guessing that the answer is because we are doing a passive transformation. But that is just a guess, and I'm not sure I even fully understand it...

Also, note that at this stage in my notes, they have only just stated what the definition of the Lorentz transformation is.

As a deeper guess, I am guessing that you can think of the passive transformations as moving from one space-time coordinate to another one, or, a "ghost" in $\mathbb R^3$ Note space-time here still refers to Newton's version.

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I think there are few areas in which confused language is more widespread and has penetrated deeper into mainstream education than relativity. I can't claim to clear up all your confusions because I have trouble following many of them, but I'm pretty sure that they will all be resolved if you start using precise language and concepts and see through the illusion that any of the handwaving stuff that relativity is often surrounded with actually means anything.

What does it mean to "have said where $P$ is w.r.t. $F$'" or to "find $P$ w.r.t. $F$"? A point $P$ is only in one place, namely at $P$; there's no such thing as where it is w.r.t. some frame. You can describe $P$ using different coordinate systems. The result is different sets of numbers, not different positions.

I'm aware that this doesn't come close to answering the myriad of questions you've posed all in one go; but I think if you start clearing up concepts in this way, the fog will start to lift and perhaps you can then express some of the questions in a clear enough form that they admit a clear answer.

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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/3140/discussion-between-adam-rubinson-and-joriki)2012-04-18