August 4, 192 1] 



NATURE 



717 



his present position at distance x\ so as to be moving 

 away from the light instead of towards it. The light 

 will now have to catch him up ; and he may think, at 

 first, that the ray which left the source at the instant 

 he began his return journey will take the original 

 time t to reach him, since it now has to travel the 

 full distance x. But he will have to travel a little 

 further than the original position, and take a little 

 longer time, before he is overtaken ; and he cannot 

 write the reciprocal equations 



and 



x =x - t/t, 



because they are inconsistent with the previous ones. 

 To make the two pairs of equations agree (as rela- 

 tivity demands), either x must equal x' , which frus- 

 trates the whole experiment, or a common factor must 

 be introduced, say /3, such that 



t'-mt 



and 





/3/(i 



'■(-")■ 



This will render them harmonious, and a suitable 

 value (the only right value) of fi is easily reckoned — 

 again mechanically, without further hypothesis — 

 namely, 



^ 



(-":)= 



If that is satisfied, the reversal of the journey will not 

 give any different result ; there is perfect reciprocity. 

 You cannot by an experiment of reversing your motion 

 with regard to light, or reflecting back the light with 

 regard to the observer, discriminate between c—u and 

 c+u ; nor can you discriminate either from c. 



Now this fi factor is the FitzGerald-Lorentz contrac- 

 tion ; the experiment thus neutralised is the Michel- 

 son-Morley experiment ; and the direct supposition 

 that an observer must find c — ii, and c, and c+u all the 

 same, or at least indistinguishable by observation, 

 and that there is nothing more to be said, is the point 

 of view of Einstein. 



These names must suffice to suggest a flood of ideas 

 to those who have read about the subject. 



To sum up compactly : — 



Assume that you cannot help measuring the same 

 speed of light whether you be moving or stationary, 

 so that xjt and x' jt' both equal c (the accented letters 

 referring to the measurements made when you were 

 moving with speed u to meet the light), then allow 

 that xjt' is not equal to c+u, as you would expect, 

 nor x' It equal to c-u (for in that case xx' /tf would 

 equal c^-u* instead of c'^), but that, instead, 



^=fiic+u) and ^ = ^(c~u), 

 which, together, require that 



then all the rest follows. 



The Contraction. 



A customary and older interpretation of the 

 introduction of the factor j8 — to complete and 

 make accurate what then became the Larmor- 

 Lorentz transformation — is that the measuring 

 rod with which you are hypothetically supposed to 

 measure x or x' shrinks to 1//8 of its normal 



NO. 2701, VOL. 107] 



length if the experimenter is moving either to or 

 fro with speed u, so that all distances in the direc- 

 tion of motion measure out a little bigger than 

 they otherwise would ; more steps of the yard 

 measure having to be taken. Note that space or 

 aether does not shrink, but only the matter in 

 space. The distance x has not changed, but only 

 the instrument with which you hypothetically 

 measure it. That having shrunk, the fixed dis- 

 tance measures out longer. The same thing 

 happens with the instrument whereby you are sup- 

 posed to measure time. Both distance and time 

 of journey are abbreviated by approach, but, to 

 measurement, not so much as an unchanged 

 measurer would give. They are both lengthened 

 by recession, and the measurements give rather 

 more increment than might have been expected. 



The ratio between measurements made during 

 uniform approach, and the same made during rela- 

 tive rest, is 



X / V cJ V \c + u/ 



This line, with the definition c^xjt, is the 

 briefest possible summary of the transformation 

 equations. 



A short and easy way of getting, or at least of 

 recording, the essence of the transformation is to 

 allow for the contraction of the hypothetical measur- 

 ing rod by niultiplying any distance across space sup- 

 posed to be measured by a flying observer — flying 

 towards or away from a distant event which really 

 occurred at the instant he started to fly — by an un- 

 defined numerical coefficient ^, and omitting this 

 factor from any distance which he could have 

 measured at rest before starting. 



Thus let an event occur at the origin, and let an 

 observer at x and t immediately begin travelling 

 towards it, so as to meet the light at a place which 

 appears to him to be x' and t' , the combined velocity 

 over the original distance being c+u, he can correct 

 his x' measurement, which has been traversed by the 

 light alone, and write 



X _^x\ 

 c+u c 



while if' he started from the leisurely measured x' 

 and t' position, directly the event occurred at the 

 origin, and receded so that the light overtook him at 

 what appears to him to be a place x and t, coming 

 with the relative velocity c — u, he can correct his x 

 measurement for the whole distance traversed by the 

 light, and write 



x' _/SU- , 



saying, if he likes, that it is just the same as if he 



had stood still and the light had come to him with 



diminished speed. (Or he might time his own 



Qx — x' 3x\ 



journey as , and equate that to ^— ). 



u c I 



Com- 



bining these equations with the definition 



xf V 



and not troubling about the y and z co-ordinates, 

 which remain unchanged and need no attention, we 

 get the Lorentz transformation complete (and inci- 

 dentally we see that the usual differential invariant 

 ds^ = dx^+dy*+dz^-c^df 'is always zero for light). 



