(not relative acceleration), there is no sense in enquiring whether it is
or
which is in motion. This, indeed, constitutes the essence of the special principle of relativity. Hence, when we consider the moving rod, we can ascribe no absolute significance to its motion. Whatever motion exists between frame and rod might with equal justification be attributed to the motion of the frame gliding beneath a motionless rod, there being no absolute term of comparison like the stagnant Lorentzian ether to decide the issue. And so we cannot claim that the so-called moving rod has suffered a physical change by reason of its motion, a change rendering it unfit for measurement purposes.
And yet, if we retain all our rods as equally valid, whether in motion or at rest, we obtain conflicting results which are such as to deprive the geometry of space of all significance. When we analyse this anomalous situation we find that the following facts stand out clearly: Rods remain unmodified whether in motion or at rest, and yet their length varies. The truth is that length is not a definite characteristic of a body; it appears as a shadow of a something else projected into the space of our frame. When the rod is at rest in our frame, its length, or the shadow, presents a maximum; it is as though the something and its shadow coincided. But when relative motion is present, the something becomes tilted—tilted along a fourth dimension out of the space of our frame; and its shadow is shortened in proportion. Yet this rod which is moving in our frame is at rest in some other frame, and in this other frame, therefore, the something again coincides with its shadow. And so we must assume that the various Galilean frames and the spaces they serve to define are variously tilted, rotated in a fourth dimension with respect to one another.
There is then no longer one all-embracing Euclidean space which may be defined with respect to one Galilean frame or another. In its place we must conceive of an indefinite member of Euclidean spaces variously tilted, yet fused together.[61] To each separate Galilean frame one of these spaces will correspond, so that when we change our motion we are also changing our space.
The situation is analogous to that which arises when we consider various verticals drawn to the earth’s surface.
John, in London, will assert that his vertical is truly vertical and that Peter’s at New York is slanting, and Peter will return the compliment. Owing to these discrepancies, we cannot regard the two verticals as defining one same direction, though they both may be truly vertical for John and Peter respectively. Rather must we say that there exist an indefinite number of different vertical directions, no one of which is more truly vertical in any absolute sense than any other. Now replace the concept of vertical by that of three-dimensional space, and we have a picture of the spaces of relativity.
Very similar conclusions apply to time. Thus, consider two observers