Lorentz Boost of an Event

Source

Collection: Wiki SR simulations — engine-generated raw sim spacetime-boost-hinge-v2, converted to the shared shell.
Concept: A Lorentz boost tilts the primed axes (ct′, x′) toward the 45° light line, yet the spacetime interval s² = (ct)² − x² of a fixed event is unchanged.
Serves: Reading an event's coordinates off the boosted (skewed) axes, and seeing that (ct′)² − x′² stays pinned at the rest-frame value 3.750 for every boost β.
Inquiry: As β grows, x′ and ct′ each swing a lot — what happens to the combination (ct′)² − x′²?
Misconception: That because the projected coordinates change so much, the interval must grow or shrink with the boost.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "interval grows / shrinks with β" errors are wrong prediction choices; the rebuttal appears only if the student picks one.
A1	Physics honesty — no false animation	The boost β is a static slider (SR frames are inertial); dragging relabels S→S′ and redraws the tilted axes. No auto-swept velocity, so Speed/Play are dropped.
J	Invariant panel is scoped to an invariance sim	This sim's lesson IS invariance, so the s′² interval readout is shown (gold) and classified timelike/spacelike/lightlike.
A2	Fewest objects	Opens with one fixed event at (ct=2, x=0.5) on its diagram; no vestigial counters.
C	Every number earns its place	β, γ, x′, ct′ and s′² are each read or compared by a step.

Minkowski diagram

ct, x (rest frame) ct′, x′ (boosted frame) light line (45°) s² = 3.75 hyperbola (invariant) the event

Guided Inquiry

1 · Predict the invariant

In the rest frame the event sits at (ct, x) = (2.000, 0.500), so s² = (ct)² − x² = 3.750. As you increase the boost β, what happens to the same quantity computed from the boosted coordinates, s′² = (ct′)² − x′²?

2 · Tilt the axes

Drag the β slider in the sidebar. The primed axes ct′ and x′ scissor symmetrically toward the 45° light line. The event stays put in the rest frame; only the tilted axes you read it against change. Keep an eye on the s² readout.

3 · What did the interval do?

As you swept β across its whole range, the boosted interval s′² = (ct′)² − x′² of the event …

4 · The invariant hyperbola

The gold curve is the set of events with s² = 3.750 — a Minkowski "circle". The event sits on it for every boost: each β simply relabels its coordinates while it stays glued to the curve. That preserved number, (ct)² − x², is the invariant interval. Turn on ∑ Formal (top bar) for the algebra, then hit Finish to explore freely.

Controls

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β = 0.60 · γ = 1.250

00.90

Readouts

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x′ (boosted frame)—

ct′ (boosted frame)—

s² = (ct′)² − x′²—

Lorentz boost (c = 1)

Lorentz factor γ

Invariant interval

A boost mixes t and x, so t′ and x′ each change with β — but the combination t² − x² is left unchanged. That preserved number is the invariant interval.