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SCIENCE 



[N. S. Vol. U. No. 1315 



meaning. It is scarcely fair to call Min- 

 kowski's development a picture; for to us a 

 picture can never have more than three 

 dimensions, our senses limit us; while his 

 picture calls for perception of iova dimen- 

 sions. It is this fact that renders any even 

 semi-popular discussion of Minkowski's work 

 so impossible. We can all see that for us to 

 describe any event a knowledge of four 

 coordinates is necessary, three for the space 

 specification and one for the time. A com- 

 plete picture could be given then by a point 

 in four dimensions. All four coordinates are 

 necessary: we never observe an event except 

 at a certain time, and we never observe an 

 instant of time except with reference to space. 

 Discussing the laws of electromagnetic phe- 

 nomena, Minkowski showed how in a space of 

 four dimensions, by a suitable definition of 

 axes, the mathematical transformation of 

 Lorentz and Einstein could be described by 

 a rotation of the set of axes. We are all 

 accustomed to a rotation of our ordinary 

 cartesian set of axes describing the position 

 of a point. We ordinarily choose our axes at 

 any location on the earth as follows: one 

 vertical, one east and west, one north and 

 south. So if we move from any one labora- 

 tory to another, we change our axes; they 

 are always orthogonal, but in moving from 

 place to place there is a rotation. Similarly, 

 Minkowski showed that if we choose four 

 orthogonal axes at any point on the earth, 

 according to his method, to represent a space- 

 time point using the method of measuring 

 space and time intervals as outlined by Ein- 

 stein; and, if an observer on Arcturus used a 

 similar set of axes and the method of meas- 

 urement which he naturally would, the set of 

 axes of the latter could be obtained from 

 those of the observer on the earth by a pure 

 rotation (and naturally a transfer of the 

 origin). This is a beautiful geometrical re- 

 sult. To complete my statement of the 

 method, I must add that instead of using as 

 his fourth axis one along which numerical 

 values of time are laid off, Minkowski defined 

 his fourth coordinate as the product of time 

 and the imaginary constant, the square root 



of minus one. This introduction of imagi- 

 nary quantities might be expected, ixsssibly, 

 to introduce difficulties; but, in reality, it is 

 the very essence of the simplicity of the geo- 

 metrical description just given of the rotation 

 of the sets of axes. It thus appears that 

 different observers situated at different points 

 in the universe would each have their own set 

 of axes, all different, yet all connected by the 

 fact that any one can be rotated so as to 

 coincide with any other. This means that 

 there is no one direction in the four dimen- 

 sional space that corresponds to time for all 

 observers. Just as with reference to the 

 earth there is no direction which can be 

 called vertical for all observers living on the 

 earth. In the sense of an absolute meaning 

 the words " up and down," " before and after," 

 " sooner or later," are entirely meaningless. 

 This concept of Minkowski's may be made 

 clearer, perhaps, by the following process of 

 thought. If we take a section through our 

 three dimensional space, we have a plane, i. e., 

 a two-dimensional space. Similarly, if a sec- 

 tion is made through a four-dimensional 

 space, one of three dimensions is obtained. 

 Thus, for an observer on the earth a definite 

 section of Minkowski's four dimensional space 

 will give us our ordinary three-dimensional 

 one; so that this section will, as it were, 

 break up Minkowski's space into oxa space 

 and give us our ordinary time. Similarly, a 

 different section would have to be used for 

 the observer on Arcturus; but by a suitable 

 selection he would get his own familiar three- 

 dimensional space and his own time. Thus 

 the space defined by Minkowski is completel.y 

 isotropic in reference to measured lengths 

 and times, there is absolutely no difference 

 between any two directions in an absolute 

 sense; for any particular observer, of course, 

 a particular section will cause the space to 

 fall apart so as to suit his habits of measure- 

 ment; any section, however, taken at random 

 will do the same thing for some observer 

 somewhere. From another point of view, 

 that of Lorentz and Einstein, it is obvious 

 that, since this four dimensional space is 

 isotropic, the expression of the laws of elec- 



