December ii, 1919] 



NATURE 



375 



earth notes objects falling- away from him towards 

 the earth. Ordinarily, he attributes this to the 

 earth's attraction. If he falls with them, his sense 

 of gravitation is lost. His watch ceases to press 

 on the bottom of his pocket ; his feet no longer 

 press on his boots. To this falling observer there i 

 is no gravitation. If he had time to think or make 

 observations of the propagation of light, according 

 to the principle of equivalence he would now find 

 nothing gravitational to disturb the rectilinear 

 motion of light. In other words, a ray of light 

 propagated horizontally would share in his vertical 

 motion. To an observer not falling, and, there- 

 fore, cognisant of a gravitational field, the path 

 of the ray would" therefore be bending downward 

 towards the earth. 



The systematic working out of this idea 

 requires, as has been remarked, considerable 

 mathematics. .\11 that can be attempted here is 

 to give a faint indication of the line of attack, 

 mainly by way of analogy. 



It is no new discovery to speak of time as a 

 fourth dimension. Every human mind has the 

 power in some degree of looking upon a period 

 of the history of the world as a whole. In doing 

 this, little difference is made between intervals of 

 time and intervals of space. The whole is laid 

 out before him to comprehend in one glance. He 

 can at the same time contemplate a succession of 

 events in time, and the spatial relations of those 

 events. He can, for instance, think simultane- 

 ously of the growth of the British Empire chrono- 

 logically and territorially. He can, so to speak, 

 draw a map, a four-dimensional map, incapable 

 of being drawn on paper, but none the less a 

 picture of a domain of events. 



Let us pursue the map analogy in the familiar 

 two-dimensional sense. Imagine that a map of 

 some region of the globe is drawn on some 

 material capable of extension and distortion with- 

 out physical restriction save that of the preserva- 

 tion of its continuity. No matter what distortion 

 takes place, a continuous line marking a sequence 

 of places remains continuous, and the places 

 remain in the same order along that line. The 

 map ceases to be any good as a record of distance 

 travelled, but it invariably records, certain facts, 

 as, for example, that a place called London is in 

 a region called England, and that another place 

 called Paris cannot be reached from London with- 

 out crossing a region of water. But the common 

 characteristic of maps of correctly recording the 

 shape of any small area is lost. 



The shortest path from any place on the earth's 

 surface to any other place is along a great circle ; 

 1 on all the common maps, one series of great 

 ' circles, the meridians, is mapped as a series of 

 straight lines. It might seem at first sight that our 

 extensible map might be so strained that all great 

 circles on the earth's surface might be represented 

 by straight lines. But, as a matter of fact, this 

 is not so. We might represent the meridians and 

 the great circles through a second diameter of the 

 earth as two sets of straight lines, but then every 



NO. 2615, VOL. 104] 



other great circle would be represented as a 

 curve. 



The extension of this to four dimensions gives 

 a fair idea of Einstein's basic conception. In a 

 world free from gravitation we ordinarily con- 

 ceive of free particles as being permanently at 

 rest or moving uniformly in straight lines. We 

 may imagine a four-dimensional map in which the 

 history of such a particle is recorded as a straight 

 line. If the particle is at rest, the straight line is 

 parallel to the time axis ; otherwise it is inclined 

 to it. Now if this map be strained in any manner, 

 the paths of particles are no longer represented as 

 straight lines. Any person who accepts the 

 strained map as a picture of the facts may inter- 

 pret the bent paths as evidence of a "gravitational 

 field," but this field can be explained right away 

 as due to his particular representation, for the 

 paths can all be made straight. 



But our two-dimensional analogy shows that we 

 may conceive of cases where no amount of strain- 

 ing will make all the lines that record the history 

 of free particles simultaneously straight ; pure 

 mathematics can show the precise geometrical 

 significance of this, and can write down expres- 

 sions which may serve as a measure of the devia- 

 tions that cannot be removed. The necessary cal- 

 culus we owe to the genius of Riemann and 

 Christoffel. 



Einstein now identifies the presence of curva- 

 tures that cannot be smoothed out with the pres- 

 ence of matter. This means that the vanishing of 

 certain mathematical expressions indicates the 

 absence of matter. Thus he writes down the laws 

 of the gravitational field in free space. On the 

 other hand, if the expressions do not vanish, they 

 must be equal to quantities characteristic of matter 

 and its motion. These equalities form the expres- 

 sion of his law of gravitation at points where 

 matter exists. 



The reader will ask : What are the quantities 

 which enter into these equations? To this only a 

 very insufficient answer can here be given. If, in 

 the four-dimensional map, two neighbouring points 

 be taken, representing what may be called two 

 neighbouring occurrences, the actual distance 

 -between them measured in the ordinary geo- 

 metrical sense has no physical meaning. If the 

 map be strained, it will be altered, and therefore 

 to the relativist it represents something which is 

 not in the external world of events apart from the 

 observer's caprice of measurement. But Einstein 

 assumes that there is a quantity depending on the 

 relation of the points one to the other which is 

 invariant -that is, independent of the particular 

 map of events. Comparing one map with another, 

 thinking of one being strained into the other, the 

 relative positions of the two events are altered as 

 the strain is altered. It is assumed that the strain 

 at any point may be specified by a number of 

 quantities (commonly denoted fi„), and the invari- 

 able quantity is a function of these and of the 

 relative positions of the points. 



It is these quantities /; which characterise the 



