What Is the Relativistic Velocity Addition Equation?

The relativistic velocity addition equation calculates an object's velocity in one inertial reference frame from its velocity in another. Unlike Galilean velocity addition, it ensures that transforming subluminal velocities never produces a speed greater than the speed of light, $c$.

This is HL-only content from A.5 Galilean and special relativity.

The Equation and Its Mechanism

Suppose reference frame (S') moves at velocity $v$ relative to frame $S$, while an object has velocity $u$ in $S$. The velocity (u') measured in (S') is

$$u' = \frac{u-v}{1-\frac{uv}{c^2}}.$$

The equation follows from the Lorentz transformations for position and time. Since both spatial intervals and time intervals transform between inertial frames, dividing transformed displacement by transformed time produces the denominator $1-uv/c^2$. Velocities therefore cannot simply be subtracted as $u'=u-v$ when their magnitudes are comparable to .

To transform in the opposite direction, rearrange the equation or replace $v$ with (-v):

$$u = \frac{u'+v}{1+\frac{u'v}{c^2}}.$$

For example, a spacecraft travels at $0.80c$ relative to Earth and launches an object forward at $0.70c$ relative to the spacecraft:

$$u=\frac{0.70c+0.80c}{1+(0.70)(0.80)}=0.962c.$$

The common misconception is to add the speeds directly and obtain $1.50c$. Relativistic addition instead gives $0.962c$, consistent with the postulate that $c$ has the same value in every inertial reference frame. At low speeds, $uv/c^2$ is negligible, so the equation reduces to Galilean addition.

Exam Technique

In an IB Calculate question, define both reference frames, choose one direction as positive, assign signed velocities, select the correct equation, and show substitution. Do not treat speeds as unsigned: an object moving in the opposite direction has a negative velocity. Finally, check that the result has a sensible direction and remains below $c$.