654 SCIENCE PROGRESS 



representation, which can be obtained by imagining the /-axis drawn through 

 perpendicular to the paper, and rotating the whole two-dimensional figure about 

 the axis WOE. The transformation consists in passing from one to another set 

 of three semidiameters of the conjugate hyperboloids x % +y* - c 2 l* =- + i. The four- 

 dimensional transformation required for motion in three dimensions consists in a 

 process which may be described analogically as passing from one set consisting of 

 one semidiameter of the first of the pair of conjugate four-dimensional hyperboloids 

 v t +y 2 + z i -c i P= ± i and a pencil of semidiameters of the second to another 

 corresponding set. 



If the specification of physical phenomena were complete in terms of observed 

 coincidences, they could be expressed by means of intersections of world-lines 

 filling the whole world. Any one observer's (correct) results would be expressed 

 in terms of a. field-figure in the world, composed of a net of world lines of which 

 the point-instants determined by their intersections would alone be significant, 

 and correct observations of all other observers would give rise to field figures 

 apparently differing widely from each other, yet all essentially identical in so far 

 as they record identical phenomena, for any such would be transformable one into 

 the other by mere deformations of the world with its world lines. The deforma- 

 tions can be followed up analytically by introducing coordinates by attaching to 

 each point-instant P in a field-figure F four numbers, say x x , x%, x 3 , x 4 , such that 

 along any line in F they change continuously and no two points have the same 

 four numbers. Passage from tetrad to tetrad of numbers by means of the Lorentz 

 transformation will then carry us from point to point of F. The deformations 

 can also be followed geometrically by imagining as constructed around each 

 point-instant an infinitesimal four-dimensional pair of conjugate hyperboloids 

 xl + xl + xl-\-x^= +e* where e is a real infinitesimal quantity, and so mapping out 

 the world X\, x% xs, x\ by these indicatrices, very much as an electric field is 

 mapped out by Faraday lines (unit tubes) of electric force. 



Gravitation impresses the physicist as being something more general and more 

 fundamental than other natural forces. At a given point in a gravitational field, 

 every material substance receives the same gravitational acceleration whatever its 

 physical or chemical state. Moreover, no satisfactory theory of its nature has yet 

 been found, and every hypothesis that appeared, even temporarily, likely to lead 

 to a solution has involved seeking its source in interaction between the known 

 universe and a greater unknown universe in which ours would be included. Now, 

 in the Minkowski world, any point-instant in a gravitational field, representing a 

 mateiial particle moving under the acceleration due to the field, may be trans- 

 formed to rest, and so will apparently cease to be subject to acceleration. And it 

 is only through the acceleration to which it gives rise that a gravtational field is 

 known to us. Such considerations led Einstein to formulate what is known as 

 "The Equivalence Principle," that there is no distinction between a gravitational 

 field and a field of acceleration. Its admission would involve the conclusion that 

 electromagnetic acceleration must be included as dependent on the gravitational 

 field, a statement which I think summarises the theoretical objections that might, 

 a priori, be raised against it. If true, gravitation must be determined by the 

 coordinates. , 



Einstein's procedure in developing the result of this hypothesis is first to define 

 a Minkowski world with three imaginary space-axes, ix\, ix>x, ix,t, and a real time- 

 axis, x^ so that the invariant element of length joining two adjacent points, PQ, is 

 given by the equation ds*= - dx 1 ]- dx'\- dx'\ + dx'\, where dx\, etc., are the changes 

 jn the coordinates in passing from P to Q. There is then no gravitational field. 



