Page:Eddington A. Space Time and Gravitation. 1920.djvu/86

Our attention is thus directed to the natural tracks of unconstrained bodies, which appear to be marked out in some absolute way in the four-dimensional world. There is no question of an observer here; the body takes the same course in the world whoever is watching it. Different observers will describe the track as straight, parabolical, or sinuous, but it is the same absolute locus.

Now we cannot pretend to predict without reference to experiment the laws determining the nature of these tracks; but we can examine whether our knowledge of the four-dimensional world is already sufficient to specify definite tracks of this kind, or whether it will be necessary to introduce new hypothetical factors. It will be found that it is already sufficient. So far we have had to deal with only one quantity which is independent of the observer and has therefore an absolute significance in the world, namely the interval between two events in space and time. Let us choose two fairly distant events $P_{1}$ and $P_{2}$. These can be joined by a variety of tracks, and the interval-length from $P_{1}$ to $P_{2}$ along any track can be measured. In order to make sure that the interval-length is actually being measured along the selected track, the method is to take a large number of intermediate points on the track, measure the interval corresponding to each subdivision, and take the sum. It is virtually the same process as measuring the length of a twisty road on a map with a piece of cotton. The interval-length along a particular track is thus something which can be measured absolutely, since all observers agree as to the measurement of the interval for each subdivision. It follows that all observers will agree as to which track (if any) is the shortest track between the two points, judged in terms of interval-length.

This gives a means of defining certain tracks in space-time as having an absolute significance, and we proceed tentatively to identify them with the natural tracks taken by freely moving particles.

In one respect we have been caught napping. Dr A. A. Robb has pointed out the curious fact that it is not the shortest track, but the longest track, which is unique^[1]. There are any number

1. It is here assumed that $P_{2}$ is in the future of $P_{1}$ so that it is possible for a particle to travel from $P_{1}$ to $P_{2}$. If $P_{1}$ and $P_{2}$ are situated like $O$ and $P$ in Fig. 3, the interval-length is imaginary, and the shortest track is unique.