Page:Eddington A. Space Time and Gravitation. 1920.djvu/118

102 from the true map to the customary picture of $$r$$ and $$t$$ as perpendicular space and time, we must strain Fig. 14 until all the meshes become squares as in Fig. 15.

Now in the map the geometry is Euclidean and the tracks of all material particles will be straight lines. Take such a straight track $$PQ$$, which will necessarily be nearly vertical, unless the velocity is very large. Strain the figure so as to obtain the customary representation of $$r$$ and $$t$$ (in Fig. 15), and the track $$PQ$$ 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 $$m/r^2$$, 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. 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 $$m/r^2$$ 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, $$G_{\mu\nu} = 0$$, 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