That is, why did he use signals whose speed is invariant?
Because 99.99999% of the world we deal with involves electromagnetic interaction.
What else would he use?
One advantage in using light signals to synchronize clocks, is that the procedure is easy to analyze from any frame. But you don't have to use light to synchronize clocks--you can use any kind of signal, even sound if you wanted (and could arrange it). But then the analysis from different frames is complicated by the fact that speed of the signal is frame--and direction--dependent (and must be computed using the relativistic addition of velocities formula).
There's two reasons.
1) it agrees with experiment
2) Making the speed of light isotropic also makes "physics" (i.e. momentum) isotropic. Thus synchronizing clocks with light-beams also makes the momentum, p, of an object moving left to right with a velocity v equal and opposite to the momentum, p, of an object moving right to left with a velocity v.
Synchronizing clocks in a non-standard manner destroys this important relationship.
We measure most everything by EM signals, visible light being just one small subset of the EM spectrum. And for astronomical distances or very fast moving bodies, what else could be used besides EM? Even when you use a ruler, your eyes make the judgement via receipt of the light signals.
Before Einstein's era, everyone assumed that light's speed would add & subtract like billiard balls. But by Einstein's era, theory and evidence both were making a convincing argument that light's speed was invariant, that it did not add & subtract like material bodies do. That even though you and I are traveling at half the speed of light wrt each other, that the sole beam of light passing us both by must be recorded at speed c per each of us.
That said, the entire point of it was to determine how you and I could make predict the other's observations, while at the same time we both record speed c for the sole beam of light. No one needed to build a model for light acting as billiard balls, since that model had long existed. The new buzz of the day was invariant light speed, and so the race was on the determine how a universe could exist under such a context, while preserving Newtonian mechanics for everyday experience known to be true and accurate.
Well, Albert won the race. His insight was the realization that time's rate must slow down for others of relative motion. The higher the speed, the more the time rate contraction. Since x/t=c=X/T, then so too must distances contract given t<>T. That bodies do not contract in and of themselves in some unwaivering aether (Lorentz's view), but rather because of the way in which we measure the very dimensions of space & time.
The electromagnetic szar of the day was Henrich Lorentz who published his theory first. However about 6 months later in 1905, a 3rd class swiss patent clerk nobody trumped him and the szar ditched his own theory in favor of Einstein's. This took some 4 years to happen since no one understood Einstein's theory. Max Planck read Einstein's 1905 paper, understood it, and made him famous. The irony is that this leading physicist of the day (Planck) would eventually have many debates with Einstein over his own specialty area of quantum mechanics, over the completeness of quantum mechanics and the validity of quantum uncertainty and locality. It seems that Einstein was eventually proven wrong on these matters years later.
Because light speed is invariant, it is a cosmic speed limit for all motion. The perspectives of two observers of relative motion are related via the common reference of invariant light speed.
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