February 17, 192 1] 



NATURE 



783 



Now in order that the special principle of rela- 

 tivity may hold, it is necessary that all the equa- 

 tions of physics do not alter their form in the 

 transition from one inertial system to another, 

 when we make use of the Lorentz transformation 

 for the calculation of this chang'e. In the lan- 

 guage of mathematics, ail systems of equations 

 that express physical laws must be co-variant with 

 respect to the Lorentz transformation. Thus, 

 from the point of view of method, the special prin- 

 ciple of relativity is comparable to Carnot's prin- 

 ciple of the impossibility of perpetual motion of 

 the second kind, for, like the latter, it supplies us 

 with a general condition which all natural laws 

 must satisfy. 



Later, H. Minkowski found a particularly 

 elegant and suggestive expression for this 

 condition of co-variance, one which reveals a 

 formal relationship between Euclidean geometry 

 of three dimensions and the space-time continuum 

 of physics. 



Eudiiean Geometry of 



Three Dimensions. 

 Corresponding to two 

 neighbourinf; points in 

 space, there exists a 

 numerical measure (dis- 

 tance ds) which conforms 

 to the equation 



ds'=dx,'+dx,'-^dx,'. 



It is independent of the 

 system of co-ordinates 

 chosen, and can be 

 measured with the unit 

 measuring-rod. 



The permissible trans- 

 formations are of such a 

 character that the expres- 

 sion for ds' is invariant, 

 i.e. the linear orthogonal 

 transformations are per- 

 missible. 



With rMp«ct to these 



I ' -. the laws 



fjeorfHrtry 



.-in.' iiivari.iiu. 



Special Theory of 

 Relativity. 



Corresponding to two 

 neighbouring points in 

 space-time (point events), 

 there exists a numerical 

 measure (distance ds) 

 which conforms to the 

 equation 



ds'=dx,'+dx,'+dx,'+dx,' 



It is independent of the 

 inertial system chosen, 

 and can be measured 

 with the unit measuring- 

 rod and a standard clock. 

 X,, X,, X, are here 

 rectangular co-ordinates, 

 whilst Xt= >/ — 1 ct is the 

 time multiplied by the 

 imaginary unit and by 

 he velocity of light. 



The permissible trans- 

 formations are of sucJi a 

 character that the cxpres- 

 .sion for ds' is invariant, 

 I.e. those linear ortho- 

 gonal .substitutions are 

 permissible which main- 

 tain the semblance of 

 reality of x,, x,, x„ x,. 

 These substitutions are 

 the Ix>rent2 transforma- 

 tions. 



With respect to these 

 transformations, the laws 

 of physics are invariant. 



From this it follows that, in respect of its rdle 

 in the equations of physics, though not with regard 

 to its physical significance, time is equivalent to 

 the space co-ordinate.s (apart from the relations 

 of reality). From thi» point of view, physics is, 

 as it were, a Fuclidcan geometry of four dimen- 



NO. 2677, VOL. 106] 



sions, or, more correctly, a statics in a four- 

 dimensional Euclidean continuum. 



The development of the special theory of rela- 

 tivity consists of two main steps, namely, the 

 adaptation of the space-time " metrics " to 

 Maxwell's electro-dynamics, and an adaptation of 

 the rest of physics to that altered space-time 

 " metrics." The first of these processes yields 

 the relativity of simultaneity, the influence of 

 motion on measuring-rods and clocks, a modifica- 

 tion of kinematics, and in particular a new theorem 

 of addition of velocities. The second process 

 supplies us with a modification of Newton's law 

 of motion for large velocities, together with 

 information of fundamental importance on the 

 nature of inertial mass. 



It was found that inertia is not a fundamental 

 property of matter, nor, indeed, an irreducible 

 magnitude, but a property of energy. If an 

 amount of energy E be given to a body, the 

 inertial mass of the body increases by an amount 

 E/c', where c is the velocity of light in vacuo. 

 On the other hand, a body of mass m is to be 

 regarded as a store of energy of magnitude tnc*. 



Furthermore, it was soon found impossible to 

 link up the science of gravitation with the special 

 theory of relativity in a natural manner. In this 

 connection I was struck by the fact that the force 

 of gravitation possesses a fundamental property, 

 which distinguishes it from electro-magnetic 

 forces. All bodies fall in a gravitational field with 

 the same acceleration, or — what is only another 

 formulation of the same fact — the gravitational 

 and inertial masses of a body are numerically 

 equal to each other. This numerical equality 

 suggests identity in character. Can gravitation 

 and inertia be identical? This question leads 

 directly to the General Theory of Relativity. Is it 

 not possible for me to regard the earth as free 

 from rotation, if I conceive of the centrifugal 

 force, which acts on all bodies at rest relatively 

 to the earth, as being a " real " field of gravita- 

 tion, or part of such a field? If this idea can be 

 carried out, then we shall have proved in very 

 truth the identity of gravitation and inertia. For 

 the same property which is regarded as inertia 

 from the point of view of a system not taking 

 part in the rotation can be interpreted as gravita- 

 tion when considered with respect to a system that 

 shares the rotation. According to Newton, this 

 interpretation is impossible, because by Newton's 

 law the centrifugal field cannot be regarded as 

 being produced by matter, and because in 

 Newton's theory there is no place for a " real 

 field of the " Koriolis-field " type. But perhaps 

 Newton's law of field could be replaced by another 

 that fits in with the field which holds with respect 

 to a "rotating" system of co-ordinates? My 

 conviction of the identity of inertial and gravita- 

 tional mass aroused within me the feeling of abso- 

 lute confidence in the correctness of this interpre- 

 tation. In this connection I gained encourage- 

 ment from the following idea. We arc familiar 

 with the " apparent " fields which are valid rcia- 



