Relativity: The Special and General Theory

by Albert Einstein

ds² = g₁₁du² + 2g₁₂du dv + g₂₂dv²,

where g₁₁, g₁₂, g₂₂, are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g₁₁, g₁₂ and g₂₂, determine the behaviour of the rods relative to the u-curves and v-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and to attach numbers to them, in such a manner, that we simply have:

ds² = du² + dv²

Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian coordinates are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points “in space.”

So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum, we associate arbitrarily four numbers, x₁, x₂, x₃, x₄, which are known as “co-ordinates.” Adjacent points correspond to adjacent values of the coordinates. If a distance ds is associated with the adjacent points P and P′, this distance being measurable and well defined from a physical point of view, then the following formula holds:

ds² = g₁₁dx₁² + 2g₁₂dx₁dx₂ . . . . + g₄₄dx₄²,

where the magnitudes g₁₁, etc., have values which vary with the position in the continuum. Only when the continuum is a Euclidean one is it possible to associate the co-ordinates x₁ . . x₄. with the points of the continuum so that we have simply

ds² = dx₁² + dx₂² + dx₃² + dx₄².

In this case relations hold in the four-dimensional continuum which are analogous to those holding in our three-dimensional measurements.

However, the Gauss treatment for ds² which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry. Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table.