Two observers in relative motion do not share a clock or a ruler — and yet they agree on one number. The boost x′=γ(x−vt), t′=γ(t−vx/c²) re-mixes space into time and time into space, but it leaves the interval (ct)²−x² untouched. Down the center an event goes in, the boost re-mixes it, the invariant comes out. The blue team builds and defends it; the red team tries to break it.
source Einstein, Zur Elektrodynamik bewegter Körper (1905), Ann. Phys. 17, 891–921 — doi:10.1002/andp.19053221004; after Lorentz (1904), geometrized by Minkowski (1908). Rendered, not quoted.
Working in units where c=1 (space and time both measured in light-seconds), the transform is a hyperbolic rotation:
x′ = γ(x − βt), t′ = γ(t − βx), with β=v/c and γ=1/√(1−β²) ≥ 1.
For the current boost, the live numbers:
| quantity | value |
|---|
γ → ∞ as β → 1: no massive frame reaches light speed.
The boost is not arbitrary re-mixing: it is exactly the family of linear maps that fixes (ct)²−x². That one preserved quantity is the neighbour sphere the-spacetime-interval.
From it, two consequences fall out with no extra assumption: moving clocks run slow by dt=γ dτ (the-time-dilation) and moving rulers shorten by L=L₀/γ (the-length-contraction). Each sphere is the next one’s premise. Lorentz 1904 → Einstein 1905 → Minkowski 1908.
The blue team’s live check: re-run the boost over a deterministic sweep of events and confirm the interval is unchanged and light stays at c. If red swaps in the Galilean transform, this badge is where it shows.
Feed the engine one event — a place-and-time (x, t) in the rest frame — and one relative velocity β=v/c for the moving observer. The engine returns the same event’s coordinates (x′, t′) in the moving frame, plus the interval both observers must agree on.
Two events on a light ray satisfy x=±ct; those are the walls of the light cone, and the boost must slide events along the cone, never across it.
The transform is applied live from γ(x−βt), γ(t−βx) — never looked up. The diagram shows the boosted x′ / t′ axes tilting toward the light cone.
Move β or the event — x′, t′ and the interval are recomputed on the spot from the four expressions above.
What the machine produces, proven: for every event and every boost, (ct′)²−x′² = (ct)²−x² to 1e-9, a photon worldline x=ct maps to x′=ct′, and composing v with −v returns the original event to 1e-12. The current event’s two intervals are shown above; their equality is the output.
The blue team’s witness (left) confirms this live; the red team (right) tries to make it false.
It is also strictly 1+1 here (motion along x). The full transform mixes only the boost direction; transverse coordinates y,z are untouched, and a general boost + rotation lives in the six-parameter Lorentz group. The panel renders the load-bearing 1+1 slice, not the whole group.
“Moving clocks slow because motion is absolute.” Cut. There is no preferred frame — each observer sees the other clock run slow. The symmetry is real; the resolution is the relativity of simultaneity, built into t′=γ(t−βx).
“Length contraction is an optical illusion.” Cut. It is a real coordinate fact, L=L₀/γ; what you see (Penrose–Terrell rotation) is a separate effect and differs from what you measure.
“Lorentz derived it first, so it is his.” Kept, corrected. Lorentz (1904) had the algebra as a contraction of the ether; Einstein (1905) removed the ether and made it kinematics. Both names, honestly.
The red team’s move: swap the Lorentz boost for the pre-1905 Galilean one — x′=x−vt, t′=t; drop γ and the time-mixing. The blue team’s witness (window 7) is watching.
Under Galilean addition the interval is no longer invariant and light changes speed between frames — the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.