This is not the first time that I have tried to explain how to get from very basic (Galilean) relativity (the si = so + vt sort of thing) to special relativity. What I have done here, however, is try to boil it down to one page that could conceivably be printed out and pored over. Here goes:
So, what I am wondering is – does this make any sense to anyone other than me?
I should, perhaps, explain some things.
First, priming notation in the uppermost box. The prime indicates that the quantities in question are not fixed, the distance (x') between M and the little cloud (notionally an explosion at some location) changes with time (t').
This notation is carried through to the second box. In this box, however, it's not necessarily the case that we know the location of the explosion. Without external assistance, we can't immediately know where it happened without knowing when it happened, and we can't immediately know when it happened without knowing where it happened. Therefore, rather than consider the explosion per se, we consider the photons from the explosion at the moment that S and M were collocated.
In the third box, it becomes clear that prime notation is relative. For S, primed means "in motion relative to me" but for M, primed means "stationary relative to me". Another way to think of it is that x' is always the distance between M and the event and t' is always the time taken for a photon to reach M from the event, while x and t relate to similar values associated with S, irrespective of which is moving (S or M).
Finally, in the lowermost box, I mention "inequalities". I think that this is pretty obvious, but the point is that if S is in motion away from location at which S and M met, then it will take more time for those photons to reach S than it would if S were stationary. In each case the other's measure of an invariant distance is one's own measure multiplied by some factor (which we go on to calculate).
The remainder is, I believe, pretty self-explanatory.
A further step one can take is to simply divide the resultant equation by c. This gives us:
x'/c = (x/c - vt/c)/√(1 - v2/c2)
And, since x = ct and x' = ct'
t' = (t - v/c2.x)/√(1 - v2/c2)
This is the standard Lorentz Transformation in the time domain.
It's all quite simple really.
It might be worth noting that, in the process above, I make no assumptions or claims about the value or nature of c. We do know that light from an explosion consists of photons that travel at some speed. We could even consider one photon that passes M and continues on to S rather than two separate photons. There's nothing in the scenario which would drive a change in speed of the photon. We don't need to know more than this to arrive at our equations and the odd features of relativity (like photons always travelling in vacuo at c compared to individual observers irrespective of any motion they might have relative to other observers and having two spaceships travel towards the other at 0.75c each [relative to a third observer] but at 0.96c combined [relative to each other, rather than 1.5c]) fall out of those equations. The value of c becomes (in this derivation) merely something that needs to be measured.
Interestingly, it was Galileo of Galilean relativity fame who is credited with first attempting to measure the speed of light in 1638. He concluded that it was very fast, if not instantaneous and gave it a figure of at least 10 times the speed of sound (so therefore at least 3403 m/s). Given that he had concluded that light has a speed, Galileo could have conceivably arrived at special relativity with no more than the information that he had to hand. To be fair to him, when he attempted to measure the speed of light Galileo has been under house arrest for five years, having been found guilty of heresy for his support of Copernicus, and he died four years later. And even if he had arrived at special relativity, such an idea may have been considered too heretical for him to pursue with the church looking over his shoulder.
I thought it might also be worth mentioning the last step, in which I drop the subscripts. I only do this to arrive at the standard equations. I would keep the subscripts, if that were permitted, because I think they make some things clearer. For example, it might not be immediately obvious by (4) and (5) are actually the equations for length contraction.
Consider xS and xM. The former, xS is a measurement made by S who is considered by M to be in motion. What is xS in terms of xM? This is given by (4), xM = γ.xS so xS = xM/γ. Using standard length contraction terms, this is L = L0/γ (occasionally and, rather confusingly, written as Δx' = Δx/γ – although this usage appears to have declined over the past decade or so).