Page:Eddington A. Space Time and Gravitation. 1920.djvu/122

106 longer be accurately mappable on a plane sheet of paper, because they do not conform to Euclidean geometry.

Now suppose a heavy particle wishes to cross this field, passing near but not through the centre. In Euclidean space, with the hurdles uniformly distributed, it travels in a straight line, i.e. it goes between any two points by a path giving the fewest hurdle jumps. We may assume that in the non-Euclidean field with rearranged hurdles, the particle still goes by the path of least effort. In fact, in any small portion we cannot distinguish between the rearrangement and a distortion; so we may imagine that the particle takes each portion as it comes according to the rule, and is not troubled by the rearrangement which is only visible to a general survey of the whole field.

Now clearly it will pay not to go straight through the dense portion, but to keep a little to the outside where the hurdles are sparser—not too much, or the path will be unduly lengthened. The particle's track will thus be a little concave to the centre, and an onlooker will say that it has been attracted to the centre. It is rather curious that we should call it attraction, when the track has rather been avoiding the central region; but it is clear that the direction of motion has been bent round in the way attributable to an attractive force.

This bending of the path is additional to that due to the Newtonian force of gravitation which depends on the second appearance of $$\gamma$$ in the formula. As already explained it is in general a far smaller effect and will appear only as a minute correction to Newton's law. The only case where the two rise to equal importance is when the track is that of a light-wave, or of a particle moving with a speed approaching that of light; for then $$dr^2$$ rises to the same order of magnitude as $$dt^2$$.

To sum up, a ray of light passing near a heavy particle will be bent, firstly, owing to the non-Euclidean character of the combination of time with space. This bending is equivalent to that due to Newtonian gravitation, and may be calculated in the ordinary way on the assumption that light has weight like a material body. Secondly, it will be bent owing to the