VIII
By analogy with water waves and sound waves, and
more specifically because of Maxwell's equations, we
can expect light waves to travel through free aether
with a fixed speed. If, in our laboratory, we find that
light waves have different speeds in different directions,
we can conclude that we are moving through the
aether. Suppose, for example, that we find that their
greatest speed is 186,600 miles per second in this di-
rection → and their least speed 186,000 miles per
second in this direction ←. Then we can say that our
laboratory is moving through the aether in this direc-
tion → at 300 miles per second (half the difference
of the speeds), and that the light waves are travelling
through the aether at 186,300 miles per second (half
the sum). Thus we shall have discovered our absolute
velocity, and this despite Fresnel. But in speaking of
Fresnel's so-called aether drag, we said it implied that
every feasible first order experiment would fail. The
above is not feasible. The direct laboratory methods
of measuring the speed of light have involved not
one-way but round-trip speeds.
Shortly before his death, Maxwell outlined a way
to measure the earth's velocity through the aether by
comparing not one-way but round-trip speeds of light
in various directions in the laboratory. But since there
would be only a residual effect of the second order—if
v is the earth's orbital speed and the sun is at rest v2/c2
is about 10-8—he dismissed the effect as “far too small
to be observed.”
In 1881, however, Michelson succeeded in perform-
ing the experiment with borderline accuracy for de-
tecting the orbital speed. And in 1887, with Morley,
he repeated the experiment, this time with ample
accuracy. It gave a null result, and thereby precipitated
a crisis. For it suggested, and this was indeed Michel-
son's own interpretation, that the earth carries the
nearby aether along with it. But aberration implied
that the earth does not.
To resolve the conflict, FitzGerald, and later Lorentz
independently, proposed that objects moving through
the aether contract by an amount of the second order
in the direction of their motion.
Lorentz, assuming a fixed aether, untrapped and
undragged, had nevertheless obtained an electromag-
netic derivation of Fresnel's formula far more con-
vincing than that given by Fresnel. Thus Lorentz could
account for the null results of the feasible first-order
experiments to detect absolute motion. His task was
to express Maxwell's equations in a reference frame
moving uniformly through the aether with velocity v,
and to do so in such a way that, to the first order,
the v did not show up. But the Maxwell equations were
far from being pliable. In the moving frame they more
or less forced Lorentz to replace the t representing
the time by a new mathematical quantity that he called
“local time” because it was not the same everywhere.
By incorporating the contraction of lengths, he was
able to account for the null result of the Michelson-
Morley experiment without spoiling the theory of ab-
erration. But again the Maxwell equations forced his
hand, causing him to introduce with the contraction
a corresponding dilatation, or slowing down, of the
local time. Specifically, he found in 1904 what we now
call the Lorentz transformation, a name given it by
Poincaré in 1905. Consider two reference frames simi-
larly oriented, one at rest in the aether and the other
moving with uniform speed v in the common x-direc-
tion. Ordinarily one would have related the coordinates
(x, y, z) of the former to the coordinates (x′, y′, z′) of
the latter by what P. Frank named the Galilean trans-
formation:
xʹ = x - vt, yʹ = y, zʹ = z
.
But the Lorentz transformation relates (
x, y, z) and the
true time
t to (
x′,
y′,
z′) and the local time
t′ of the
moving frame as follows:
x′ = (x - vt)/√1 - v2/c2, y′ = y, z = z
t′ = (t - vx/c2/√1 - v/c2.
By means of these equations, Lorentz succeeded, ex-
cept for a small blemish removed by Poincaré in 1905,
in transferring the Maxwell equations to the moving
reference frame in such a way that they remained
unchanged in form. Since no trace of the
v survived,
neither the Michelson-Morley nor any other electro-
magnetic experiment could now be expected to yield
a value for
v.
It is of interest that the Lorentz transformation, (2),
had already been obtained on electromagnetic grounds
by Larmor in 1898, and its essentials by Voigt on the
basis of wave propagation as early as 1887, the very
year of the Michelson-Morley experiment.
