According to the Galilean system of reference we may adopt, we divide space-time into space and time in one way or in another. Hence the resolution of the drawings of phenomena into space and time components is effected in various ways. We thus obtain variations in the spatio-temporal appearances of phenomena. Notwithstanding these variations, dependent on our relative motion, the absolute space-time drawings themselves, with their intersections, remain unchanged, and a direct study of the absolute world would consist in a study of these absolute drawings without reference to their varying spatio-temporal appearances.
It should be noted that inasmuch as space-time appears as an absolute continuum, existing independently of our measurements, there is no reason to limit ourselves to Cartesian mesh-systems when we wish to explore it. The mesh-system is, as we know, a mere mathematical device facilitating our investigations; and from a purely mathematical point of view we may always with equal justification split up a given continuum with one type of mesh-system or another. However, it must be remembered that in the case of space-time, when we wish to express the results of our measurements in terms of the space and time proper to our frame of reference, the mesh-system we must adopt is no longer arbitrary.
All Galilean observers, as we have seen, must split up space-time with their particular Cartesian mesh-systems. Certain curvilinear mesh-systems will correspond with more or less precision to the partitioning of space-time by accelerated or rotating observers. But in the most general case, arbitrary curvilinear mesh-systems do not correspond to the space and time partitions of any possible observer, though this fact does not detract from their utility.
So far as the essential characteristics of space-time itself are concerned, these Gaussian mesh-systems are just as legitimate as the Cartesian ones. As a matter of fact, when space-time becomes curved, owing to the presence of gravitation, Cartesian co-ordinates cannot be realised, being incompatible with the curvature of the continuum.
Yet, regardless of the mesh-system we use, certain general characteristics of our space-time drawings remain absolute. Thus, we have seen that the interval between two space-time points has a value which is invariant to a change of mesh-system. While it is true that a world-line which measures out as straight from a Cartesian system may appear curved when referred to a curvilinear or Gaussian one, yet, on the other hand, intersections or non-intersections of world-lines are absolute and are in no wise affected by our choice of a mesh-system. It is for this reason that phenomena such as coincidences are absolute, in contrast to simultaneities of events at spatially separated points. These are relative.
Thus, if two billiard balls kiss in the observation of one man, they will continue to kiss in the observation of all other men, regardless of the relative motion of these men. The kissing of the balls constitutes a coincidence, an intersection of the world-lines of the two balls; hence it is an absolute. On the other hand, if two billiard balls hit different cushions simultaneously in the observation of one man, they will not in general hit the cushions simultaneously in the observation of a man in motion with respect to the first. The reason is that our second observer will have adopted a new mesh-system, oriented differently from the first; and in this new mesh-system the two space-time point-events represented by the instantaneous impacts of the two balls against the two cushions will no longer lie necessarily at the same distance along the new time direction.
Owing to the common use of curvilinear mesh-systems in the general theory, we must recall that in a curvilinear mesh-system the square of the interval adopts the more complicated form
expressed more concisely by
