x1 = wt1

By means of the first and fourth equations of the Galilei transformation we can express x1 and t1 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,

W=v+w A)

But we can carry out this consideration just as well on the basis of the theory of relativity.  In the equation

x1 = wt1 B)

we must then express x1and t1 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

eq. 09:  file eq09.gif

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, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity v ?

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 the railway embankment or of the co-ordinate system K, the liquid plays the part of the carriage or of the co-ordinate system K1, and finally, the light plays the part of the

Figure 03:  file fig03.gif

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 (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts.  Experiment * decides in favour of equation (B) 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 (B) to within one per cent.