Relativistic Velocity Addition

Ship Velocity (v)

Relative to station 0.60c

Probe Velocity (u′)

Fired forward, relative to ship 0.60c

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Newtonian sum v+u′

1.20c

Relativistic sum

0.88c

Elapsed station time

0.0 s

Gap: ghost vs light

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How it works

Naive Newtonian intuition says velocities just add: if a ship moves at speed v relative to a station and fires a probe forward at speed u′ relative to itself, the probe's speed relative to the station should be v + u′. That formula quietly assumes an absolute, shared notion of time — one Einstein's postulates rule out.

Special relativity replaces it with the velocity-addition law, derived directly from the Lorentz transformation:

u = (v + u′) / (1 + v·u′/c²)

Because the denominator grows whenever both v and u′ are positive, the relativistic sum u is always ≤ c, even as v → c and u′ → c — the formula caps itself out at exactly the speed of light. The Newtonian sum has no such limit: at v = u′ = 0.9c it predicts 1.8c, an outright violation.

Ship (cyan cone) — moves at v relative to the station, on a real, physically consistent trajectory.
Probe (magenta sphere) — fired from the ship at u′ in the ship's own frame; its station-frame trajectory uses the correct relativistic sum u, always trailing the expanding light front.
Newtonian ghost (dashed red sphere) — the same launch, but advanced using v + u′ instead. At high v and u′ it visibly overtakes the light front, which is the whole point: that outcome is physically impossible, and the relativistic law is what prevents it.
Expanding light front (thin cyan ring) — a pulse of light released from the station at t = 0, marking the true causal boundary no massive object can cross.

Real-world relevance: this exact correction is why particle accelerators, GPS timing, and relativistic jets from black holes and quasars never observe faster-than-light kinematics no matter how many speeds you naively chain together.

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

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