Boost & Skewing Axes on a Spacetime Diagram

Source

Collection: Wiki SR simulations (v1) — raw sim boost-skewing-axes-v2.
Concept: Under a Lorentz boost the moving frame's ct′ and x′ axes do not rotate rigidly — they scissor toward the 45° light line, each tilting from its rest axis by arctan(β).
Serves: Geometric intuition that a boost preserves the light line (and the invariant hyperbola), not right angles — so the axes skew rather than turn.
Inquiry: How do ct′ and x′ move as β grows? Do they stay perpendicular? What stays invariant?
Misconception: That the axes rotate rigidly together (staying perpendicular) like a Euclidean rotation, or that only the labels change.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "rigid rotation" and "only labels change" errors are wrong prediction choices; the rebuttal appears only if the student picks one. An optional red overlay shows how the wrong rigid rotation would swing x′.
D6/D7	Guided inquiry reveals in stages	Sidebar step-cards run predict → observe → verify → resolve, driving β directly so the axes visibly scissor toward the light line.
J	Invariant panels are scoped to invariance sims	The lesson is what a boost preserves, so the invariant hyperbola and the closing angle between ct′ and x′ are shown; 90° would mean a (wrong) rotation.
A2	Fewest objects	Opens with the rest axes, light line and one hyperbola; the wrong-rotation overlay is opt-in, not standing.
Physics honesty	No auto-swept velocity	β is a static slider the student sets; the diagram updates live on drag. A ramping β would imply an accelerating (non-inertial) frame, so it is not animated in time.

Boost & Skewing Axes

Speed

Minkowski diagramrest frame: horizontal x, vertical ct

rest axes x, ct boosted x′, ct′ light line (45°) invariant hyperbola rigid-rotation x′ (wrong)

Guided Inquiry

1 · Predict how the axes move

Set the boost speed β above 0 with the slider. As β increases from 0, how do the moving frame's ct′ and x′ axes move relative to the rest-frame axes?

2 · Watch the axes skew

Drag the β slider up from 0. The time axis ct′ (worldline x = βct) and the space axis x′ (line ct = βx) each tilt from their rest axes by arctan(β), both leaning toward the gold 45° light line. Keep an eye on the angle between ct′ and x′ readout.

3 · Do they stay perpendicular?

As β grew, the angle between the ct′ and x′ axes …

4 · Compare with a rigid rotation

Tick Show the WRONG rigid-rotation version in the controls. The red x′ swings the opposite way (staying perpendicular to ct′) — clearly not what the boost does. A rotation preserves right angles and the circle x² + ct²; a boost preserves the light line and the hyperbola x² − ct² instead.

5 · What is invariant

The gold light line stays pinned at 45° and every frame's unit tick lands on the same hyperbola x² − ct² = 1. A boost is a hyperbolic "rotation" by the rapidity φ = artanh(β), using cosh φ and sinh φ — which is why the axes shear toward the light line rather than turning rigidly. Turn on ∑ Formal for the algebra, or hit Finish to explore freely.

Controls

▾

β 0.60c

00.99c

Wrong rigid x′

Readouts

▾

boost speed β—

axis skew arctan(β)—

rapidity φ = artanh(β)—

Lorentz factor γ—

angle between ct′ and x′—

Lorentz boost (c = 1)

The ct′ and x′ axes

Axis skew toward the light line

Hyperbolic rotation by rapidity φ