Minkowski Boost ↔ Euclidean Rotation

Source

Collection: Wiki SR simulations (v1) — converted from the raw engine sim lorentz-boost-hyperbolic-rotation.
Concept: A Lorentz boost is the hyperbolic analogue of a Euclidean rotation: cosh/sinh replace cos/sin, and it preserves s² = (cΔt)² − (Δx)² the way a rotation preserves r² = X² + Y².
Serves: Side-by-side geometry: the same event runs along an unbounded invariant hyperbola under a boost, versus a bounded invariant circle under a rotation, both driven by the one angle variable φ.
Inquiry: What is the boost's analog "angle"? How big does it get as β → 1? Why does the hyperbola run to infinity while the circle only goes around once?
Misconception: That the boost angle is atan(β) (which saturates below 90°), rather than the rapidity φ = atanh(β), which grows without bound.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The atan(β) mix-up and the "90° ceiling" error are wrong prediction choices; the rebuttal appears only if the student picks one.
D6/D7	Guided inquiry reveals in stages	Five sidebar step-cards run fundamental → full (predict the analog angle → observe the hyperbola vs circle → verify the interval → introduce atanh vs atan → resolve why the hyperbola is unbounded).
J	Invariant panels are scoped to invariance sims	This sim's lesson IS invariance, so both the hyperbolic interval s² and the circular r² invariant readouts are shown.
A2	Fewest objects	One event / point, its transform, and each invariant curve — no vestigial counts no step reads.
C	Every number earns its place	β, φ, γ, both coordinate pairs and both invariants are each read or compared by a step; the atan(β) row exists precisely to be contrasted with φ.

Boost ↔ Rotation

Speed

Minkowski boost — hyperbolathe event runs to ∞ as β → 1

Invariant: s² = (cΔt)² − (Δx)² = —

Euclidean rotation — circlethe point stays on a bounded circle

original event / point transformed by φ invariant curve φ swept light line ct=±x

Invariant: r² = X² + Y² = —

Guided Inquiry

1 · Predict the analog angle

A boost is a "rotation" by an analog angle set by β. At β = 0.99, roughly how big is that analog rotation angle φ? (You can just drag β and check.)

2 · Watch both "rotations"

Drag the β slider. On the left the event slides along its invariant hyperbola; on the right the same coordinates ride a bounded circle. The pink arc is the φ sweep in each geometry. Notice one runs off toward infinity while the other just goes around.

3 · What stays invariant?

As β changes, the event's individual coordinates (cΔt′, Δx′) move. What about the interval s² = (cΔt)² − (Δx)² on the left?

4 · atanh(β), not atan(β)

The circle's angle is bounded by 2π; the boost's rapidity is φ = atanh(β). Compare the "atan(β)" row with the "rapidity φ" row in the readouts. As β → 1, which one runs off to infinity?

5 · Why the hyperbola is unbounded

Because atanh(β) → ∞ as β → 1, the boost "angle" is unbounded — the hyperbola runs off to infinity while the circle can only ever go around once. Same matrix structure, one sign and cosh/sinh vs cos/sin apart. Turn on ∑ Formal for the algebra, or use the event presets to make the interval timelike, spacelike, or lightlike. Hit Finish to explore freely.

Controls

▾

β0.60

−0.99+0.99

φ—

ct₀2.0

0.53.0

x₀1.0

−3+3

Presets

Live quantities c = 1, natural units

▾

β—

γ = cosh φ = 1/√(1−β²)—

rapidity φ = atanh(β)—

φ in degrees—

event E₀ = (cΔt₀, Δx₀)—

transformed E′ = (cΔt′, Δx′)—

interval s² = (cΔt)²−(Δx)²—

common error: atan(β) ≠ φ—

—

The two transformations are structurally identical — swap the trig functions and one sign. Both are driven by the same angle variable φ; for the boost that variable is the rapidity, and it is atanh(β), not atan(β).

Minkowski boost (cosh/sinh)

Euclidean rotation (cos/sin)

Invariants preserved

Rapidity φ

Because atanh(β) → ∞ as β → 1, the boost "angle" is unbounded — that is why the hyperbola runs off to infinity while the circle can only ever go around once.