bei48482_FM

The Lorentz Transformation 41

y   y (1.42)

z   z (1.43)

t    (1.44)

These equations comprise the Lorentz transformation.They were first obtained

by the Dutch physicist H.A. Lorentz, who showed that the basic formulas of

electromagnetism are the same in all inertial frames only when Eqs. (1.41) to (1.44)

are used. It was not until several years later that Einstein discovered their full

significance. It is obvious that the Lorentz transformation reduces to the Galilean

transformation when the relative velocity is small compared with the velocity of

light c.

t



c^2

x





 1 ^2 c^2

Example 1.9

Derive the relativistic length contraction using the Lorentz transformation.

Solution

Let us consider a rod lying along the x axis in the moving frame S. An observer in this frame

determines the coordinates of its ends to be x  1 and x     2 , and so the proper length of the rod is

L 0 x 2 x (^1)
Hendrik A. Lorentz (1853–1928)
was born in Arnhem, Holland, and
studied at the University of Leyden.
At nineteen he returned to Arnhem
and taught at the high school there
while preparing a doctoral thesis that
extended Maxwell’s theory of elec-
tromagnetism to cover the details of
the refraction and reflection of light.
In 1878 he became professor of the-
oretical physics at Leyden, the first
such post in Holland, where he remained for thirty-four years
until he moved to Haarlem. Lorentz went on to reformulate
and simplify Maxwell’s theory and to introduce the idea that
electromagnetic fields are created by electric charges on the
atomic level. He proposed that the emission of light by atoms
and various optical phenomena could be traced to the mo-
tions and interactions of atomic electrons. The discovery in
1896 by Pieter Zeeman, a student of his, that the spectral
lines of atoms that radiate in a magnetic field are split
into components of slightly different frequency confirmed
Lorentz’s work and led to a Nobel Prize for both of them in
1902.
The set of equations that enables electromagnetic quantities
in one frame of reference to be transformed into their values in
another frame of reference moving relative to the first were
found by Lorentz in 1895, although their full significance was
not realized until Einstein’s theory of special relativity ten years
afterward. Lorentz (and, independently, the Irish physicist G. F.
Fitzgerald) suggested that the negative result of the Michelson-
Morley experiment could be understood if lengths in the
direction of motion relative to an observer were contracted. Sub-
sequent experiments showed that although such contractions
do occur, they are not the real reason for the Michelson-
Morley result, which is that there is no “ether” to serve as a
universal frame of reference.
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