ri PRESIDENTS ADDRESS — SECTION A. 



Now, if we assume that at an infinite distance from gravitating 

 masses a free particle moves with uniform velocity, and consequently 

 has a straight world line, it follows that at an infinite distance from 

 gravitating masses the general expression iov ds^ can be reduced to 

 dxi^ + dx-i^ -\- dx^^ -|- dxi^ by a suitable choice of co-ordinates. 



The Absolute Differential Calculus of Levi-Civita and Ricci gives 

 in the form of co- variant equations the conditions under which the 

 general quadratic differential form can be reduced to the sum of squares 

 of differentials. These conditions are given by the vanishing of the 

 Riemann-ChristofEel tensor. 



Einstein takes this as his Law of Gravitation at an infinite distance 

 from the attracting masses and, noticing that the. vanishing of the 

 contracted tensor gives a less stringent set of equations among the g 



which are satisfied when the equations given by the vanishing of the 

 uncontracted tensor are satisfied, he ado])ts the vanishing of the 

 contracted tensor G , as his statement of the Law of Gravitation. 



The equations 6^ = give six independent relations among 



the g in the form of second order partial differential equations. 



It should be noticed that the Einstein law is not a necessary con- 

 sequence of his original assumption. His analysis of the implications 

 of his assumption indicates certain general characteristics of the law, 

 and he finally adopts the simplest law of this type that he can find. 

 Its final justification must come from the experimental evidence. 



Before considering this evidence I would direct your attention to 

 the geometrical interpretation of Einstein's work. 



Riemann has shown that the metric })roperties of an n-dimen- 

 sional continuum de))end on the expression for the differential 

 element of length in terms of the differentials of the n co-ordinates. 

 Further, he has pointed out that in a continuum specified by 

 ds^ = Eq„ dx,dx , the q„ , hnin -I- 1) in number, can be made to satisfy 



n given conditions by suitable choice of co-ordinates. There remain 

 then |w{« — 1) functions of the g which express intrinsic properties 



of the continuum. These properties, he shows, are connected with 

 the curvature and can be ex]:)ressed in terms of the curvature. In the 

 case in which the length element can be expressed as the sum of squares, 

 he calls the continuum flat. 



Einstein's assum])tion that the g^^^, in his four-dimensional space 



time manifold satisfy six specified conditions is thus equivalent to the 

 assumption that the manifold is warped in a specified way. The further 

 assumption that at an infinite distance from gravitating matter the 

 square of the linear element takes the form Edx'^ amounts to the 

 assumption that this warping is characteristic of the presence of material 

 bodies and that at an infinite distance from then\ the continuum is 

 flat. 



