Page:The Meaning of Relativity - Albert Einstein (1922).djvu/82

space-time continuum of physics. In the immediate neighbourhood of an observer, falling freely in a gravitational field, there exists no gravitational field. We can therefore always regard an infinitesimally small region of the space-time continuum as Galilean. For such an infinitely small region there will be an inertial system (with the space co-ordinates, $X_{1},X_{2},X_{3}$, and the time co-ordinate $X_{4}$ relatively to which we are to regard the laws of the special theory of relativity as valid. The quantity which is directly measurable by our unit measuring rods and clocks,

$dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2}-dX_{4}^{2}$

or its negative,

$ds^{2}=-dX_{1}^{2}-dX_{2}^{2}-dX_{3}^{2}+dX_{4}^{2}$

(54)

is therefore a uniquely determinate invariant for two neighbouring events (points in the four-dimensional continuum), provided that we use measuring rods that are equal to each other when brought together and superimposed, and clocks whose rates are the same when they are brought together. In this the physical assumption is essential that the relative lengths of two measuring rods and the relative rates of two clocks are independent, in principle, of their previous history. But this assumption is certainly warranted by experience; if it did not hold there could be no sharp spectral lines; for the single atoms of the same element certainly do not have the same history, and it would be absurd to suppose any relative difference in the structure of the