Lorentz Transformations

Relativistic Addition of Velocities

Imagine that you are standing between two space-ships moving away from you. One space-ship moves to the left with a speed of 0.75 c (relative to you) and the other one moves to the right also with a speed of 0.75 c (relative to you).

At what speed will each space-ship see the other moving away? 0.75 c + 0.75 c = 1.5 c?
No, their relative speed will be 0.96 c (according to the relativistic addition of velocities), and it cannot, of course, be faster than the speed of light c.

In classical Newtonian mechanics, two different velocities and are added together by the formula

where is the sum of the two velocities. However, in special relativity, the velocities are added together as

This formula is called the relativistic addition of velocities.

Note that if = c and/or = c, then = c, and for small velocities , << c, then the classical formula is regained.