Arthur Eddington

“ It can now be deduced that the space-time in which we live is not quite flat. If it were, a mesh-system could be drawn for which the 's have the Galilean values, and the geometry with respect to these partitions of space and time would be that discussed in Chapter iii. For that geometry the geodesics, giving the natural tracks of particles, are straight lines.
Thus in flat space-time the law of motion is that (with suitably chosen coordinates) every particle moves uniformly in a straight line except when it is disturbed by the impacts of other particles. ”

“ No doubt it seems a very clumsy way of approaching these intrinsic differences of kinds of space to introduce potentials which specifically refer to a particular mesh-system, although the mesh-system can have nothing to do with the matter. It is worrying not to be able to express the differences of space in a purer form without mixing them up with irrelevant differences of potential. But we have neither the vocabulary nor the imagination for a description of absolute properties as such. All physical knowledge is relative to space and time partitions ”

“ The choice of 's in any special problem is thus arrived at by a three-fold sorting out: (1) many sets of values can be dismissed because they can never occur in nature, (2) others, while possible, do not relate to the kind of space-time present in the problem considered, (3) of those which remain, one set of values relates to the particular mesh-system that has been chosen. We have now to find the law governing the first discrimination. What is the criterion that decides what values of the 's give a kind of space-time possible in nature? ”

“ There is some complicated mathematical property common to all values of the potentials which belong to the same space-time, which is not shared by those which belong to a different kind of space-time. ”

“ For centuries man was in doubt whether the earth was flat or round—whether he was using plane rectangular coordinates or some kind of spherical coordinates. In some cases an observer adopts his mesh-system blindly and long afterwards discovers by accurate measures that does not fit the formula he assumed—that his mesh-system is not exactly of the nature he supposed it was. ”

“ It is the way these values link on to those at other points—the gradient of the 's, and more particularly the gradient of the gradient. Or, as has already been said, the kind of space-time is fixed by differential equations.
Thus, although a gravitational field of force is not an absolute thing, and can be imitated or annulled at any point by an acceleration of the observer or a change of his mesh-system, nevertheless the presence of a heavy particle does modify the world around it in an absolute way which cannot be imitated artificially. Gravitational force is relative ”

“ If the potentials have these values everywhere, space-time may be called "flat," because the geometry is that of a plane surface drawn in Euclidean space of five dimensions. Recollecting what we found for two dimensions, we shall realise that a quite different set of values of the potentials may also belong to flat space-time, because the meshes may be drawn in different ways. ”

“ 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. ”

“ If we deform the jelly in any way, the intersections will still occur in the same order along each world-line and no additional intersections will be created. The deformed jelly will represent a history of the world, just as accurate as the one originally drawn; there can be no criterion for distinguishing which is the best representation. ”

“ The distance between two points can be stated without referring to any mesh-system. For this reason, if we use a mesh-system, it is important to find formulae connecting the absolute distance with the particular system that is being used.
In the more complicated kinds of mesh-systems it makes a great simplification if we content ourselves with the formulae for very short distances. The mathematician then finds no difficulty in extending the results to long distances by the process called integration. ”

“ 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. ”