November 20, 1914] 



SCIENCE 



725 



ttat identical clocks in the moving system must 

 go slower. In such a case, any natural phe- 

 nomenon, preferably a vacuum phenomenon 

 like the velocity of light, is the same in all 

 systems, moving or at rest. One system is 

 as good as another. All observation is rela- 

 tive. The equations of this celebrated prin- 

 ciple of relativity, culminating in Einstein's 

 famous addition theorem of velocities belong- 

 ing to different systems — an ultimate break 

 vfith the Galileo transformation, where time 

 has the same absolute value everywhere — ^have 

 been the very focus of discussion for the last 

 ten years. 

 ' In its original form, the principle is as yet 

 rather a detached statement, adapted to defi- 

 nite purposes but lacking in mathematical 

 elegance. It was left to the genius of Minkowski 

 (1908) to mould this flotsam of ideas into a 

 philosophical system of extraordinary sym- 

 metry and breadth, the promise of which it is, 

 as yet, too soon to adequately appreciate. In 

 fact, the untimely death of Minkowski was an 

 irreparable loss to science, even if with Hilbert 

 we resignedly conclude to be grateful for what 

 he has done for us. Minkowski's world, as he 

 himseK remarks, is a response of modern 

 mathematical culture to the urgent demands 

 of the laboratory, and therein lies its strength. 

 In the minds of prominent thinkers it is a 

 philosophical revolution, an inversion of 

 thought, as far-reaching in scope as the simi- 

 lar revolution of Copernicus. " Let space and 

 time be submerged," cries Minkowski in an 

 impassioned utterance, " Sie sollen in den 

 Schatten versinken," to make way for a single 

 unified world; in other words, let the incanta- 

 tion ring in a world in which the variables 

 ^, y, ^, t, are linked with ties as inherent and 

 indissoluble as the variables x, y, z, in common 

 space. So understood, every point in space, 

 even if at rest, describes a world line, which 

 may be referred to and is contained between 

 the two extremities of the time axis. Uniform 

 motion is a straight world line. Any other 

 motion an appropriately curved world line. 

 World time is the length of a world line in 

 relation to the speed of light. These world 

 lines are thus a veritable warp and woof of the 



Deity. With Goethe we may say " Sie weben 

 der Gotheit ewig Gewand" — or recall the 

 curious passage of Wagner's Parsifal "Du 

 siehst mein Sohn, zum Raum wird hier die 

 Zeit." 



To establish the connection between the 

 four variables which shall be invariant in case 

 of linear time transformations as is the case 

 in Newton's dynamics, or that shall embrace 

 the Einstein transformations as a special case, 

 Minkowski postulates a fouj-dimensional 

 hyperboloid with a single parameter c, the 

 velocity of light, given by the reciprocal of 

 the time axis. The other parameters are one. 

 The hyperboloid is now usually made equi- 

 lateral by calling the time variable d,. The 

 intersection of the a;i-plane with this hyper- 

 boloid, thus cuts out two hyperbolas symmet- 

 rically above and below the a;-axis, the former 

 (for positive time) alone being considered. 

 The major axis is again the reciprocal of c, 

 the minor axis a unit. 



Now if the hyperbola in question with its 

 parameter c is referred to conjugate diameters, 

 it is easily shovra that the oblique time and 

 a;-axes imply all the transformations of the 

 theory of relativity, for the same c. The equa- 

 tion of the hyperbola is an invariant with 

 relation to the new axes. The axes, or units 

 of measurement, are proportionately increased, 

 the specifications or numerics decreased, but 

 the ties of the variables are exactly the same 

 as before. Minkowski calls this the group Gte- 

 Velocities greater than c are imaginary and 

 are thus essentially excluded. 



On the other hand, if the parameter c be 

 supposed to increase to infinity, the symmet- 

 rical hyperbola eventually coincides with the 

 a;-axis, eliminating the time axis, and referring 

 the whole system back to Newton's dynamics. 

 This is the transitional group G°°. 



The generalized time is then the new varia- 

 ble of which X, y, z and i are all functions. 

 Every translational vector now has four com- 

 ponents and the rotational vectors six com- 

 ponents, corresponding to the six pairs of 

 variables or planes of rotation. One may even 

 add that the new world, like Csesar's Gaul, 

 is divided into three parts by the asymptotic 



