March 12, 1920] 



SCIENCE 



257 



tromagnetic phenomena take identical mathe- 

 matical forms when expressed by any observer. 



The question of course must be raised as 

 to what can be said in regard to phenomena 

 which so far as we know do not have an 

 electromagnetic origin. In particular what 

 can be done with respect to gravitational 

 phenomena? Before, however, showing how 

 this problem was attacked by Einstein; and 

 the fact that the subject of my address is 

 Einstein's work on gravitation shows that 

 ultimately I shall explain this, I must empha- 

 size another feature of Minkowski's geometry. 

 To describe the space-time characteristics of 

 any event a point, defined by its four coordi- 

 nates, is sufficient; so, if one observes the life- 

 history of any entity, e. g., a particle of mat- 

 ter, a light-wave, etc., he observes a sequence 

 of points in the space- time continuum; that 

 is, the life-history of any entity is described 

 fully by a line in this space. Such a line was 

 called by Minkowski a " world-line." Further, 

 from a different point of view, all of our 

 observations of nature are in reality observa- 

 tions of coincidences, e. g., if one reads a 

 thermometer, what he does is to note the 

 coincidence of the end of the column of 

 mercury with a certain scale division on the 

 thermometer tube. In other words, thinking 

 of the world-line of the end of the mercury 

 column and the world-line of the scale divi- 

 sion, what we have observed was the inter- 

 section or crossing of these lines. In a 

 similar manner any observation may be 

 analyzed; and remembering that light rays, 

 a point on the retina of the eye, etc., all have 

 their world lines, it will be recognized that it 

 is a perfectly accurate statement to say that 

 every observation is the perception of the in- 

 tersection of world-lines. Further, since all 

 we know of a world-line is the result of ob- 

 servations, it is evident that we do not know 

 a world-line as a continuous series of points, 

 but simply as a series of discontinuous points, 

 each point being where the particular world- 

 line in question is crossed by another world- 

 line. 



It is clear, moreover, that for the descrip- 

 tion of a world-line we are not limited to the 



particular set of four orthogonal axes adopted 

 by Minkowski. We can choose any set of 

 four-dimensional axes we wish. It is further 

 evident that the mathematical expression for 

 the coincidence of two points is absolutely 

 independent of our selection of reference 

 axes. If we change oiu- axes, we will change 

 the coordinates of both points simultaneously, 

 so that the question of axes ceases to be of 

 interest. But our so-called laws of nature 

 are nothing but descriptions in mathematical 

 language of our observations; we observe only 

 coincidences ; a sequence of coincidences when 

 put in mathematical tenns takes a form which 

 is independent of the selection of reference 

 axes; therefore the mathematical expression 

 of our laws of nature, of every character, 

 must be such that their form does not change 

 if we make a transformation of axes. This is 

 a simple but far-reaching deduction. 



There is a geometrical method of picturing 

 the effect of a changie of axes of reference, i. e., 

 of a m'athematical transformiation. To a man 

 in a railway coach the path of a drop of water 

 does not appear vertical, i. e., it is not parallel 

 to the edge of the window ; still less so does it 

 appear vertical to a man performing manoevres 

 in an airplane. This means that whereas with 

 reference to axes fixed to the earth the path of 

 the drop is vertical; with reference to other 

 axes, the path is not. Or, stating the conclu- 

 sion in general language, changing the axes of 

 reference (or effecting a mathematical trans- 

 formation) in general changes the shape of any 

 line. If one imagines the line forming a part 

 of the space, it is evident that if the space is 

 deformed by compression or expansion the 

 shape of the line is changed, and if sufficient 

 care is taken it is clearly -possible, by deforming 

 the sipace, to make the line take any shape de- 

 sired, or better stated, any shape ^specified by 

 the previous change of axes. It is thus possible 

 to picture a mathematical transformation as a 

 deformation of space. Thus I can draw a line 

 on a sheet of paper or of rulbbesr and by bending 

 and stretching the sheet, I can make the line 

 assume a great variety of shapes ; each of these 

 new shapes is a picture of a suitable transfor- 

 mation. 



