In order to make these points clearer, let us understand that when we speak of the instantaneous spaces of two observers in relative motion as being tilted one with respect to the other, hence as lying outside of each other (except for their common two-dimensional cross-section), we do not mean that the space of one is some incomprehensible entity in the opinion of the other. The tilted instantaneous spaces will enter into our perception, but not as spaces alone: they will appear as a succession of two-dimensional surfaces moving with uniform speed.
All these spaces are equally justified. No one of them stands out more prominently than any other on the ground of symmetry with respect to the four-dimensional world. Only when we specify the frame of the observer will one particular space and one particular duration be allotted. Hence we may say that practical congruence exists for space and for time, as in classical science, provided the Galilean observer effects his measurements with rods and periodic mechanisms, such as clocks, which do not move about in his Galilean frame while performing measurements. There is thus a perfectly definite physical meaning in stating that the distances between two point-pairs at rest in the frame are congruent or unequal; but we must specify that this distance is measured according to the standards of the frame; and the same applies to time-stretches. Likewise, as in classical science, the Galilean observer will discover upon measurement with his congruent rods that his space is Euclidean. It is only when the rods and clocks are in relative motion or when, while at rest in our frame, the frame happens to be accelerated or submitted to gravitational action, that they cease to measure congruent stretches. It is only, therefore, when we reason in an impersonal way, without specifying any particular frame, only when we reason from the standpoint of all possible observers, whatever be their motion, that space and time fade away into shadows; and it is only then that we are compelled, whether we like it or not, to reason in terms of the common objective world of relativity, that is, in terms of four-dimensional space-time.
CHAPTER XVII
THE MATHEMATICAL EXPRESSION OF EINSTEIN’S FUNDAMENTAL PREMISES
CONSIDER a Galilean system, and two points
and
in this system. If a ray of light is propagated from
to