Relativistic Addition of Velocities

A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.

As Galileo Galilei observed in 17th century, if a ship is moving relative to the shore at velocity $v$, and a fly is moving with velocity $u$ as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities $v$ and $u$. When both the fly and the ship are moving slowly compared to speed of light, it is accurate enough to use the vector sum $s = u + v$ where $s$ is the velocity of the fly relative to the shore.

According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed.

Since special relativity dictates that the speed of light is the same in all frames of reference, light shone from the front of a moving car can't go faster than light from a stationary lamp. Since this is counter to what Galileo used to add velocities, there needs to be a new velocity addition law.

This change isn't noticeable at low velocities but as the velocity increases towards the speed of light it becomes important. The addition law is also called a composition law for velocities. For collinear motions, the velocity of the fly relative to the shore is given by the following equation:

$s = \frac{v + u}{1 + vu/c^{2}}$.

Composition law for velocities gave the first test of the kinematics of the special theory of relativity. Using a Michelson interferometer, Hyppolite Fizeau measured the speed of light in a fluid moving parallel to the light in 1851 . The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. The speed of light in a collinear moving fluid is predicted accurately by the collinear case of the relativistic formula.

Setup of the Fizeau Experiment

A light ray emanating from the source S' is reflected by a beam splitter G and is collimated into a parallel beam by lens L. After passing the slits O1 and O2, two rays of light travel through the tubes A1 and A2, through which water is streaming back and forth as shown by the arrows. The rays reflect off a mirror m at the focus of lens L', so that one ray always propagates in the same direction as the water stream, and the other ray opposite to the direction of the water stream. After passing back and forth through the tubes, both rays unite at S, where they produce interference fringes that can be visualized through the illustrated eyepiece. The interference pattern can be analyzed to determine the speed of light traveling along each leg of the tube.