Theorem of the Addition of Velocities. The Experiment of FizeauTheoremOfTheAdditionOfVelocitiesTheExperimentOfFizeau2006-03-11 23:29:532006-03-11 23:29:53TopicRelativity: The Special and General Theory% this is the default PlanetMath preamble. as your knowledge
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% define commands here\subsection{Theorem of the Addition of Velocities.\\
The Experiment of Fizeau}
Now in practice we can move clocks and measuring-rods only with
velocities that are small compared with the velocity of light; hence
we shall hardly be able to compare the results of the previous section
directly with the reality. But, on the other hand, these results must
strike you as being very singular, and for that reason I shall now
draw another conclusion from the theory, one which can easily be
derived from the foregoing considerations, and which has been most
elegantly confirmed by experiment.
In Section 6 we derived the theorem of the addition of velocities
in one direction in the form which also results from the hypotheses of
classical mechanics---This theorem can also be deduced readily horn the
Galilei transformation (Section 11). In place of the man walking
inside the carriage, we introduce a point moving relatively to the
co-ordinate system $K'$ in accordance with the equation
$$x' = wt'.$$
~
By means of the first and fourth equations of the Galilei
transformation we can express $x'$ and $t'$ in terms of $x$ and $t$, and we
then obtain
$$x = (v + w)t.$$
~
This equation expresses nothing else than the law of motion of the
point with reference to the system $K$ (of the man with reference to the
embankment). We denote this velocity by the symbol $W$, and we then
obtain, as in Section 6,
\begin{equation}
W=v+w
\label{eqnA}
\end{equation}
But we can carry out this consideration just as well on the basis of
the theory of relativity. In the equation
\begin{equation}
x'=wt'
\label{eqnB}
\end{equation}
\noindent we must then express $x'$and $t'$ in terms of $x$ and $t$, making use of the
first and fourth equations of the Lorentz transformation. Instead of
the equation \ref{eqnA} we then obtain the equation
$$W = \frac{v+w}{I+\frac{vw}{c^2}}$$
~
\noindent which corresponds to the theorem of addition for velocities in one
direction according to the theory of relativity. The question now
arises as to which of these two theorems is the better in accord with
experience. On this point we axe enlightened by a most important
experiment which the brilliant physicist Fizeau performed more than
half a century ago, and which has been repeated since then by some of
the best experimental physicists, so that there can be no doubt about
its result. The experiment is concerned with the following question.
Light travels in a motionless liquid with a particular velocity $w$. How
quickly does it travel in the direction of the arrow in the tube T
(see the accompanying diagram, Figure \ref{fig:3}) when the liquid above
mentioned is flowing through the tube with a velocity $v$?
% Figure 3
%
% T
% /
% --------------------------------------
% v --------->
% --------------------------------------
%
\begin{figure}[hbtp]
\centering
\caption{}
\label{fig:3}
\begin{picture}(200,75)(0,0)
\thicklines
\put(0,15){\line(1,0){200}}
\put(0,35){\line(1,0){200}}
\put(100,35){\line(1,3){5}}
\put(107,52){T}
\thinlines
\put(40,25){\vector(1,0){50}}
\put(60,26){$v$}
\end{picture}
\end{figure}
In accordance with the principle of relativity we shall certainly have
to take for granted that the propagation of light always takes place
with the same velocity w \emph{with respect to the liquid}, whether the
latter is in motion with reference to other bodies or not. The
velocity of light relative to the liquid and the velocity of the
latter relative to the tube are thus known, and we require the
velocity of light relative to the tube.
It is clear that we have the problem of Section 6 again before us. The
tube plays the part of the railway embankment or of the co-ordinate
system $K$, the liquid plays the part of the carriage or of the
co-ordinate system $K'$, and finally, the light plays the part of the
man walking along the carriage, or of the moving point in the present
section. If we denote the velocity of the light relative to the tube
by $W$, then this is given by the equation \ref{eqnA} or \ref{eqnB}, according as the
Galilei transformation or the Lorentz transformation corresponds to
the facts. Experiment\footnotemark decides in favour of equation \ref{eqnB} derived
from the theory of relativity, and the agreement is, indeed, very
exact. According to recent and most excellent measurements by Zeeman,
the influence of the velocity of flow $v$ on the propagation of light is
represented by formula \ref{eqnB} to within one per cent.
Nevertheless we must now draw attention to the fact that a theory of
this phenomenon was given by H. A. Lorentz long before the statement
of the theory of relativity. This theory was of a purely
electrodynamical nature, and was obtained by the use of particular
hypotheses as to the electromagnetic structure of matter. This
circumstance, however, does not in the least diminish the
conclusiveness of the experiment as a crucial test in favour of the
theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
which the original theory was based, in no way opposes the theory of
relativity. Rather has the latter been developed trom electrodynamics
as an astoundingly simple combination and generalisation of the
hypotheses, formerly independent of each other, on which
electrodynamics was built.
% Notes
\footnotetext{Fizeau found $W=w+v\left(I-\frac{I}{n^2}\right)$, where $n=\frac{c}{w}$
is the index of refraction of the liquid. On the other hand, owing to
the smallness of $\frac{vw}{c^2}$ as compared with $I$,
we can replace (B) in the first place by $W=(w+v)\left(I-\frac{vw}{c^2}\right)$, or to the same order
of approximation by
$w+v\left(I-\frac{I}{n^2}\right)$, which agrees with Fizeau's result.}