Einstein's Formula for Adding Velocities
According to the special theory of relativity, the composition of two collinear velocities and is given by a well-known formula derived by Einstein: , where is the speed of light, 2.9979× m/sec. For , this reduces to the simple Galilean formula . The result can be derived by successive application of collinear Lorentz boosts, but it can be shown more intuitively by an argument outlined in the details. In the Demonstration the speeds are scaled as (also known as ) to keep the two rocket ships within the graphic. Einstein's formula is analogous to the law of addition for hyperbolic tangents, , where corresponds to the rapidity, defined by .
Suppose a baseball team is traveling on a train moving at 60 mph. The star fastball pitcher needs to tune up his arm for the next day’s game. Fortunately, one of the railroad cars is free, and its full length is available. If his 90 mph pitches are in the same direction the train is moving, the ball will actually be moving at 150 mph relative to the ground. The law of addition of velocities in the same direction is relatively straightforward, . But according to Einstein’s special theory of relativity, this is only approximately true and requires that and be small fractions of the speed of light, ≈ 3 × m/sec (or 186,000 miles/sec). Expressed mathematically, we can write if According to special relativity, the speed of light, when viewed from any frame of reference, has the same constant value . Thus, if an atom moving at velocity emits a light photon at velocity , the photon will still be observed to move at velocity , not .
Our problem is to deduce the functional form of consistent with these facts. It is convenient to build in the known asymptotic behavior for by defining . When , we evidently have , so , and likewise . If both and equal , . A few moments' reflection should convince you that a function consistent with these properties is , which gives Einstein's velocity addition law .
Reference:S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 pp. 5–6.