PRESIDENT S ADDRESS — SECTION A . 7 



As a first step towards the problem of modifying the Minkowski four- 

 fold in such a way as to explain gravitation as an illusion due to choice 

 of co-ordinates, Einstein notes that along the path of a particle moving 



freely in the absence of a gravitational field, -^, -f-, and — are 

 •^ * (h fit (It 



constant. Hence the world line of the particle is straight. Its equation is 



therefore given by 



8fds = 0, 



where ds is the element of length in the Minkowski world, and con- 

 sequently 



ds' = dx' + dy' + dz^ - c'dt\ 



He takes this property as characteristic of a freely moving particle 

 and investigates the form to be given to the length element ds in the 

 neighbourhood of attracting matter under which the world line of a 

 moving particle will be given by 



S/ds = 0. 



If the element is given bv 



ds" = I^g^,Xf,x^, 



where the x denote the co-ordinates, the </ factors have two inter- 

 pretations. 



From one j)oint of view they specif y,the form of the four-dimensional 

 manifold ; from the other they determine the curvature in the manifold 

 given by rectangular co-ordinates (.Xj, Xn, Xa, x^) of the world line 

 Sfds = ; that is, they determine the gravitational field to which 

 an observer using these co-ordinates will attribvite the acceleration 

 of the moving particle. 



Now, from an abstract mathematical standpoint we can give the g 



any form that we please. We believe, however, that in the gravi- 

 tational fields occurring in nature the g' are connected by certain 



relationships. The statement of these relationships is the Law of 

 Gravitation. 



In seeking for this Law of Gravitation Einstein notes that a change 

 of co-ordinates will change the (f ; that is, will change the 

 gravitational field. 



By the Principle of Equivalence this geometrical change is in no way 

 distinguishable from a gravitational change in the ordinary sense. 

 Hence the Law of Gravitation expressed in terms of the new g^^^, and 



the new co-ordinates is the same as that exjjressed in terms of the 

 old g and the old co-ordinates. It follows that the Law of 



Gravitation must be expressed by a set of co-variant equations, 

 lost.— 4 



