A length which is about 100,000 times smaller than the radius of an atom.
     Thus the hypothesis that the mass of the electron is electromagnetic in origin does not conflict with the known facts. But this does not prove the hypothesis.
     At this stage the theory found b support in refined observa-tions of cathode rays and of the -rays of radioactive substances, which are also ejected electrons. We explained above how electric and magnetic action on these rays allows us to determine the ratio of the charge to mass,   e  , and also their velocity v, and that at first a definite valuefor   e   was obtained,
el                                                                                                  mel
which was independent of   v.  But, on proceeding to higher velocities, a decrease of  e  was found.  This effect was particularly clear and could be measured quantitatively in the case of the -rays of radium, which are only slightly slower than light.  The assumption that an electric charge should depend on the velocity is incompatible with the ideas of the electron theory. But that the mass should depend on the velocity was certainly to be expected if the mass was to be electromagnetic in origin. To arrive at a quantitative theory, it is true, definite assumptions had to be made about the form of the electron and the distribution of the charge on it. M.
Abraham (1903) regarded the electron as a rigid sphere, with a charge distributed on the one hand, uniformly over the interior, or, on the other, over the surface,and he showed that both assumptions lead to the same dependence of the electromagnetic mass on the velocity, namely, to an increase of mass with increasing velocity. The faster the electron travels, the more the electromagnetic field resists a further increase of velocity. The increase of  mel  explains the observed decrease of  e  , and Abraham's theory agrees quantitatively very well with the
results of measurement  

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of Kaufmann (1901) if it is assumed that there is no "ordinary" mass present.
    Thus, the object of tracing the inertia of electrons back to electro-magnetic fields in the ether was attained. At the same time a further perspective presented itself. Since atoms are the carriers of positive electricity, and also contain numerous electrons, perhaps their mass is also electromagnetic in origin? In that case, mass as the measure of the inertial persistence would no longer be a primary phenomenon, as it is in elementary mechanics, but a secondary consequence of the ether. Therefore, Newton's absolute space, which is defined only by the mechanical law of inertia, becomes super-fluous; its part is taken over by the ether whose electromagnetic properties are well known.
     We shall see (V, 15, P. 221) that new facts contradict this view. But the relationship between mass and electromagnetic energy, which was first discovered in this way, constitutes a fundamental discovery the deep significance of which was brought into prominence only when Einstein proposed his theory of relativity".
The Zed Aliz Zed made a nine from the anagram   EI
NSTEIN     "We have yet to add that, besides Abraham's theory of the rigid electron, other hypothesi were set up and worked out mathematically. The most important is that of Lorentz (1904) which is closely connected with the theory of relativity. Lorentz assumed that every moving electron contracts in the direction of motion, so that from a sphere it becomes a flattened    
spheroid of revolu-tion, the amount of flattening depending in a definite way on this velocity. This hypothesis seems at first sight strange. It certainly gives a simpler formula for the way electromagnetic mass depends on velocity than does Abraham's theory, but this in itself does not justify it. The actual confirmation came from the course Lorentz's theory of electrons took when it had to consider quantities of the second order in the discussion of experimental researches, to which we shall presently direct our attention. Lorentz's formula then turned out to have a universal significance in the theory of relativity. The experimental decision between it and Abraham's theory will be discussed later (VI, 7, p 278).     At the beginning of the new century, after the theory of electrons had reached the stage above described, the possibility of forming a uniform physical picture of the world seemed at hand - a picture  

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which would reduce all forms of energy, including mechanical inertia, to the same root, to the electromagnetic field in the ether. Only one form of energy - gravitation - seemed still to remain outside the system; yet it could be hoped that that, too would allow itself to be interpreted as an action of the ether.


14. Michelson and Morley's Experiment

     Twenty years before this period, however, the base of the whole structure had already cracked and, while the building was going on above, the foundations needed repairing and strengthening.
     We have several times emphasized that any decisive experiment in regard to the theory of the stationary ether had to be precise enough to determine quantities of the second order in . Only then could it be ascertained whether or not a
fast- moving body is swept by an ether wind which blows away the light waves as is demanded by theory.
     Michelson and Morley (
1881) were the first sucecessfully to carry out the most important experiment of this type They used Michel-son's interferometer (IV, 4, p. 102) which they had refined to a precision instrument of unheard-of efficiency.
     In investigating the influence of the earth's motion on the velocity of light (IV,
9, P. 130), it has been found that the time taken by a ray of light to pass back and forth along a distance l parallel to the earth's motion differs only by a quantity of the second order from the value it has when the earth is at rest. We found earlier that this time was                                                                     t1= l (     1     +     1   ) =    21c  ,
                                                                                c + v       c - v     c
2 - v2
for which we may also write
1 = 21      1  .  
                                                                                            c    1-
   If this time could be so accurately measured that the fraction   1    could be distinguished from 1 in spite of the extremely
                                                                                                    1- 2
small value of the quantity 2, we should have means of proving the exis-tence of an ether wind.  

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It is by no means possible, however, to measure the short time taken by a light ray to traverse a certain distance.  Interferometric methods give us rather only differences of the times taken by light to traverse different routes between two given points. But they give these with amazing accuracy.
     For this reason Michelson and Morly caused a second ray of light to traverse a path AB of the same length l backwards and forwards, but perpendicular to the earth's orbit (Fig, 109). While

Fig. 109  The path of the light in Michelson's experiment.

the light passes from Ato B, the earth moves a short distance forward so that the point B arrives at the point B' of the ether. Thus the true path of the light in the ether is AB', and if it takes a time t to cover this distance, then AB, = ct. During the same time A has moved on to the point A' with the velocity v, thus AA' = vt. If we now apply Pythagoras' theorem to the right-angled triangle  AA'B, we get  
2t2 =l 2 + v2t2
2 (c2- v2) =l2,       t2   =    l2        =    l2        1    ,                                                                                                              c2 - v2          c2   1- 2
                                                                                      t  =  l        1    .
                                                                                             c   /  1
- 2

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The light requires exactly the same time to make the return journey, for the earth shifts by the same amount, so that the initial point  A moves from  A' to A ".
      Thus the light takes the following time for the journey backwards and forwards:
                                                                                        t  =  2l        1    .
                                                                                                c    /  1
- 2
The difference between the times taken to cover the same distance parallel and perpendicular to the earth's motion is thus
1- t2  = 21  (      1   .  -           1     ).
                                                                                                c      1-
2           / 1- 2

Now, by neglecting terms of higher order than the second in   (similar to what was done on p. 126) we may approximate by replacing     1     by    1 +
2, and        1       by 1 +  2*  .
                                                         /1 -
2               2
                                                                       Hence we may write to a sufficient degree of approximation  t1- t2  = 2l      [ ( 1 + 2) - ( 1 + 2 ) ] = 21   2  = 1  2 .
                                                                                                                        c                                    2           c     2       c
     The retardation of the one light wave compared with the other is thus a quantity of the second order.
     This retardation may be measured with the help of Michelson's interferometer (Fig 110).  In this the light coming from the source Q is divided at the half silvered-plate P into two rays which run in perpendicular directions to the mirrors S
1 and S2,                                                                                                                        
At which they are reflected and sent back to the plate  P. From P onwards they run parallel into the telescope F where they interfere. If the distances S
1P and S2P are equal and if one arm of the apparatus is placed in the direction of the earth's motion, one duplicates the    /                                                                                                                  
* To show the approximate validity of the equation       1     = 1 + 2    write it   1 = (1-  2) ( 1 + 2) - 4,
- 2
which is correct if  
4 is neglected. In the same way squaring        1      = 1 +  2 , one obtains      1     =  1 + 2 + 4;  if the                                                                                                       / 1- 2             2                         1- 2                     4
last term is neglected one has the same formula as above.
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case just discussed. Thus the two rays reach the field of vision with a difference of time   l  2.  Hence the                                                                                                                                                                  c
interference fringes are not situated precisely where they would be if the earth were at rest. But if we now turn the apparatus through
90º until the other arm is paral-lel to the direction of the earth's motion, the interference fringes will be displaced by the same amount but in the opposite direction.

Fig. 110   Michelsons's interfero-meter.                          Fig.  111   Two waves of the same wave length , one shifted
                                                                                                            against the other  by 2  l
Hence if we observe the position of the interference fringes while the apparatus is being rotated, a displacement should be measured which corresponds to the double retardation 2  1
2.                                                                                                                                    c
    If T is the period of vibration of the light used, the ratio of the retardation to the period is 2l  
2, and                                                                                                                                  cT
since by formula (35)  (p.
99) the wave length = Ct, we may write this ratio as 2  1  2.
     Hence, when the apparatus is rotated, the two interfering trains of waves experience a relative displacement whose ratio to the wave length is given by   2l
2  (Fig. 111). The interference fringes them-selves arise because the rays which leave the                                                  
source in different  

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directions have to traverse somewhat different paths. The distance between two fringes corresponds to a path difference of one wave length, hence the observable displacement of the fringes is the fraction   2l 2 of the width of the fringe.                                                                                                                                                  
    Now Michelson, in a repetition of his experiment with Morley (1887) carried out on a larger scale, extended the length of the path traversed by the light by means of several reflections forward and back to 11m. = 1.1 x 10  
3 cm. The wave length of the light used was about = 5.9 x 10-5 cm. We know that   is approximately equal to 10 - 4, and hence 2 = 10 - 8.
So we get
2  = 2 x 1.1 x 103 x 10 - 8  = 0.37,                                                                                                       5.9 x 10 -5
that is, the interference fringes must be displaced by more than one-third of their distance apart when the apparatus is turned through
90º. Michelson was certain that the one-hundredth part of this displacement would still be observable.
      When the experiment was carried out, however, not the slightest sign of the expected displacement manifested itself, and later repe-titions with still more refined means led to no other result. From this we must conclude that the ether wind does not exist. The velocity of light is not influenced by the motion of the earth even to the extent involving quantities of the second order.

15. The Contraction Hypothesis

     Michelson and Morley concluded from their experiment that the ether is carried along completely by the moving earth, as is main-tained in the elastic theory of Stokes and in the electromagnetic theory of Hertz. But this conclusion contradicts the numerous experiments which prove partial convection. Michelson then investigated whether it was possible to establish a difference in the velocity of light at different heights above the earth's surface, but without a positive result. He concluded from this that the motion of the ether that is carried along by the earth must extend to very great heights above the eart's surface. Thus, then, the ether would be  

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influenced by a moving body at considerable distances. But this is in fact not the case, for Oliver Lodge showed (1892) that the velocity of light in the neighborhood of rapidly moving bodies is not influen-ced in the slightest, not even when the light passes through a b electric or magnetic field, carried along by the body. But all these efforts seem superfluous, for even if they led to an unobjection-able explanation of Michelson's experiment, the rest of electrodyna-mics and optics of moving bodies which speaks in favour of partial convection would remain unexplained.                          
      We see now Lorentz's theory of electrons placed in a very difficult position by Michelson and Morley's experiment. The doctrine of the stationary ether seems to demand that an ether wind exist on the earth, and hence stands in contradiction to the results of Michelson and Morley's experiment. The fact that it did not at once succumb to this challenge proves the inherant strength of the theory, a strength deriving from the consistency and completeness of its physical picture of the world. Finally, it overcame even this difficulty to a certain extent, although by a a very strange hypothesis, which was proposed by Fitzgerald (
1892) and at once taken up and elaborated by Lorentz.
    Let us recall the reflections on which Michelson and Morley's experiment were based. We found that the time taken by a light ray to travel to and fro along a distance  l differs according to whether the ray travels parallel or perpendicular to the earth's motion. In the former case          
1 = 2l     1  ,                          
                                                                c   1
- 2
in the second,
2 = 2l       1      .                          
                                                                c    / 1
- 2

If we now assume that the arm of the interferometer which is directed parallel to the direction of the earth's motion is shortened in the ratio  
/  1- 2 : 1,  the time t1 would become reduced in the same ratio, namely,                      t1 =  21 / 1- 2  = 21     1   .                              c(1- 2        c  / 1- 2                                                                  
Thus we should have t
1 =t2.     

 / Page 220  /  

This suggests the following general hypothesis, the crudeness and boldness of which is startling indeed:
Everybody which has the velocity v with respect to the ether contracts in the direction of motion by the fraction
                    .       .                        .       .                                            
/  1- 2         =          / 1 -  v2 .      
                                                   /          c
     Michelson and Morley's experiment must actually, then, give a negative result, since for both positions of the interferometer  t
1 = t2. Furthermore - and this is the important point - such a contraction could not be ascertained by any means on earth, for every earthly measuring rod would be contracted in just the same way. An observer who was at rest in the ether outside the earth would, it is true, observe the contraction. The whole earth would be flattened in the direction of motion and likewise all things on it.
     The contraction hypothesis seems so remarkable
- indeed, almost absurd - because the contraction is not a consequence of any forces but appears only as a companion circumstance to motion. Lorentz, however, did not allow this objection to keep him from absorbing this hypothesis into his theory, particularly as new experiments con-firmed that no second-order
effect of the earth's motion through the ether could be detected.
     We cannot desc
ribe all these experiments or even outline them. They are partly optical and concern the events involved in re-flection and refraction, double refraction, rotation of the plane of polarization, and so forth; and they are partly electromagnetic and concern induction phenomena, the distribution of the current in wires and the like. The improved technique of physics allows us nowadays to establsh unambiguously the existence or
absence of second-order effects in these phenomena. A particularly note-worthy experiment is that of Trouton and Noble (
1903), which was intended to detect a torsional force which should occur in a suspended plate condenser in consequence of the ether wind.
     These experiments produced without exception a negative result. There could no longer be any doubt that a motion of translation through the ether cannot be detected by an observer sharing in the motion. Thus the principle of relativity which holds for mechanics is also valid for all electromagnetic phenomena.  /
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Lorentz next proceeded to bring this fact into harmony with his ether theory. To do this there seemed no other way than to assume the contraction hypothesis and to fuse it into the laws of the electron theory so as to form a consistent whole free from inner contradictions. He first observed that a system of electric charges which keep in equilibrium only through the action of their electrostatic forces con-tracts of itself as soon as it is set into motion; or, more accurately the electromagnetic forces that arise when the system is moving uni-formly change the configuration of equilibrium in such a way that every length is contracted in the direction of its motion by the factor