Why classical physics breaks

Galilean addition can’t survive a constant $c$ .

Take the simplest classical rule: if you’re in a moving train and you throw a ball forward, its speed in the ground frame is your speed plus the ball’s speed in the train. Postulate 2 says light disobeys that rule. Watch what happens.

The thought experiment

A spaceship moves through the lab at velocity $v$ . From inside the ship, the captain fires a photon directly forward. In the captain’s frame, the photon moves at $c$ (postulate 2).

Question: what speed does the lab observer measure for the photon?

Galilean intuition shouts $c + v$ . Postulate 2 insists on plain $c$ . They cannot both be right.

Spaceship velocity v / c β = 0.60 γ = 1.250

The numbers

Rule	Lab-frame photon speed	Verdict
Galilean: $u + v$	1.600 c	contradicts c
Einstein: $\frac{u + v}{1 + uv / c ^{2}}$	1.000 c	obeys c

With the photon’s ship-frame speed $u = c$ (i.e. $u / c = 1$ ), the Einstein rule reduces algebraically to exactly $c$ for any $v$ . That is the postulate, baked into the velocity-addition law.

So we have to redo the transformations

The Galilean transform that produced $u + v$ is:

x^{'} = x - v t, t^{'} = t .

Time is universal; positions just shift. Differentiate the first and you get $u^{'} = u - v$ , equivalently $u = u^{'} + v$ . Plug a photon in ( $u^{'} = c$ ) and you get the offending $u = c + v$ .

To fix this we look for a linear transform that (a) reduces to Galilean at low speed, (b) treats the two frames symmetrically, and (c) sends a light pulse to a light pulse — i.e. preserves $x = c t$ and $x^{'} = c t^{'}$ :

x^{'} = γ (x - v t), t^{'} = γ (t - v x / c^{2}) .

Demand both $x = c t$ and $x^{'} = c t^{'}$ hold simultaneously. Substitute the first into the right-hand side of each equation and divide:

\frac{x ^{'}}{t ^{'}} = \frac{x - v t}{t - v x / c ^{2}} =! c .

Set $x = c t$ on the right-hand side, simplify, and solve for the unknown scaling factor $γ$ :

γ = \frac{1}{1 - v ^{2} / c ^{2}}

At your current setting, $v = 0.60 c$ ⇒ $γ = 1.250$ . Note that γ → 1 as v → 0 (Galilean limit), and γ → ∞ as v → c.

The cost of saving postulate 2

The replacement transform — the Lorentz transform — keeps $c$ sacred. But the price is steep:

Time is no longer universal. The $t^{'} = γ (t - v x / c^{2})$ piece mixes $x$ into $t^{'}$ . Different observers carve spacetime into different "now"s.
Lengths along motion contract. A rod with proper length $L_{0}$ shows up as $L_{0} / γ$ in any frame in which it’s moving.
Clocks moving relative to you tick slow by the same factor: proper time $Δ τ$ stretches to $γ Δ τ$ in your frame.

These are not optional decorations. They are forced on us by the act of rescuing the constancy of $c$ . The next three lessons walk through each one in turn.

What if we kept Galilean?

If you refused to give up Galilean addition, then either:

Light’s speed must depend on the source’s motion (which would make $c + v$ the lab-frame answer above) — directly contradicting decades of experiments from Michelson–Morley onward.
Or there must be a privileged "ether" rest frame — but every attempt to detect motion through it has come up null.

Postulate 2 isn’t arbitrary. It’s what experiment kept handing back. Lorentz wrote down the math; Einstein supplied the framing.