Now, though we have stated that the precise shape of the co-ordinate system, or mesh-system, is arbitrary, there are mathematical reasons which render the use of Cartesian systems especially convenient. So long as we are dealing with Euclidean space, in which, as we know, Cartesian mesh-systems are always permissible, we need not be concerned with Gaussian co-ordinate systems. It was for this reason that classical science, when dealing with physical problems, confined itself to Cartesian co-ordinates; and by a change of co-ordinate system it meant a rotation or a displacement of the Cartesian frame of reference from one position to another.
Nevertheless, we always had to view the possibility of having to investigate problems in which Cartesian co-ordinate systems would be impracticable, owing to the curvature or non-Euclideanism of the continuum. Accordingly, even before the advent of the theory of relativity, mathematicians had endeavoured to perfect a form of calculus which would free them from all dependence on any particular type of co-ordinate system. This task was accomplished chiefly by Levi-Civita and Ricci, and the mathematical instrument evolved was called the absolute calculus. And so, when in the general theory Einstein was led to his conception of a curved or non-Euclidean space-time, in which Cartesian co-ordinate systems were impossible, the mathematical instrument he required was at hand. All he had to do was to put it to practical use.
Although, in the Euclidean space of classical physics which we now propose to discuss, there was no need to consider Gaussian or curvilinear co-ordinate systems, yet, in view of the greater generality of the method, we shall treat the subject matter of this chapter from the standpoint of all systems of co-ordinates, whether Gaussian or Cartesian. The conclusions we shall reach must, however, be construed as holding for all manner of co-ordinate systems. If these preliminary points are understood, we may proceed to a more detailed discussion of the problem.
Let us consider a straight rod situated in this space. If we agree on a unit of measurement, we can assign some definite length to the rod. Obviously, this length, whatever it may be, will remain unaffected by a change in our co-ordinate system in space. Magnitudes of this sort, which transcend in every respect our choice of a co-ordinate system, are known as scalars, numbers or invariants.
But suppose now that we wish to determine the exact orientation of our rod. Orientation, in contrast to length, has no meaning until some mesh-system has been prescribed; for we can define a direction in empty space only by referring it to some accepted system of standard directions at each point. The meshes of our mesh-system at each point will, of course, define such standard directions for us. When, therefore, we attempt to determine the orientation of the rod, our procedure will be to project its length upon the three meshes that pass through one of its extremities. We thereby obtain what are known as the three components of the rod; and when these three components are known, the orientation and the length of the rod are fully determined.[86]
From this example we see that a considerable difference exists between the directed magnitude we are here considering and the invariant one which was given by the mere length of the rod irrespective of its orientation. Whereas the invariant magnitude possessed no components, the directed one has as many components as the space has dimensions, three in the present case. A directed magnitude of this sort is termed a vector.
Suppose now that we change our system of reference, rotate it slightly, for instance. We shall thereby obtain three new components for our vector in the new co-ordinate system, and these new components will differ in magnitude from our former ones. However, the change in the value of the components for a definite change of frame of reference is not arbitrary, but will be submitted to well-defined mathematical rules.[87] This fact enables us to discover a criterion of objectivity.
For, suppose that in a first co-ordinate system certain components,
,