The Mathematical Theory of RelativityEddington, Arthur Stanley, Sir
Science
The Mathematical Theory of Relativity
Eddington, Arthur Stanley, Sir
Relativity (Physics)
When this proof is compared with the statement commonly (and correctly)
made that the equality of the forward and backward velocity of light cannot
\PageSep{21}
be deduced from experiment, regard must be paid to the context. The use
of the Michelson-Morley experiment to fill a particular gap in a generally
deductive argument must not be confused with its use (e.g.\ in \Title{Space, Time
and Gravitation}) as the basis of a pure induction from experiment. Here we
have not even used the fact that it is a second-order experiment. We have
deduced the Lorentz transformation from the fundamental hypothesis of \SecRef{1},
and have already introduced a conventional system of time-reckoning explained
in \SecRef{4}. The present argument shows that the convention that time is defined
by the slow transport of chronometers is equivalent to the convention that
the forward velocity of light is equal to the backward velocity. The proof of
\index{Velocity of light!in moving matter}%
this equivalence is mainly deductive except for one hiatus---the connection
of the propagation of light and the fundamental velocity---and for that step
appeal is made to the Michelson-Morley experiment.
The law of composition of velocities~\Eq{(6.2)} is well illustrated by Fizeau's
\index{Addition of velocities}%
\index{Composition of velocities}%
experiment on the propagation of light along a moving stream of water. Let
the observer~$S'$ travel with the stream of water, and let $S$~be a fixed observer.
The water is at rest relatively to~$S'$ and the velocity of the light relative to
him will thus be the ordinary velocity of propagation in still water, viz.\
$v' = c/\mu$, where $\mu$~is the refractive index. The velocity of the stream being~$w$,
$-w$~is the velocity of~$S$ relative to~$S'$; hence by~\Eq{(6.2)} the velocity~$v$ of the
light relative to~$S$ is
\begin{align*}
v = \frac{v' + w}{1 + wv'/c^{2}}
&= \frac{c\mu + w}{1 + w/\mu c} \\
&= c/\mu + w(1 - 1/\mu^{2})\quad\text{approximately,}
\end{align*}
neglecting the square of~$w/c$.
Accordingly the velocity of the light is not increased by the full velocity
of the stream in which it is propagated, but by the fraction $(1 - 1/\mu^{2}) w$. For
water this is about $0.44 w$. The effect can be measured by dividing a beam
of light into two parts which are sent in opposite directions round a circulating
stream of water. The factor $(1 - 1/\mu^{2})$ is known as Fresnel's convection-coefficient;
\index{Fresnel's convection-coefficient}%
it was confirmed experimentally by Fizeau in~1851.
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