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

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 $d{x_{1}}^{2}+{x_{1}}^{2}d{x_{2}}^{2}=0$. 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 $ds^{2}=g_{11}d{x_{1}}^{2}+2g_{12}dx_{1}d{x_{2}}+g_{22}d{x_{2}}^{2}$, 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.