(ii.) The Newtonian law may be untrue in its original form, but may become true when amended so as to conform to the relativity hypothesis.
(iii.) Neither of the foregoing possibilities may be true.
Alternative (i.) was explored by Sir Oliver Lodge, who, assuming the exact truth of the Newtonian law of gravitation, deduced that the observed motion of the perihelion of Mercury could be accounted for if the sun were moving through space with a velocity of about 70 km. a second in a certain direction. This investigation had to be abandoned when it was shown by Eddington that a similar discussion of the motions of the other planets would lead to vastly different values for the sun's velocity. Alternative (ii.) was explored by Einstein and others, but was found to lead to a motion of the perihelion of Mercury equal only to one-sixth part of that actually observed.
Alternative (iii.) remained with its innumerable possibilities. Einstein commenced his attack on the problem by eliminating all possibilities which did not conform to two general principles. Thi first of these was the principle of relativity. Inasmuch as all physical phenomena except gravitation were believed to conform to this principle, it was natural to try, as a working hypothesis, the effect of assuming gravitation also to conform. Th: second principle was the so-calbd principle of equivalence, and this demands a word of explanation.
To our children we explain that an apple falls to the ground because a force of gravitation inherent in the earth's mass impels the apple towards the centre of the earth. Most schoolboys know that this is not quite the whole story; the path of the apple is more accurately determined by supposing the apple to be acted on simultaneously by two forces a gravitational force of'attraction towards the earth's centre and the centrifugal force arising frpm the earth's rotation. It is only because the earth's rotation is comparatively slow that the conception of an attraction towards the earth's centre gives a tolerably plausible account of the fall of the apple. If the earth rotated at 17 times its present rate objects would not fall, even approximately, towards the earth's centre; they would fall always parallel to the earth's axis, and the inhabitants of the northern hemisphere might explain this as arising from a force of repulsion inherent in the pole star. If the earth rotated many times faster even than this, bodies would fall always perpendicularly away from the earth's axis, and this might be interpreted as arising from a gravitational repulsion residing in the earth's axis.
These illustrations will show that it is easy to confuse accelera- tion arising from the earth's rotation with gravitational attrac- tion. We may go further and say that it is impossible to dis- tinguish between the effects of gravitational attraction and the effects of acceleration of any kind whatever. Every aeroplanist knows this to his sorrow; it is inherently impossible to devise any instrument which shall show the direction of the vertical in an aeroplane, since an acceleration of the aeroplane produces on any instrument whatever, effects which are indistinguishable from those of gravity. From such considerations Einstein was led to his principle of equivalence, which may be enunciated as follows:
" A gravitational field of force at any point of space is in every way equivalent to an artificial field of force resulting from accelera- tion, so that no experiment can possibly distinguish between them."
Guided by these two principles relativity and equivalence Einstein was led to the view that all gravitational " fields of force " must be illusions. The apparent " force " arises solely from acceleration and there is no other kind of gravitational force at all. In this statement, as in the statement of the principle of equivalence above, the word acceleration is used in its widest sense. Acceleration results not only from change in the amount of a velocity, but from a change in its direction also. For instance a motor-cyclist riding in a circle at a uniform speed of 60 miles an hour will be the subject of an acceleration towards the centre of the circle. He knows that the apparent force so produced is just as real in its effects as gravitation, and to save himself from falling as a result of its influence he must incline the direction of his machine to the vertical.
It is clear that the acceleration or curvature of path which figures as gravitation cannot be an acceleration or curvature in ordinary three-dimensional space. Before the apple starts to fall from the tree there is neither acceleration nor curvature, and yet the apple is undoubtedly acted on by gravitation. Moreover, this three-dimensional space is, as we have seen, different for different observers it is a subjective and not an objective conception, and the gravitation resulting from such a curvature could not conform to the relativity condition. Einstein was accordingly led to suppose that gravitation arose from curvature in the four-dimensional space, or continuum, in which time formed the fourth dimension. This continuum, as has been seen, is objective and if the path of the particle can also be made objective, the resulting gravitation will conform to the relativity principle. The path of the particle in the continuum is, however, simply its " world line," which we have already had under discussion. This world line is determined by natural laws, and if these are to be objective the specification of the world h'ne must also be objective. There is, however, only one specification of world lines in the continuum which is objective in the sense that the same specification will give the same world lines to observers moving with different velocities. It is that every world line must be so drawn as to represent the shortest path between any two points on it. Mathematically, lines which satisfy this condition are known as geodesies. Thus Einstein was led to suppose that world lines must be geodesies in the four-dimensional continuum.
Consider for a moment a page of this volume as presenting a two dimensional analogy of the continuum. The shortest distance between any two points is of course the straight line joining them, so that the geodesies are simply straight lines. These possess no curvature of path and if they formed a true analogy to the geodesies in the continuum there could clearly be no explanation of gravitation of the type we have been contemplating. There is, however, another type of two-dimen- sional surface. It is represented by the surface of a solid body such as a sphere say the earth. On the earth's surface the geodesies are the great circles; every mariner or aeronaut who desires to sail the shortest course between two points sails along a great circle. To take a definite instance, the shortest course from Panama to Ceylon is not along the parallel of lat. (about 9 N.) which joins them the aeronaut wishing to fly the shortest course between the two countries will fly N.E. from Panama, he will pass over England and finally reach Ceylon from the north-west. The reader may rapidly verify this by stretching a thread tightly over the surface of an ordinary geographical globe. Let him now trace out the course on an ordinary Mercator chart, and it will be found to appear very curved indeed the course of the aeronaut will look surprisingly like that of a comet describing an orbit under the attraction of a sun situated somewhere near the middle of the Sahara.
The reader who performs these simple experiments will understand how Einstein was led to suppose that gravitation could be explained by a curvature inherent in the continuum. The world lines of particles are geodesies but the space itself, so to speak, provides the curvature. The curvature of path is thrust upon the particle by the nature of the continuum, but we, who until recently have been unaware even of the existence of the continuum, have been tempted to ascribe it to the action of a special agency which we have invented ad hoc and called " gravitation." According to Einstein, it is no more accurate to say that the earth attracts the moon than to say that the pockets of an uneven billiard table repel the balls.
This train of thought may seem artificial. If so, the reason is that we have not been able to explore the other possibilities which have branched off our main line of thought. In point of fact, Einstein found himself practically limited to the conclusion we have stated. Not only so, but the actual type and degree of curvature in the continuum prove to be uniquely fixed in terms of the masses of the gravitating bodies. Thus Einstein, knowing the mass of the sun, found himself in a position to predict absolutely what the motion of the perihelion of Mercury ought
