THEOREM OF THE ADDITION OF VELOCITIES.
THE EXPERIMENT OF FIZEAU Relativity: The Special and General Theory | ||
13. 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 VI 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 XI). 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
But we can carry out this consideration just as well on the basis of
the theory of relativity. In the equation
x' = wt' (B),
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 (A) we then obtain the equation
[Description: Equation]
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
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 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
[Description: Equation]
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 theman walking along the carriage, or of the moving point in the present
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
Fizeau found [Description: Equation] , where [Description: Equation] is the index of refraction of the liquid. On the other hand, owing to the smallness of [Description: Equation] as compared with I, we can replace (B) in the first place by [Description: Equation], or to the same order of approximation by [Description: Equation], which agrees with Fizeau's result.
THEOREM OF THE ADDITION OF VELOCITIES.
THE EXPERIMENT OF FIZEAU Relativity: The Special and General Theory | ||