From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.
Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these became meaningless if we choose values of v greater than c.
If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to K, then we should have found that the length of the rod as judged from K1 would have been eq. 06 ;
this is quite in accordance with the principle of relativity which forms the basis of our considerations.
A Priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galileian transformation we should not have obtained a contraction of the rod as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks :
t = 0
and
eq. 07: file eq07.gif
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but
eq. 08: file eq08.gif
seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.
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 cla.s.sical 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 K1 in accordance with the equation
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.
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 circ.u.mstance, 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
*) Fizeau found eq. 10 , where eq. 11
is the index of refraction of the liquid. On the other hand, owing to the smallness of eq. 12 as compared with I,
we can replace (B) in the first place by eq. 13 , or to the same order of approximation by
eq. 14 , which agrees with Fizeau"s result.
THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
Our train of thought in the foregoing pages can be epitomised in the following manner. Experience has led to the conviction that, on the one hand, the principle of relativity holds true and that on the other hand the velocity of transmission of light in vacuo has to be considered equal to a constant c. By uniting these two postulates we obtained the law of transformation for the rectangular co-ordinates x, y, z and the time t of the events which const.i.tute the processes of nature. In this connection we did not obtain the Galilei transformation, but, differing from cla.s.sical mechanics, the Lorentz transformation.
The law of transmission of light, the acceptance of which is justified by our actual knowledge, played an important part in this process of thought. Once in possession of the Lorentz transformation, however, we can combine this with the principle of relativity, and sum up the theory thus:
Every general law of nature must be so const.i.tuted that it is transformed into a law of exactly the same form when, instead of the s.p.a.ce-time variables x, y, z, t of the original coordinate system K, we introduce new s.p.a.ce-time variables x1, y1, z1, t1 of a co-ordinate system K1. In this connection the relation between the ordinary and the accented magnitudes is given by the Lorentz transformation. Or in brief : General laws of nature are co-variant with respect to Lorentz transformations.
This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental a.s.sumptions of the theory would have been disproved. Let us now examine what general results the latter theory has. .h.i.therto evinced.
GENERAL RESULTS OF THE THEORY
It is clear from our previous considerations that the (special) theory of relativity has grown out of electrodynamics and optics. In these fields it has not appreciably altered the predictions of theory, but it has considerably simplified the theoretical structure, i.e. the derivation of laws, and -- what is incomparably more important -- it has considerably reduced the number of independent hypothese forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour.
Cla.s.sical mechanics required to be modified before it could come into line with the demands of the special theory of relativity. For the main part, however, this modification affects only the laws for rapid motions, in which the velocities of matter v are not very small as compared with the velocity of light. We have experience of such rapid motions only in the case of electrons and ions; for other motions the variations from the laws of cla.s.sical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity the kinetic energy of a material point of ma.s.s m is no longer given by the well-known expression