Velocity-addition Formula - Special Theory of Relativity

Special Theory of Relativity

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. 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

This is also the law of addition of hyperbolic tangents

\tanh(\alpha + \beta) = {\tanh(\alpha) + \tanh(\beta) \over 1+ \tanh(\alpha) \tanh(\beta) }

where

{v\over c} = \tanh(\alpha) \, \quad {u \over c}=\tanh(\beta) \, \quad\, {s\over c}=\tanh(\alpha +\beta) ,

which shows that the composition of collinear velocities is associative and commutative. The quantities α and β (equal to the artanh of the velocities divided by c) are known as rapidities. The reason that the velocities are hyperbolic tangents is because the Lorentz transformation can be thought of as the application of a hyperbolic rotation through a hyperbolic angle which is the rapidity. Suppose the velocity of a line in space-time is the slope of the line, which is the hyperbolic tangent of the rapidity, just as the slope of the x-axis after a rotation is given by the tangent of the rotation angle. When a plane is successively rotated by two angles, the final rotation is by the sum of the two angles. So the final slope of the x-axis is the tangent of the sum of the two angles. In the same way, the slope of the time axis after two boosts is the hyperbolic tangent of the sum of the two rapidities.

The colinear law of composition of velocities gave the first test of the kinematics of the special theory of relativity. Using a Michelson interferometer, Fizeau measured the speed of light in a fluid moving parallel to the light. 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 colinear moving fluid is predicted accurately by the colinear case of the relativistic formula.

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