Relativistic Velocity Addition — Spacetime & Rapidity Diagram

Ship Velocity (v)

Relative to station 0.60c

Probe Velocity (u′)

Fired forward, relative to ship 0.60c

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Time scale 1.0×

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Newtonian sum v+u′

1.20c

Relativistic sum

0.88c

Rapidity φv + φu′

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tanh(φ total)

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Elapsed station time

0.0 s

Gap: ghost vs light

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How it works

This is the same physics as the 3D version, drawn a completely different way: instead of animating objects moving through simulated 3D space, it plots their worldlines directly on a Minkowski spacetime diagram — horizontal axis = position x (light-seconds), vertical axis = time t (seconds), with the station sitting still at x = 0.

Naive Newtonian intuition says velocities just add: v + u′. Einstein's velocity-addition law, derived from the Lorentz transformation, replaces it with:

u = (v + u′) / (1 + v·u′/c²)

The lower panel plots the equivalent statement in rapidity φ = artanh(v/c) — the hyperbolic angle of a boost. Rapidities add exactly like ordinary numbers, φ = φv + φu′, and only get converted back to a velocity at the very end via u = c·tanh(φ). Because tanh saturates at 1, no sum of finite rapidities can ever reach c — the speed limit falls straight out of the shape of the function, with no extra rule bolted on.

Cyan line — the ship's worldline, slope x = v·t; this line is the ship's own time axis t′ in the boosted frame.
Magenta line — the probe's worldline using the correct relativistic sum u; always inside the light cone.
Dashed red line — the Newtonian "ghost" worldline using v + u′; at high v and u′ it visibly crosses outside the light cone, which is the whole point: that region of the diagram is causally forbidden.
Cyan diagonals — the light cone, x = ±c·t; nothing massive may have a worldline flatter than this.
Dotted amber line — the ship's line of simultaneity (t = v·x/c²), showing why events the ship considers simultaneous are not simultaneous in the station's frame.
Rapidity strip — a plain number line in φ; φv, then φu′ appended nose-to-tail from it, land at φ total. Converting that point back through tanh reproduces the exact same relativistic sum shown above — a second, independent route to the identical number.

Real-world relevance: this is the exact correction used to compose successive boosts in particle accelerators, GPS timing and relativistic astrophysical jets — the rapidity picture is precisely why physicists prefer to add boosts as rapidities rather than as raw velocities.

Ship worldline (v) Probe (relativistic sum) Newtonian ghost (v+u′) Light cone (c)