432 



NA TURE 



[September 23, 1922 



Summary of the Theory of Relativity. 



By Prof. H. T. H. Piaggio, University College, Nottingham. 



I. Breakdown of Older Theories. — The older 

 electromagnetic theory of moving bodies did not 

 agree with experiment, or even with itself. For 

 example, the theory of a magnet moving in a straight 

 line towards a fixed conductor gave results quite 

 different from those of the theory of a conductor 

 moving in a straight line with the same velocity 

 towards a fixed magnet. Yet experiment showed 

 that the results should be the same, depending only 

 on the relative velocity. Again, the aether was 

 assumed to be at the same time quite unaffected by 

 the earth's motion (to explain aberration), partly 

 affected (to explain Fizeau's water-tube experiment), 

 and entirely affected (to explain the experiments of 

 Michelson and Morley, Lodge, Rowland, Rayleigh and 

 Brace, Trouton and Noble, and others). 



II. Fundamental Assumptions of Einstein's 

 Restricted Theory (1905). — This takes over Max- 

 well's theory so far as it applies to bodies at rest 

 relative to the earth and deals with other systems by 

 the two following assumptions : 



(1) All electrodynamical and optical equations 

 which hold for a system S hold also for another 

 system S' which, relative to S, moves with uniform 

 velocity v in a straight line. 



(2) Light is propagated in a vacuum with a velocity 

 c which appears the same for observers in S and S'. 



Kinematical deductions from these assumptions. — 

 These imply that the measures of time and space in 

 S and S' must be such that 



.\ 2 + y 2 + z 2 -c 2 t 2 = x' 2 + y' 2 + z' 2 -cH' 2 , 

 from which, taking the corresponding axes in each 

 system to be parallel and the relative velocity to be 

 along Ox (or Ox'), we can prove that 



x'=B{x-vt),y'=y,z'=z,t' = 8[t-~ ) 



where 8= ( I- 



(A): 



hence two observers, one in S and one in S', will each 



imagine 

 (i.) that a rod along Ox (or Ox') in the other's 



system has contracted in the ratio 8 : 1 ; 

 (ii.) that the other's clocks (supposed controlled by 

 light signals) lose, taking 8 seconds instead of 

 1 for a beat ; 



(hi.) that the events which the other takes as simul- 

 taneous are not so. 



What they will agree about is the velocity of light, 



c, their own relative speed, and the interval between 



two sets of values, x, y, z, t, for two events, this 



interval being defined as 



>Jic*(h-h)*-(% i -% 1 ) !l -[y. i ~y 1 Y-{z i -z 1 y}, 



which may be written, 



^{c 2 dt 2 -dx 2 -dy 2 -dz 2 }. 

 It is generally denoted by ds. 



(lx_ t[ 



From equations (A) 



dt' 



v d 1 ' 

 ' <■- dt 



so that if the 



velocity of the body moving along 0.v"(or^Ox') is V in 

 the system S and V' in the system S' 



\t> V ~ v \r V ' +v 

 V = TT, or V= J77. 



1 5- IH — sr 



c* c- 



This is confirmed by Fizeau's water -tube experi- 

 ment, and (it is claimed) by Majorana's moving 

 mirror experiment. From this formula we see that 



by combining two velocities V and v, each of which 

 is smaller than c, we obtain a velocity V which is 

 always smaller than c. (The statement that " no 

 velocity can exceed c " is too sweeping; the velocity 

 of li.uhi in ,i thin metal prism exceeds c.) 



Electrodynamical deductions from these assumptions. 

 — Transforming Maxwell's equations for free space 

 in which electrons move with velocity V along Ox we 

 get from assumption (1) and equations (A) that 

 E',.= E„ 



e'.=/»(s.+Jh,), 



/ = 8 P (i~ V ^). 



H'. r = H.,. 



h',=b(h v 



+ -E. 



(B). 



The expression for p' gives the remarkable result 

 that the charge on an electron appears the same in 

 both systems. From these we can deduce : 

 (i.) Doppler's effect in the modified form — 



/'=/ 



i + - 



where v is the relative velocity 



in the line of sight, /and/' the frequencies ; 

 (ii.) a modified law of aberration ; 

 (iii.) the force exerted by light on a moving mirror ; 

 (iv.) the electric and magnetic fields due to a uni- 

 formly moving electron. 

 The differences between these forms and those 

 given by older theories are too small to be detected 

 l>v experiment. 



Dynamics of an electron (slowly accelerated). — With 

 tie additional assumption that every electron has 

 a constant m associated with it, such that force = 

 m x acceleration at the instant when the electron is at 

 rest in the system of co-ordinates used (and only at 

 that instant), we deduce that in any other system the 

 equations of motion are 



iu i 3 ' ,,l = eE„ 



dt 2 



d*y 



d 2 z 



where e is the charge on the 

 electron and the axis of x is 

 taken in the direction of its 

 velocity v. The second and 

 third of these equations are 

 confirmed by Bucherer's ex- 

 periments. 



If, witii Lorentz, we take the right-hand sides as 



the components of the force, and retain the old law 



foi 1 e mass x acceleration, we find it necessary to speak 



of a longitudinal mass mB 3 and a transverse mass mB. 



But we may rewrite the left-hand sides in the 



mB 



dt- 



■(e.-Jh.). 



symmetrical form 



MS- 



This suggests the definitions : 



mass (M) = mass at low speeds x 8 (both for 

 longitudinal and transverse mass) ; 

 momentum = mass x velocity ; 



force = rate of change of momentum. 

 Defining work in the usual way from force and dis- 

 placement, we can further deduce : 

 Work done on an electron = increase of its kinetic 

 energy, provided that kinetic energy is defined as 

 Mc 2 + a constant = m8c 2 + a constant. 



If we take the constant equal to - me'-, this new 

 definition reduces to huv- approximately for small 

 values of vjc. From Maxwell's equations we can derive 

 four relations for an isolated system of electrons which 



NO. 2760, VOL. I IO] 



