Page:Eddington A. Space Time and Gravitation. 1920.djvu/113

always be added, if desired, by conditions of symmetry. The result of long algebraic calculations ^[1] is that, round a particle $ds^{2}=-{\frac {1}{\gamma }}\,dr^{2}-r^{2}d\theta ^{2}+\gamma \,dt^{2}$ ..................(6) where $\gamma =1-{\tfrac {2m}{r}}$.

The quantity $m$ is the gravitational mass of the particle—but we are not supposed to know that at present. $r$ and $\theta$ are polar coordinates, the mesh-system being as in Fig. 11 ; or rather they are the nearest thing to polar coordinates that can be found in space which is not truly flat.

The fact is that this expression for $ds^{2}$ is found in the first place simply as a particular solution of Einstein's equations of the gravitational field; it is a variety of hummock (apparently the simplest variety) which is not curved beyond the first degree. There could be such a state of the world under suitable circumstances. To find out what those circumstances are, we have to trace some of the consequences, find out how any particle moves when $ds^{2}$ is of this form, and then examine whether we know of any case in which these consequences are found observationally. It is only after having ascertained that this form of $ds^{2}$ does correspond to the leading observed effects attributable to a particle of mass $m$ at the origin that we have the right to identify this particular solution with the one we hoped to find.

It will be a sufficient illustration of this procedure, if we indicate how the position of the matter causing this particular solution is located. Wherever the formula (6) holds good there can be no matter, because the law which applies to empty space is satisfied. But if we try to approach the origin ($r=0$), a curious thing happens. Suppose we take a measuring-rod, and, laying it radially, start marking off equal lengths with it along a radius, gradually approaching the origin. Keeping the time $t$ constant, and $d\theta$ being zero for radial measurements, the formula (6) reduces to $ds^{2}=-{\frac {1}{\gamma }}\,dr^{2}$ or $dr^{2}=-\gamma \,ds^{2}$.

1. Appendix, Note 7.