Tensors for Physics

18.2 Lorentz-Vectors and Lorentz-Tensors 377

yields
KnKn=−k^2 +ω^2 /c^2 = 0. (18.32)

This 4D version of the dispersion relation is equivalent to (7.65).

18.2.4 Some Historical Remarks.

The appropriate rules for the transformations underlying the Maxwell equations were
studied by a number of scientists from about 1890 to 1910. Hendrik Lorentz (1853–
1928) published his findings in 1899 and 1904. The distancertravelled by light with
speedcduring the timetis determined byr^2 =c^2 t^2. Then the invariance of the
speed of light implies the invariance condition

(r/t)^2 =(r′/t′)^2 , (18.33)

which is a special case of (18.1). Lorentz noticed that the more general rule

x′=γ (x−vt), y′=y, z′=z, t′=γ

(

t−

v

c^2

x

)

, (18.34)

with an yet unspecified scale function=(v^2 )guarantees the validity of (18.33). It
is understood that the velocity has the componentsvx=vandvy=vz=0. In 1905,
Henri Poincaré pointed out that the scale function should be=1, then (18.34)
reduces to the special transformation (18.15). Poincaré coined the term ‘Lorentz-
transformation’ for this transformation rule. In the same year, Albert Einstein gave
an alternative derivation of the transformation rules and elucidated their meaning. In
particular, he postulated that the transformation rule should also apply for the motion
of particles, not just for the propagation of light. He also introduced a scale function,
similar to, and presented arguments for=1. In 1905, Einstein did not refer to the
work of Lorentz and Poincaré, later he also used the term Lorentz transformation.
Woldemar Voigt, who introduced in 1898 the word and the notion tensor in the
sense we still use it nowadays, had already noticed in 1887: the Maxwell equations
and the invariance of the speed of light require the invariance condition (18.5). For
the case where the primed coordinate system moves inx-direction, as in Sect.18.1.4,
Voigt proposed the transformation rule

x′=x−vt, y′=γ−^1 y, z′=γ−^1 z, ct′=ct−

v

c

x. (18.35)

This corresponds to the more general transformation (18.34) with the choice=γ−^1
for the scale function. It was by analogy to a volume conserving elastic deformation
of a solid body, which shrinks in two directions, when it is stretched in one, which led
Voigt to assume also a change of they- andz-components. This is in contradistinction