Page:Eddington A. Space Time and Gravitation. 1920.djvu/144

now to examine whether experimental evidence can be found as to the exactness of this correspondence.

It seems reasonable to suppose that a vibrating atom is an ideal type of clock. The beginning and end of a single vibration constitute two events, and the interval $ds$ between two events is an absolute quantity independent of any mesh-system. This interval must be determined by the nature of the atom; and hence atoms which are absolutely similar will measure by their vibrations equal values of the absolute interval $ds$. Let us adopt the usual mesh-system ($r$, $\theta$, $t$) for the solar system, so that $ds^{2}=-\gamma ^{-1}dr^{2}-r^{2}d\theta ^{2}+\gamma \,dt^{2}$. Consider an atom momentarily at rest at some point in the solar system; we say momentarily, because it must undergo the acceleration of the gravitational field where it is. If $ds$ corresponds to one vibration, then, since the atom has not moved, the corresponding $dr$ and $d\theta$ will be zero, and we have $ds^{2}=-y\,dt^{2}$. The time of vibration $dt$ is thus $1/{\sqrt {\gamma }}$ times the interval of vibration $ds$.

Accordingly, if we have two similar atoms at rest at different points in the system, the interval of vibration will be the same for both; but the time of vibration will be proportional to the inverse square-root of $\gamma$, which differs for the two atoms. Since ${\begin{aligned}\gamma \;&=1-{\frac {2m}{r}}\\1/{\sqrt {\gamma }}\;&=1+{\frac {m}{r}}{\text{, very approximately.}}\\\end{aligned}}$

Take an atom on the surface of the sun, and a similar atom in a terrestrial laboratory. For the first, $1+m/r=1.00000212$, and for the second $1+m/r$ is practically 1. The time of vibration of the solar atom is thus longer in the ratio 1.00000212, and it might be possible to test this by spectroscopic examination.

There is one important point to consider. The spectroscopic examination must take place in the terrestrial laboratory; and we have to test the period of the solar atom by the period of the waves emanating from it when they reach the earth. Will they carry the period to us unchanged? Clearly they must.