At a glance
The sim puts a Euclidean rotation of the x–y plane next to a Lorentz boost of the ct–x plane, with c = 1. Both change coordinates by turning a pair of axes, and setting them side by side shows what the two have in common and where they part.
What you'll learn
- What it means to relabel one fixed point with a second set of axes.
- How each transformation changes the angle between its primed axes.
- Which combination of coordinates each one leaves unchanged.
- How v/c, γ and the rapidity φ are related.
Each panel starts with its primed axes lying on the unprimed ones (θ = 0, v/c = 0).
What's on screen
- Unprimed axes (grey): x and y on the left, x and ct on the right, with unit ticks at 1 and 2.
- Primed axes (amber): x′ and y′ on the left, x′ and ct′ on the right.
- Light cone (dashed yellow, right panel): the lines ct = ±x.
- Sample points P₁, P₂ (left) and events E₁, E₂ (right), with dashed lines down to the primed axes.
- Invariant curves (teal) and the amber primed grid and ticks, when their buttons are on.
- Strip under each panel: the points in both coordinate systems, an invariant row, and the angle between the primed axes.
See itReturn both panels to their start, with the sample points and strips showing. Watch for: the amber axes lying exactly on the grey ones, and the strips listing P₁, P₂ and E₁, E₂.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| θ (Euclidean rotation) | The angle through which the x′ and y′ axes are turned in the left panel. | 0°–80° opens at 0° | |
| v/c (Lorentz boost) | The speed of the primed frame as a fraction of c, in the right panel. Its readout also shows γ. | 0–0.80 opens at 0 | |
| ⌗ Primed grid + ticks | Draws the primed grid lines and unit ticks 1 and 2 on each primed axis, in both panels. | on / off opens off | |
| ◯ Invariant curves | Draws the unit circle on the left and the unit hyperbolae on the right. With the grid on, the n = 2 curves appear too, fainter. | on / off opens off | |
| ▦ Readouts | Shows or hides the strip under each panel. | on / off opens on | |
| ◈ Sample events | Shows or hides P₁, P₂ and E₁, E₂ with their dashed projections. | on / off opens on | |
| Read the symbols | A key to θ, γ and the colours used in the panels. | panel opens folded | |
| ↻ Reset | Sets θ and v/c back to 0, turns the grid and curves off, turns the readouts and events on, and clears the Guided Inquiry answers. | top bar |
Fixed: c = 1 in the right panel, so ct and x share one scale. Nothing moves on its own, so Play, Pause and Speed change nothing on screen.
A boost tilts both axes toward the light line
Both panels relabel one fixed point with a second set of axes; only the axes move. Take c = 1 and write β = v/c. The primed frame moves at v along +x relative to the unprimed one.
Rotation: x′ = x cos θ + y sin θ, y′ = −x sin θ + y cos θ
Boost: ct′ = γ(ct − βx), x′ = γ(x − βct), γ = 1/√(1 − β²)
The sim at θ = 39° and v/c = 0.80. Angles are measured on the diagram, with ct and x on equal scales.
- Left. x′ and y′ turn the same way through θ, so ∠(x′, y′) = 90° at every θ.
- Right. The ct′ axis is the line x = βct and the x′ axis is the line ct = βx. Each tilts by arctan β, ct′ toward x and x′ toward ct, so the pair closes on the light line: ∠(ct′, x′) = 90° − 2 arctan β.
- At v/c = 0.80 each axis tilts 38.7° and ∠(ct′, x′) = 12.7°.
- The x′ axis is every event with ct′ = 0: the events the primed observer calls simultaneous with the origin.
See itSet θ = 39° and v/c = 0.80, with the readouts on. Watch for: ∠(x′, y′) = 90.0° under the left panel and ∠(ct′, x′) = 12.7° under the right, with the two amber axes on the right leaning together.
Each transformation preserves one combination
A fixed point gets new coordinates, but one combination of them stays the same. A rotation keeps x² + y², so the point lies on a circle. A boost keeps −(ct)² + x², so the event lies on a hyperbola.
One sign separates the circle from the hyperbola.
- The interval. The sim writes it as s² = −(ct)² + x²: negative inside the light cone (timelike), positive outside (spacelike).
- Hyperbola. −(ct)² + x² = −1 has the light lines as asymptotes. E₂ sits at (x, ct) = (0.89, 1.34) on it, and its invariant row reads −1.00 at every v/c. The sim also draws −(ct)² + x² = +1.
- Ticks. A primed unit tick sits where its axis meets its curve. On the circle it stays 1 from the origin. On the hyperbola it moves outward: at v/c = 0.80 it is γ√(1 + β²) = 2.13 from the origin on the page.
See itDraw the invariant curves, the primed grid and the ticks at θ = 39° and v/c = 0.80. Watch for: a teal circle on the left and teal hyperbolae on the right, each amber tick on its curve, and both invariant rows reading 1.00 → 1.00 on the left and −1.00 → −1.00 on the right.
The axes close without reaching the light line
The angle between the boosted axes follows arctan, not a straight line. From v/c = 0.40 to 0.80 it falls from 46.4° to 12.7°, while γ rises from 1.091 to 1.667.
θ
v/c
These are the sim's own θ and v/c sliders.
- Rapidity. The parameter φ is defined by v/c = tanh φ, and then γ = cosh φ. The panel above shows φ = artanh(v/c).
- The slider stops at v/c = 0.80, where φ = 1.10 and each axis is still 6.3° from the light line.
- The angle reaches 0° only at v/c = 1. Since tanh φ stays below 1 at every finite φ, the axes approach the light line and never lie on it.
v/c = tanh φ. The slider stops at the first marked point.
See itRun v/c = 0.40 and then v/c = 0.80, one after the other. Watch for: ∠(ct′, x′) reading 46.4° and then 12.7°, which is not half, with the axes at 0.80 still apart from the light line.
A boost is a rotation through the rapidity
Put γ = cosh φ and β = tanh φ. The boost matrix then has the form of a rotation, with cosh and sinh in place of cos and sin. Lecture 10 calls this parameter ζ; the sim and this sheet call it φ.
The sim's boost matrix at c = 1.
- Rapidities add. Two boosts in the same direction make one boost of φ₁ + φ₂, as two rotations add their angles. Two boosts of v/c = 0.50 have φ = 0.549 each, so φ = 1.099 and v/c = tanh 1.099 = 0.80, not 1.00.
- Light-cone form. A boost only rescales ct + x by e−φ and ct − x by e+φ, and the two factors cancel in −(ct)² + x².
- No period. θ repeats every 360°, but cosh and sinh never repeat, so a boost never returns the axes to where they started.
- E₂ is at rapidity 0.80, (x, ct) = (sinh 0.80, cosh 0.80). A boost φ gives it rapidity 0.80 − φ, so at v/c = tanh 0.80 = 0.664 it lies on the ct′ axis.
The boost in light-cone coordinates.
See itSet v/c = 0.66, the slider step nearest 0.664, with the events on. Watch for: E₂'s boosted coordinates reading about (x′, ct′) = (0.01, 1.00), so E₂ sits on the ct′ axis.
Where it matters
- Everyday speeds. An airliner at 250 m/s has v/c = 8.3 × 10⁻⁷. Each axis tilts by 0.17 arcsec and γ − 1 = 3.5 × 10⁻¹³, so the axes look square.
- Cosmic-ray muons at v/c ≈ 0.998 have γ ≈ 15.8 and φ ≈ 3.45, far past this slider. In the ground frame γ stretches their lifetime, which is why many reach the ground.
- LHC protons at 6.5 TeV have γ ≈ 6930 and φ ≈ 9.5. ∠(ct′, x′) is about 0.002 arcsec, and the protons still move slower than light.
Go deeper: the low-speed limit
- For v/c ≪ 1, γ ≈ 1 and the boost becomes x′ ≈ x − vt and t′ ≈ t − vx/c². The second term is the only trace of relativity, and it is negligible at everyday speeds.
Summary
In one line. A rotation keeps x² + y² and a right angle between its axes; a boost keeps −(ct)² + x² and closes its axes toward the light line without reaching it.
Key points
- Both panels relabel a fixed point. The rotation keeps ∠(x′, y′) = 90° and x² + y².
- The boost tilts ct′ and x′ by arctan(v/c) each, so ∠(ct′, x′) = 90° − 2 arctan(v/c), and it keeps −(ct)² + x².
- γ = cosh φ and v/c = tanh φ. Rapidities add, and v/c stays below 1.
Common mix-ups
- "A boost is a rotation, so the primed axes stay perpendicular." Only ∠(x′, y′) stays 90°. On the page the boosted axes close up, and they remain orthogonal only in the spacetime sense that carries the minus sign.
- "Only ct′ tilts." Both axes tilt by arctan(v/c). The tilt of x′ is the frame's own simultaneity.
- "Push v/c high enough and the axes land on the light line." v/c = tanh φ stays below 1, so they only approach it.
Things to try on your own
- Turn on the grid and the invariant curves, raise v/c from 0 to 0.80, and follow the amber tick on ct′ along its hyperbola.
- Work out by hand the v/c at which ∠(ct′, x′) = 45° (√2 − 1 ≈ 0.41), then set the slider and read the strip.
- At v/c = 0.50, compute γ and φ by hand (1.155, 0.549) and check them against the live panel in section 05.