In Michelson and Morley’s experiment, the distance travelled by the beam
along OA (that is, in a direction parallel to the motion of the
platform) is 2_l_β², while the distance travelled by the beam along OB,
perpendicular to the direction of motion of the platform, is 2_l_β. Yet
the most careful experiments showed, as Eddington says, “that both parts
of the beam took the same time as tested by the interference bands
produced. It would seem that OA and OB could not really have been of the
same length; and if OB was of length _l_, OA must have been of length
_l_/β. The apparatus was now rotated through 90°, so that OB became the
up-stream. The time for the two journeys was again the same, so that 0B
must now be the shorter length. The plain meaning of the experiment is
that both arms have a length _l_ when placed along O_y_ (perpendicular
to the direction of motion), and automatically contract to a length
_l_/β, when placed along O_x_ (parallel to the direction of motion).
This explanation was first given by Fitz-Gerald.”
This Fitz-Gerald contraction, startling enough in itself, does not
suffice. Assuming this contraction to be a real one, the distance
travelled with respect to the ether is 2_l_β and the time taken for this
journey is 2_l_β/_c_. But the distance travelled with respect to the
platform is always 2_l_. Hence the velocity of light with respect to the
platform is
$$ \frac {2l}{\frac {2l\beta}{c}} = \frac {c}{\beta} $$
a variable quantity depending on the “absolute” velocity of the
platform. But no trace of such an effect has ever been found. The
velocity of light is always found to be quite independent of the
velocity of the platform. The present difficulty cannot be solved by any
further alteration in the measure of space. The only recourse left open
is to alter the measure of time as well, that is, to adopt the concept
of “local time.” If a moving clock goes slower so that one ‘real’ second
becomes 1/β second as measured in the moving system, the velocity of
light relative to the platform will always remain _c_. We must adopt two
very startling hypotheses, namely, the Fitz-Gerald contraction and the
concept of “local time,” in order to give a satisfactory explanation of
the Michelson-Morley experiment.
Public-domain text, read in full here on John Shaqi.
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