1. Setting the Stage

When Albert Einstein published his “miracle year” in 1905, he still treated space and time as separate entities. It was Hermann Minkowski (1908) who famously declared:

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

This “union” is the interval \(s\). We’ll see why it’s invariant—identical for all inertial observers—and how this forces us to redefine energy and momentum.

This interval also underpins fundamental concepts like mass–energy equivalence, explained in detail in Why 𝐸 = 𝑚𝑐².

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2. Brief History of the Interval

Cultural tidbit 🎸: while Minkowski revolutionized physics in 1908, the tango “El choclo” was sweeping Buenos Aires—another example of Latin American vanguard art and science.

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3. Lorentz Transformations in a Nutshell

For two inertial frames \(\mathcal{S}\) and \(\mathcal{S}'\) with relative velocity \(v\) along the \(x\)-axis:

$$ \begin{aligned} x' &= \gamma\,\bigl(x - vt\bigr) \\[4pt] t' &= \gamma\!\left(t - \tfrac{v\,x}{c^2}\right) \\[4pt] \gamma &= \frac{1}{\sqrt{1 - v^2/c^2}} \end{aligned} $$

Rewriting the interval in \(\mathcal{S}'\) gives: