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THE NEW KINEMATICS 143

The relations between the two sets of coordinates implied by these two equations

are assumed to be linear, in accordance with the homogeneity of space and time.

Then simple arithmetic yields

where « is an arbitrary scale factor depending on v only. Since the product of this

transformation and its inverse should yield the identity, one has

Symmetry demands that the transformations on y and z should not change if v

—>• — v, and hence

Thus (.(v) = 1 (since e(0) = 1) and

In Chapter 6, we encountered Eqs. 7.4-7.8 in the discussion of papers by Lorentz

and Poincare. The derivation of the Lorentz transformations (Eq. 7.8) from first

principles occurs for the first time in Einstein's paper, however.*

Einstein also pointed out that transformations of the type shown in Eq. 7.8 form

a group, Vie dies sein muss,' as it should be**: two successive transformations

with velocities w,,u 2 in the same direction result in a new transformation of the

form of Eq. 7.8 with a velocity v given by

Twenty years later, Einstein heard something about the Lorentz group that

greatly surprised him. It happened while he was in Leiden. In October 1925

George Eugene Uhlenbeck and Samuel Goudsmit had discovered the spin of the

electron [Ul] and thereby explained the occurrence of the alkali doublets, but for

a brief period it appeared that the magnitude of the doublet splitting did not come

out correctly. Then Llewellyn Thomas supplied the missing factor, 2, now known

as the Thomas factor [Tl]. Uhlenbeck told me that he did not understand a word

of Thomas's work when it first came out. 'I remember that, when I first heard

about it, it seemed unbelievable that a relativistic effect could give a factor of 2

instead of something of order v/c.... Even the cognoscenti of the relativity theory

(Einstein included!) were quite surprised' [U2]. At the heart of the Thomas

precession lies the fact that a Lorentz transformation with velocity y, followed by

a second one with a velocity ~v 2 m a different direction does not lead to the same

*See [Rl] for interesting comments on the roles of postulates and observations in the special theory
of relativity.

**He did not expand on this cryptic statement.