Page:Eddington A. Space Time and Gravitation. 1920.djvu/96

80 must make a number of measures of the length $$ds$$ between adjacent points $$(x_1, x_2)$$ and $$(x_1 + dx_1, x_2 + dx_2)$$ and test which formula fits. If, for example, we then find that $$ds^2$$ is always equal to $$d{x_1}^2 + {x_1}^2 d{x_2}^2$$, we know that our mesh-system is like that in Fig. 11, $$x_1$$ and $$x_2$$ being the numbers usually denoted by the polar coordinates $$r$$, $$\theta$$. The statement that polar coordinates are being used is unnecessary, because it adds nothing to our knowledge which is not already contained in the formula. It is merely a matter of giving a name; but, of course, the name calls to our minds a number of familiar properties which otherwise might not occur to us.

For instance, it is characteristic of the polar coordinate system that there is only one point for which $$x_1$$ (or $$r$$) is equal to 0, whereas in the other systems $$x_1 = 0$$ gives a line of points. This is at once apparent from the formula; for if we have two points for which $$x_1 = 0$$ and $$x_1 dx_1 = 0$$, respectively, then The distance $$ds$$ between the two points vanishes, and accordingly they must be the same point.

The examples given can all be summed up in one general expression where $$g_{11}$$, $$g_{12}$$, $$g_{22}$$ may be constants or functions of $$x_1$$ and $$x_2$$. For instance, in the fourth example their values are 1, 0, $$\cos^2 x_1$$. It is found that all possible mesh-systems lead to values of $$ds^2$$ which can be included in an expression of this general form; so that mesh-systems are distinguished by three functions of position $$g_{11}$$, $$g_{12}$$, $$g_{22}$$ which can be determined by making physical measurements. These three quantities are sometimes called potentials.

We now come to a point of far-reaching importance. The formula for $$ds^2$$ teaches us not only the character of the mesh-system, but the nature of our two-dimensional space, which is independent of any mesh-system. If $$ds^2$$ satisfies any one of the first three formulae, then the space is like a flat surface; if it satisfies the last formula, then the space is a surface curved like a sphere. Try how you will, you cannot draw a mesh-system on a flat (Euclidean) surface which agrees with the fourth formula.