Page:Eddington A. Space Time and Gravitation. 1920.djvu/104

originally drawn; there can be no criterion for distinguishing which is the best representation.

Suppose now we introduce space and time-partitions, which we might do by drawing rectangular meshes in both jellies. We have now two ways of locating the world-lines and events in space and time, both on the same absolute footing. But clearly it makes no difference in the result of the location whether we first deform the jelly and then introduce regular meshes, or whether we introduce irregular meshes in the undeformed jelly. And so all mesh-systems are on the same footing.

This account of our observational knowledge of nature shows that there is no shape inherent in the absolute world, so that when we insert a mesh-system, it has no shape initially, and a rectangular mesh-system is intrinsically no different from any other mesh-system.

Returning to our two clues, condition (1) makes an extraordinarily clean sweep of laws that might be suggested; among them Newton's law is swept away. The mode of rejection can be seen by an example; it will be sufficient to consider two dimensions. If in one mesh-system $(x,y)$ $ds^{2}=g_{11}dx^{2}+2g_{12}dx\,dy+g_{22}dy^{2}$, and in another system $(x^{\prime },y^{\prime })$ $ds^{2}={g_{11}}^{\prime }{dx}^{\prime 2}+2{g_{12}}^{\prime }dx^{\prime }dy^{\prime }+{g_{22}}^{\prime }dy^{\prime 2}$, the same law must be satisfied if the unaccented letters are throughout replaced by accented letters. Suppose the law $g_{11}=g_{22}$ is suggested. Change the mesh-system by spacing the $y$-lines twice as far apart, that is to say take $y^{\prime }={\tfrac {1}{2}}y$, with $x^{\prime }=x$. Then ${\begin{aligned}ds^{2}&=g_{11}dx^{2}+2g_{12}dx\,dy+g_{22}dy^{2}\\&=g_{11}dx^{\prime 2}+4g_{12}dx^{\prime }dy^{\prime }+4g_{22}dy^{\prime 2}{\text{,}}\end{aligned}}$ so that ${\begin{aligned}{g_{11}}^{\prime }&=g_{11}&{g_{22}}^{\prime }&=4g_{22}\end{aligned}}$. And if $g_{11}$ is equal to $g_{22}$, ${g_{11}}^{\prime }$ cannot be equal to ${g_{22}}^{\prime }$.

After a few trials the reader will begin to be surprised that any possible law could survive the test. It seems so easy to defeat any formula that is set up by a simple change of mesh-system. Certainly it is unlikely that anyone would hit on such a law by trial. But there are such laws, composed of exceedingly complicated mathematical expressions. The theory of these is