TABLE OF CONTENTS Attempts to measure the absolute velocity of the earth through the hypothetical ether had failed. The most famous of these experiments is one performed by Michelson and Morley in 1887 [Feynman and Sands (1964)]. It was 18 years later before the null results of these experiments were finally explained by Einstein in terms of a new velocity addition law that bears his name, that he introduced in his 1905 paper that founded the special theory of relativity [Einstein (1905); Einstein (1998)].
Contrasting Newtonian velocities, which are vectors in the Euclidean 3-space ? 3, Einsteinian velocities must be relativistically admissible, that is, their magnitude must not exceed the vacuum speed of light c. Let, (3.138),
| (10.2) |  |
be the c-ball of all relativistically admissible velocities. It is the ball of radius c, centered at the origin of the Euclidean 3-space ? 3, consisting of all vectors v in ? 3 with magnitude ? v ? smaller than c. Einstein addition ? in the ball is given by the equation [Einstein (1905)] [Einstein (1998), p. 141], (3.141),
| (10.3) |  |
satisfying the gamma identity, (3.144),
| (10.4) |  |
for all u, 
, where ? u is the Lorentz factor
| (10.5) |  |
Einstein addition gives rise to the Einstein groupoid ( 
, ?) of Einsteinian velocities.
It is clear from (10.4) that ? u ? v ? = ? v ? u ?. However, it follows from (10.3) that, in general,
