1 Gaussian Co-Ordinates

From Relativity: The Special and General Theory by Albert Einstein.

According to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see the diagram above) drawn on the surface of the table. These we designate as u-curves, and we indicate each of them by means of a number. The curves u = 1, u = 2, and u = 3 are drawn in the diagram. Between the curves u = 1 and u = 2 we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying between 1 and 2. We have then a system of u-curves, and this “infinitely dense” system covers the whole surface of the table. These u-curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly definite value of u belongs to every point on the surface of the marble slab.

In like manner we imagine a system of v-curves drawn on the surface. These satisfy the same conditions as the u-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of u and a value of v belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point P in the diagram has a particular pair of Gaussian co-ordinates (u,v).

Two neighbouring points P and P′ on the surface then correspond to the co-ordinates

where du and dv signify very small changes. In a similar manner we may indicate the distance (line-interval) between P and P′, as measured with a little rod, by the very small number ds. Then, according to Gauss, the most general quadratic form for the line element in two dimensions is

where g₁₁, g₁₂, and g₂₂ are quantities that depend in a perfectly definite way on u and v. These quantities determine the metric relations of the surface relative to the u-curves and v-curves.

For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring rods, it is possible to choose the u-curves and v-curves and attach numbers to them in such a manner that

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 co-ordinates are simply Cartesian ones. It is clear that Gaussian 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.

So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can also be applied to a continuum of three, four, or more dimensions. If, for instance, a continuum of four dimensions is considered, we may associate with every point four numbers,

which are known as co-ordinates. Adjacent points correspond to adjacent values of the co-ordinates. If a distance ds is associated with adjacent points P and P′, this distance being measurable and well defined from a physical point of view, then the line element may be written

where the coefficients g_μν vary with position in the continuum.

Only when the continuum is Euclidean is it possible to choose the co-ordinates so that, throughout the region under consideration,

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

The Gaussian treatment of ds² is local: sufficiently small regions of a smooth continuum can be described by co-ordinates and a metric whose coefficients encode the local distance relations. For example, this applies to the marble-slab illustration when the temperature is nearly constant over a sufficiently small region.

We can sum this up as follows: Gauss developed a method for the mathematical treatment of continua in which “size-relations,” or distances between neighbouring points, are defined. To every point of a continuum are assigned as many numbers (Gaussian co-ordinates) as the continuum has dimensions. The Gaussian coordinate system is a generalisation of the Cartesian coordinate system and is applicable to non-Euclidean continua through the position-dependent metric coefficients g_μν.

2 References

This article is derived from the Einstein Reference Archive (marxists.org), 1999, 2002. Einstein Reference Archive which is under the FDL copyright.