This is a follow-on to posts here and here.
It is common to derive the Lorentz transformation assuming velocity reciprocity, which seems to say that if a body at rest in frame of reference S´ is observed from a frame of reference S that travels with relative velocity +v, then a body at rest in frame of reference S will be observed from the frame of reference S´ to be traveling with velocity –v. But that’s not the case.
Consider the typical scenario in which a person standing on the earth (embankment, station) with frame of reference S observes a person sitting in a railway car with frame of reference S´. Say they are both waving their right hands and their frame of reference follows a right-hand orientation: the positive direction is toward their right.
The first illustration shows the scenario from behind the observer standing on the earth in frame S, who observes the passenger sitting in the train moving to their right with velocity +v. The scenario is typically presented from only this perspective, that of an observer at rest in frame A, even if the perspective of an observer at rest in frame S´ is described.
The second illustration shows the scenario from the perspective of an observer sitting in the train in frame S´, who observes the observer standing on the earth receding to their right with velocity +v. That is, bodies at rest in both frames are observed from the other frame to be moving with velocity +v, which seems contrary to the velocity reciprocity principle.
What has happened? The frames associated with the two observers are turned around from each other. It is the frames that are moving in opposite directions, not the velocities observed. The Lorentz transformation applies to frames, not observations. A transformation of one frame into another frame is the opposite of the inverse transformation, but it applies to frames, not observations.
A more precise statement of the principle of velocity reciprocity is that if a body at rest in frame S´ is observed from frame S traveling with velocity +v, then in the “language” of frame S, the observation from frame S´ of bodies at rest in frame S will be traveling with velocity –v. That also means if a body at rest in frame S is observed from frame S´ traveling with velocity +v, then in the “language” of frame S´, the observation from frame S of bodies at rest in frame S´ will be traveling with velocity –v.
The Lorentz transformation or any other transformation is like a linguistic translation of the lingo of one frame into the lingo of another frame. It answers how does frame S translate the speech of frame S´ into their own speech? Or how does frame S´ translate the speech of frame S back into their own speech?
Galileo’s relativity considers frames pairwise; it accesses a third frame only by switching pairs.