Einstein was a big fan of deductive reasoning, meaning his theories are derived based on some first principles, rather than refining it to fit experimental data. The main principles from which he derived special relativity, were the following postulates:

• The principle of relativity: the only thing distinguishing two inertial frames is their relative velocity $v$.
• Light postulate: light travels with speed $c=1$ in all inertial frames (as follows from Maxwell’s equations; if they did not take the same form in all inertial reference frames, it would violate the relativity principle).

Let’s try and follow his footsteps, without relying on any fancy modern mathematics.

We start from a basic relativistic scenario that Newton would recognize: we fix a stationary inertial frame $S$ with coordinates $(x,t)$. An observer Alice carries another inertial frame $S'$ with coordinates $(x',t')$, moving at a constant relative speed $v\neq0$ along the common $x$-axis. We adopt the standard configuration, meaning the spatial axes are parallel, the motion is along $x$, and the origins coincide at $t=t'=0$.

In $S$, the worldline of Alice’s spatial origin is $x=vt$. In her own frame $S'$, Alice sits at rest at her origin, so $x'=0$. We thus have

$$ x=vt,\quad x'=0. $$

Working entirely within a Newtonian/Galilean context for now, we can rewrite $x=vt\iff x-vt=0$, identify this with $x'$, and we then obtain the Galilean boost with absolute time

$$ x'=x-vt,\quad t'=t. $$

Now, let’s see what happens when we invoke our second postulate (the light postulate): in $S$, a freely propagating light pulse has velocity $u=\dot{x}=\pm 1$. Under the Galilean transformation, the velocity of the same light pulse in $S'$ is

$$ u'=\frac{dx'}{dt'}=\frac{d}{dt}(x-vt)=u-v. $$

Demanding that a light pulse still has speed $\pm 1$ in $S'$ would give $u'=\pm1=u-v$. For $u=+1$, this demands $1=1-v$, which only holds when $v=0$. Similarly, for $u=-1$, we see $-1=-1-v$, which again forces $v=0$. Both contradict our assumption that $v\neq0$.

Equivalently, in $S$ a light worldline satisfies $x=\pm t$, while in $S'$ it must satisfy $x'=\pm t'$. With $x'=x-vt$ and $t'=t$, a ray with $x=t$ would obey $x'=t-vt=(1-v)t\neq t=t'$ for $v\neq 0$, and is thus doesn’t satisfy $x'=\pm t'$, as required. Hence, absolute time is incompatible with the light postulate. This is troubling, because it means our Galilean transformation laws do not hold when we assume the light postulate. One must go. Do we throw out the light postulate, or the assumption that time is absolute, together with our nice Galilean transformations?

Einstein’s decision here was influenced by both his exposure to philosophy, but also circumstance. As a young man, he had read Mach, Hume, and Kant, and became convinced of the idea that physics should never rely on metaphysical structures that can’t be observed. The principle of relativity was strongly supported in mechanics, and the constancy of light came directly from Maxwell’s equations, which Einstein considered too beautiful and successful to abandon. Absolute simultaneity, by contrast, was not an empirical fact but an inherited Newtonian convention, based on metaphysical convictions rather than observational fact.

Meanwhile, at the Patent Office in Bern, Einstein was reviewing endless inventions and methods for synchronizing clocks over long distances, which was crucial for keeping Europe’s rapidly expanding railway networks on schedule. This was becoming an important engineering challenge at the time. He saw schemes for using telegraphy and electrical signals to make sure the clocks in train stations matched, so that timetables could be trusted. That environment naturally made him think: how do you actually define “synchronized” clocks when they are far apart?

This is where a series of thought experiments came in. He imagined sending light signals between clocks, observers on moving trains, or lightning strikes hitting both ends of a train at once. Each time he played through the scenarios, the same problem emerged: if you take the speed of light to be the same in all directions and reference frames, then different observers will disagree about whether two distant events happened at the same time.

By the summer of 1905, Einstein finally recognized the significance of these reflections. The operational definition of simultaneity he had been circling around, using light signals and assuming equal travel times, made it unavoidable: simultaneity depends on the state of motion of the observer. Absolute time had no place in physics. The way forward was to let go of Newton’s absolute simultaneity and allow time to transform nontrivially.

This means we must build a completely new set of transformation laws, starting from scratch.