Page:Eddington A. Space Time and Gravitation. 1920.djvu/90

74 knowledge of the motion of particles to correct the formula, we shall find that the changes introduced are so small that they are inappreciable in any practical measures with scales and clocks. There is only one case where a possible detection of the modification is indicated; this refers to the behaviour of a clock on the surface of the sun, but the experiment is one of great difficulty and no conclusive answer has been given. We conclude then that the geometry of space and time based on the motions of particles is accordant with the geometry based on the cruder observations with clocks and scales; but if subsequent experiment should reveal a discrepancy, we shall adhere to the moving particle on account of its greater simplicity.

The proposed modification can be regarded from another point of view. Equation (1) is the synthesis of the experiences of all observers in uniform motion. But uniform motion means that their four-dimensional tracks are straight lines. We must suppose that the observers were moving in their natural tracks; for, if not, they experienced fields of force, and presumably allowed for these in their calculations, so that reduction was made to the natural tracks. If then equation (1) shows that the natural tracks are straight lines, we are merely getting out of the equation that which we originally put into it.

The formula needs generalising in another way. Suppose there is a region of space-time where, for some observer, the natural tracks are all straight lines and equation (1) holds rigorously. For another (accelerated) observer the tracks will be curved, and the equation will not hold. At the best it is of a form which can only hold good for specially selected observers.

Although it has become necessary to throw our formula into the melting-pot, that does not create any difficulty in measuring the interval. Without going into technical details, it may be pointed out that the innovations arise solely from the introduction of gravitational fields of force into our scheme. When there is no force, the tracks of all particles are straight lines as our previous geometry requires. In any small region we can choose an observer (falling freely) for whom the force vanishes, and accordingly the original formula holds good. Thus it is only necessary to modify our rule for determining the interval by two provisos (1) that the interval measured must be small,