Two observers flying past each other disagree on time and disagree on distance — but they agree on one number: s² = (cΔt)² − Δx². It is the same in every inertial frame, and its sign is the law of cause: it splits the future you can reach from the elsewhere you never can. Down the center the two events go in, the metric decides, the invariant comes out. The blue team builds and defends it; the red team tries to break it.
source H. Minkowski, Raum und Zeit, address of 21 Sep 1908, Cologne; Physikalische Zeitschrift 10 (1909) 75–88 — Wikisource: Space and Time. Rendered, not quoted.
The separation of two events is not measured by a ruler or a clock alone but by the Minkowski metric (units c=1):
L1 Invariance — s² = t² − x² is unchanged by any boost. L2 Classification — s²>0 timelike (reachable), s²<0 spacelike (elsewhere), s²=0 lightlike (a photon connects them). L3 Proper time — along a timelike interval a clock reads τ = s/c. L4 Causality — for s²≥0 no boost reverses the time order.
For the current event pair, live:
| quantity | value |
|---|
Minkowski, 1908: give spacetime a metric with signature (+,−,−,−) and relativity stops being strange. The frames disagree; the interval does not. The light cone, s²=0, splits past from elsewhere.
This is precisely the quantity that the-lorentz-transformation is built to preserve: a boost is just a hyperbolic rotation that slides an event along its own hyperbola of constant s². Each sphere is the next one's premise.
The blue team's live check: re-boost fixed events across a range of velocities and confirm the interval, the classification, and causality against the metric's law. If red tampers, this badge is where it shows.
The machine takes a separation between two events in one frame: a time gap Δt and a space gap Δx (measured along the boost axis, c=1 so time and distance share a unit). Every observer measures a different Δt and Δx — that is the whole point of relativity.
| separation | meaning | agreed? |
|---|---|---|
| Δt | time between events | frame-dependent |
| Δx | distance between events | frame-dependent |
| s² | the interval | INVARIANT |
Feed Δt and Δx into the panel below; the invariant is what survives.
Move the boost slider: Δt and Δx both change in the new frame, yet s² below does not budge.
The diagram: 45° light cone, the invariant hyperbola through the event, the white event and the boosted event — both riding the same hyperbola. Every number is computed from the metric on the spot, never looked up.
What the machine produces, proven: a single number s² the same in every frame, its sign fixing the causal class, and — when timelike — the proper time τ = s/c a real clock would read. The current pair's verdict is above; the invariant is the output.
The blue team's witness (left) confirms invariance and causality live; the red team (right) tries to make the metric lie.
The signature itself is a convention: (+,−,−,−) here flips to (−,+,+,+) in many texts, and s² changes sign with it — only the classification is physical, not the bare number. Accelerated (non-inertial) frames need the full apparatus; the interval is the first true statement, not the last.
"Moving clocks just run slow — it's an illusion of measurement." Cut. Proper time τ=s/c is frame-independent and physical; the twin who accelerates really is younger. The interval is what makes that objective.
"Nothing about the past and future is absolute in relativity." Cut. For timelike/lightlike pairs the time order is absolute — causality is preserved in every frame. Only spacelike pairs can be reordered.
"The interval is just the Pythagorean distance in 4D." Kept, corrected. It has the form of a metric but the crucial minus sign — time enters with the opposite sign to space. That one sign is the whole of causality.
The red team's move: swap the metric's minus for a plus — make spacetime Euclidean, s² = t² + x². Now boosts no longer preserve it and the light cone no longer separates the regions. The blue team's witness (window 7) is watching.
Flip the sign and the interval stops being invariant; the witness re-boosts the fixed events, finds the number changing frame-to-frame, and turns red. Nothing is faked; the attack is real and it is caught.