Arthur Eddington

“ The effect of the gravitation of the sun on a light-wave, or very fast particle, proceeding radially is actually a repulsion!
The track of a transverse light-wave, coming out from the plane of the paper, will be affected like that of a particle of zero velocity in distorting from Fig. 14 to Fig. 15. Hence the sun's influence on a transverse light-wave is always an attraction. The acceleration is simply as for a particle at rest. ”

“ The result that the expression found for the geometry of the gravitational field of a particle leads to Newton's law of attraction is of great importance. It shows that the law, , proposed on theoretical grounds, agrees with observation at least approximately. It is no drawback that the Newtonian law applies only when the speed is small; all planetary speeds are small compared with the velocity of light, and the considerations mentioned at the beginning of this chapter suggest that some modification may be needed for speeds comparable with that of light. ”

“ In an ordinary map of Europe the lines of longitude run obliquely and the lines of latitude are curved. Why is this? Because the map aims at showing as accurately as possible all distances in their true proportions [4] . Distance is the important thing which it is desired to represent correctly. In four dimensions interval is the analogue of distance, and a map of the four-dimensional world will aim at showing all the intervals in their correct proportions. Our natural picture of space-time takes and as horizontal and vertical distances, e.g. when we plot the graph of the motion of a particle ”

“ Light moves more slowly in a material medium than in vacuum, the velocity being inversely proportional to the refractive index of the medium. The phenomenon of refraction is in fact caused by a slewing of the wave-front in passing into a region of smaller velocity. We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. ”

“ In the form , Einstein's law expresses conditions to be satisfied in a gravitational field produced by any arbitrary distribution of attracting matter. An analogous form of Newton's law was given by Laplace in his celebrated expression . A more illuminating form of the law is obtained if, instead of putting the question what kinds of space-time can exist under the most general conditions in an empty region, we ask what kind of space-time exists in the region round a single attracting particle? We separate out the effect of a single particle, just as Newton did. ”

“ Secondly, it will be bent owing to the non-Euclidean character of space alone, and this curvature is additional to that predicted by Newton's law. If then we can observe the amount of curvature of a ray of light, we can make a crucial test of whether Einstein's or Newton's theory is obeyed.
This separation of the attraction into two parts is useful in a comparison of the new theory with the old; but from the point of view of relativity it is artificial. Our view is that light is bent just in the same way as the track of a material particle moving with the same velocity would be bent. ”

“ Now in the map the geometry is Euclidean and the tracks of all material particles will be straight lines. Take such a straight track , which will necessarily be nearly vertical, unless the velocity is very large. Strain the figure so as to obtain the customary representation of and (in Fig. 15) , and the track will become curved—curved towards the left, where the sun lies. In each successive vertical interval (time) , a successively greater progress is made to the left horizontally (space) . Thus the velocity towards the sun increases. We say that the particle is attracted to the sun. ”

“ The mathematical reader should find no difficulty in proving from the diagram that for a particle with small velocity the acceleration towards the sun is approximately , agreeing with the Newtonian law.
Tracks for very high speeds may be affected rather differently. The track corresponding to a wave of light is represented by a straight line at 45° to the horizontal in Fig. 14. It would require very careful drawing to trace what happens to it when the strain is made transforming to Fig. 15; but actually, whilst becoming more nearly vertical, it receives a curvature in the opposite direction. ”

“ Suppose that a plane field is uniformly studded with hurdles. The distance between any two points will be proportional to the number of hurdles that must be passed over in getting from one point to the other by the straight route—in fact the minimum number of hurdles. We can use counts of hurdles as the equivalent of distance, and map the field by these counts. The map can be drawn on a plane sheet of paper without any inconsistency since the field is plane. ”

“ If the motion of the waves is slower at one end than the other, the whole wave-front must gradually slew round, and the direction in which it is rolling must change. In the sea this happens when one end of the wave reaches shallow water before the other, because the speed in shallow water is slower. ”