Page:The Meaning of Relativity - Albert Einstein (1922).djvu/113

insures that this length shall depend only upon the place, and not upon the direction. If we had chosen a different system of co-ordinates this would not be so. But however we may choose a system of co-ordinates, the laws of configuration of rigid rods do not agree with those of Euclidean geometry; in other words, we cannot choose any system of co-ordinates so that the co-ordinate differences, $\Delta x_{1},\Delta x_{2},\Delta x_{3}$, corresponding to the ends of a unit measuring rod, oriented in any way, shall always satisfy the relation $\Delta x_{1}^{2}+\Delta x_{2}^{2}+\Delta x_{3}^{2}=1$. In this sense space is not Euclidean, but "curved." It follows from the second of the relations above that the interval between two beats of the unit clock ($dT=1$) corresponds to the "time"

$1+{\frac {\kappa }{8\pi }}\int {\frac {\sigma dV_{0}}{r}}$

in the unit used in our system of co-ordinates. The rate of a clock is accordingly slower the greater is the mass of the ponderable matter in its neighbourhood. We therefore conclude that spectral lines which are produced on the sun's surface will be displaced towards the red, compared to the corresponding lines produced on the earth, by about $2\cdot 10^{-6}$ of their wave-lengths. At first, this important consequence of the theory appeared to conflict with experiment; but results obtained during the past year seem to make the existence of this effect more probable, and it can hardly be doubted that this consequence of the theory will be confirmed within the next year.