1918] on Gravitation and the Principle of Relativity 221 



nature as the undeformed bladder. The bladder represents Min- 

 kowski's space-time world, in which the world-lines were drawn ; so 

 we can squeeze Minkowski's world in any way without altering the 

 course of events. We do not usually use the vulgar word squeeze ; 

 we call it a mathematical transformation, but it means the same 

 thing. 



The laws of nature in their most general form must describe 

 correctly the behaviour of the world-lines in either the undistorted 

 or the distorted model, because it is indifferent which we take as the 

 .true representation of the course of nature. That is a very important 

 principle ; but, being almost a truism, it does not in itself help us 

 to determine the laws of nature without making some additional 

 hypothesis. There is one law— the law of gravitation — which especially 

 attracts our attention at this point, and we shall look into it more 

 closely. 



We know that one particle attracts another particle, and so 

 influences the history of its motion. This evidently means that one 

 world-line will deflect any other world-line in its neighbourhood. 

 Apart from this influence, the world-line runs straight, bending 

 neither to the right nor to the left, provided the bladder is in its 

 undistorted state, i.e. provided we use Minkowski's original space- 

 time. That is not so much a matter of observation as of definition. 

 It defines what we are to regard as the undistorted state, though it is 

 by observation that we learn that it is possible to find a space-time 

 in which the world-lines run straight when undisturbed by gravita- 

 tional or other forces. I must own that there is a certain logical 

 difficulty in saying that a world-line runs straight when there are 

 no others near it ; because in that case there could be no inter- 

 sections, and we could learn nothing about its course by observation. 

 However, that is not a serious difficulty, though you may be reminded 

 of the sage remark, " If there were no matter in the universe, the law 

 of gravitation would fall to the ground." 



We have to admit, then, that a world-line can be bent by the 

 proximity of other woild-lines. It can also be bent, as you see, by 

 the proximity of my thumb. The suggestion arises, may not the two 

 modes of bending be essentially the same ? The bending by my 

 thumb (a mathematical transformation of space and time) is in a 

 sense spurious ; the world-line is pursuing a course which is straight 

 relative to the original material. Or we may perhaps best put it this 

 way— the world-line still continues to take the shortest path between 

 two points, only it reckons distance according to the length that 

 would be occupied in the unstretched state of the bladder. It is 

 suggested that the deflection of a world-line by gravitation is of the 

 same nature ; from each world -line a state of distortion radiates, as 

 if from a badly puckered seam, and any other world-line takes 

 the shortest course through this distorted region, which would 

 immediately become straight if the strain could be undone. The 



