I'm learning about relativity and, by extension, the classic

[JustinLevy]

Thanks Justin. Any insight into the justification for the last step of the derivation? Is this the way Einstein did it, by setting the variables that way?

You are basically asking how we solve for the Lorentz transformations. Lorentz did it through a bit of intuitive leaps. He rewrote maxwell's equations in a new frame after a Galilean transformation and found if he adjusted everything a bit, they almost looked the same (he found second order 'corrections' to the transformation to make this happen). He then pursued this further and found exact transformations to make this occur. He viewed this merely as a math trick and referred to it as something like "principle of equivalent states". He was so close to relativity. He basically found that instead of forcing yourself to to all the hard math in the aether rest frame (the frame they believed was the one Maxwell's equations and the force law were true in), you could look at an "equivalent state" which was essentially the lorentz transformed frame ('rest frame' of the lab) and do the math there and equate it back to the aether frame later.

Einstein took this the final step and said that was not a math trick, but that those two perspectives were on the same footing. He claimed that the laws of physics must be the same in all inertial frames (something people already agreed on since Galileo), but additionally stated that all observers must agree on the speed of light (it is the second one that rendered the 'medium' of light unobservable and moot).

While some people did have a view of an "absolute/'real' coordinate system" (for example Newton, even though his equations had Galilean relativity) this was more philosophy than physics. The aether as a medium didn't mean "absolute coordinate system" it merely meant there was a 'physical' medium (as with sound and water waves) so there was an inertial frame in which the physics looked the simplest (the rest frame of the medium). The presence of a medium does not demand an "absolute" coordinate system. It just meant you could possibly measure your velocity realtive to the medium (and not in some "absolute" sense).

When Lorentz stressed that the aether was immobile, constant properties, and permeated all space ... it started to look more and more like they were describing the space / vacuum / "stage of it all" itself. It is THIS sentiment that leads so many to ascribe the philosophy of "absolute coordinate system" to aether. But please remember that was a philosophy and nothing required of the physics, and in many senses a newer philosophy than the idea of aether itself.

Anyway, enough history rambling.
The point is that the historical process is not very clean. I doubt anyone would want to teach based on Lorentz's publication that showed his transformations. Nor do I get the impression that many people like Einstein's presentation of his derivation of it (I've seen 'cleaned up' versions of this that are pretty nice though. So just search for 'Lorentz transformation' and scan through a couple. I'm sure you'll find something that suits you.

I do realize that "relative motion" does not rule out an aether (in the general sense), it only rules out an aether (in the absolute, passive, stationary sense). This is why I think the stationary aether hypothesis of the time was ill-conceived. However, until MM did their experiment there was still the possibility of "absolute rest" or "absolute motion" I suppose, and this was mostly arguable on philosophical/metaphysical grounds. The MM experiment laid the debate to rest once and for all, confirming the philosophical/metaphysical notion that all motion is relative, consistent with all other empirical findings up to that time.

You could go into a frame where the aether was moving. The aether doesn't need to be stationary. All they were concretely saying was that you could measure your velocity with respect to the aether, not that there was an absolute rest (for example, the other forces such as gravity may have had a 'different medium' as far as they were concerned).

I've read (informally) discussions of the Lorentz transformations as simply a consequence of "causality". I don't really know all that's involved with that, but I've also heard that they're a consequence of "locality" and violating them doesn't necessarily mean violating causality.

If there is no finite propagation speed limit, then there is no sense of "locality" since anything can affect anything else in the universe instantaneously. With a finite propagation speed limit, there becomes a coordinate system independent method of ordering 'causal' events. If event A is inside the 'history" of things that could effect event B, then everyone will agree on that. If event A is "outside" the history of B, and similarly B for A, then there can not be any causal relationship.

So yes, many of these things are closely related.
Bohmian mechanics is a whole other bizarre issue though. It is essentially many worlds interpretation but claiming that only one of those paths are real. Since it doesn't provide any way to experimentally distinguish this, and throws out locality in the underlying laws as well as postulating other state variables ... it in some sense in the 'aether' of quantum mechanics. Sure, it is equivalent, but you have to add in extra things that we can't experimentally access.