Relativistic Addition of Velocity | Special Relativity Ch. 6

Special relativity forces a hard limit on how speeds combine across different moving perspectives: relative velocities never add in the simple Newtonian way once speeds approach light speed. The key reason is the Lorentz transformation—an effective squeeze-stretch rotation of spacetime—that preserves the experimentally verified constancy of light speed. On a spacetime diagram, light always travels along a 45° line for any observer, so changing frames cannot “rotate” a light ray into a different angle.

That constraint becomes vivid when comparing an intuitive but wrong calculation for slower-than-light objects. Suppose an observer sees a “death-pellet” moving right at 60% of light speed, and another observer is also moving right at 60% of light speed relative to the first. A naive addition would double the speed to 120% of light speed, which would violate the light-speed limit. Spacetime diagrams correct that intuition: when the frame shifts, the pellet’s worldline changes angle, but it cannot cross the 45° light-speed boundary. For a pellet at 50% of light speed, the worldline in the new frame is still rightward but not steep enough to become a 45° line; at 60%, the same frame change keeps the worldline just shy of lightlike.

The underlying geometric picture is that Lorentz transformations “squeeze and stretch” along the lightlike directions. Stretching a line on a rubber sheet can rotate its angle toward the stretching direction, but it can’t flip it past the lightlike boundary. That means even repeated “boosts” (adding 60% relative speeds again and again) can only drive the result closer to light speed—not to or beyond it. The constancy of light speed therefore implies a global rule: no sequence of perspective changes can turn a sub-light relative speed into a lightlike or superluminal one.

Mathematically, the relativistic velocity-addition formula encodes the same limit. If an object moves at speed v relative to one observer, and that observer moves at speed u relative to another, then the object’s speed relative to the second observer is

v′ = (v + u) / (1 + vu/c²).

Plugging in c for either velocity always returns c, matching the invariant light-speed postulate. If both v and u are less than c, the denominator’s 1 + vu/c² term prevents the result from reaching c. In the low-speed regime where v and u are much smaller than c, vu/c² becomes negligible, and the formula reduces to the familiar Newtonian sum v′ ≈ v + u. The takeaway is that ordinary velocity addition is an approximation that works only when speeds are far from light speed; near light speed, spacetime geometry and the invariant c reshape how motion transforms between frames.