6 president's address — SECTION A. 



knowledge can be summed up in the statement that certain world lines 

 intersect. The lines will still intersect if the Minkowski continuum is 

 deformed without tearing. Hence if one observer uses co-ordinates 

 (Xj, X2, x^, a; 4) and a second uses co-ordinates (a;/, x^' , x^' , x^') which are 

 functions of the former, they will observe the same intersections of 

 world lines, but they .will interpret the results of their observations 

 differently. For example, suppose X4, x^' are the co-ordinates depend- 

 ing on the time in the usual sense and [x^, x^, x^, (x/, x^ , x^') are 

 space co-ordinates. Suppose further that x-^ = Xi , x^ := x^ , X3 ^ x^' , 

 (Xi/ic)^ = (x^'/lc). 



Now let the first observer note the motion of a particle oscillating 

 between the origin and the point {a.0.0) reaching the origin when 



(xj/ic) = 0, ?>, 26, 36, The second observer will observe it 



oscillating over the same spacial range and reaching the origin when 

 {x//ic)=0,h\2ibi,SihK 



To the first the oscillation will have a constant amplitude and con- 

 stant period ; to the second it will have a constant amplitude and a 

 decreasing period. .This difference arises from the fact that in the case 

 of the second observer the Minkowski continuum has been distorted 

 by a strain along the time axis, but the observer, in ignorance of this, 

 ignores the consequent variable time scale in the interpretation of his 

 observations. Neither observer is in a position to assert that his 

 interpretation is more correct than that of the other. The amplitude 

 and period are in each case relationships between the oscillating particle 

 and the observer ; and the observer's estimate of the relationship 

 depends on his concept of the continuum. One looks on the continuum 

 as a fourfold characterized by the rectangular Cartesian co-ordinates 

 {Xi, X2, Xs, Xi), the other imagines it to be given by the co-ordinates 

 {Xi, X2', X3', Xi), and he also considers his co-ordinates to be a rect- 

 angular Cartesian system. From the different concepts of the 

 continuum arise the different interpretations of the observations. 



The above illustration will make clear the main idea involved in 

 Einstein's Principle of Equivalence. Our ordinary concept of the 

 Minko'wski world is a fourfold determined by rectangular co-ordinates 

 {x, y, z, ict) where x, y, z are the usual three-dimensional space co- 

 ordinates and t is the time in the ordinary sense. In terms of this 

 idea we interpret our observations of moving material bodies as indicat- 

 ing an acceleration of the bodies in the presence of other masses. We 

 accept this interpretation and attribute the acceleration to gravitational 

 forces. Einstein suggests that we should adopt the alternative course 

 of attributing the acceleration to our concept of the Minkowski world ; 

 that is, to our choice of co-ordinates. This suggestion is put forward 

 in the Principle of Equivalence, which asserts that " a gravitational 

 field of force is exactly eqmvalent to a field of force introduced by a 

 transformation of the co-ordinates of reference, so that by no possible 

 experiment can we distinguish between them." 



