Relative velocities with the Lorentz transformation I'm only a high school student so I'd appreciate a relatively simple explanation. Thanks.
I am studying special relativity and the textbook I'm referring to (Kleppner and Kolenkow) derives the Lorentz transformation by first assuming the transformation is linear, and then solves for the required constants by observing various situations in two reference frames (The frame $S'$ is assumed to move with a velocity $v$ along the positive $x$-axis relative to $S$).

The second statement is clear to me. However I do not understand the first. I can see how under the Galilean transformation the relative velocity of $S$ relative to $S'$ would be $-v$, but I have no reason to think it would be the same under the Lorentz transformation. Why have we taken for granted that if $S'$ moves with velocity $v$ relative to $S$, that $S$ would move with a velocity $-v$ relative to $S'$? I tried showing it with my high-school level knowledge of vectors but that proof would only work with the Galilean transformation (just a basic shifting of origin and using the concept of absolute time).
Is there a deeper explanation here or am I missing something simple?
 A: Your hesitation is quite reasonable and indeed some subtle points can arise in connection with this. However, the main point is that one is seeking a transformation that is

*

*consistent with principle of relativity


*consistent with a finite maximum speed for signals such as light in vacuum


*as simple as possible
What this means is that we can assert whatever we like, in the first instance at least, and as long as it fits all the above 3 points, and then worry afterwards about whether or not our solution was unique. The guess that if $S'$ is observed to move at $v$ by $S$ then $S$ will be observed to move at $-v$ by $S'$ is a reasonable guess, and it is found to be consistent with the above points. So that is enough for a first derivation. I recommend proceeding like this in the first instance.
Having adopted that first approach, outlined above, then, afterwards, we  can try considering what would happen if instead of $-v$ we put something else such as $-kv$ with the value of $k$ to be determined. One then finds a way of looking at events that is sort of weird, but hard to rule out entirely. It is because in the end assigning coordinates (values of $x$, $y$, $z$, $t$) to events in spacetime is somewhat of a human choice rather than a procedure forced on us by the nature of spacetime. This means that the way to proceed in practice is influenced by which way is more useful, or leads to simpler equations. If two different ways of assigning coordinates lead to the same physical predictions, then both may be said to be allowed or "correct" in what they affirm about events, but if one way is mathematically simpler than the other then the simpler one is generally preferred and judged to offer greater insight. So that's how we get the Lorentz transformation: it is the simplest way to assign coordinates in a way which is mutually consistent between different inertial frames, and captures or expresses the nature of spacetime correctly, including the finite maximum speed for signals.
