different kinds of geometry, involving different properties of the 's. It is no explanation to say that this is because the surfaces are differently curved in a real Euclidean space of five dimensions. We should naturally ask for an explanation why the space of five dimensions is Euclidean; and presumably the answer would be, because it is a plane in a real Euclidean space of six dimensions, and so on ad infinitum.
The value of the picture to us is that it enables us to describe important properties with common terms like "pucker" and "curvature" instead of technical terms like "differential invariant." We have, however, to be on our guard, because analogies based on three-dimensional space do not always apply immediately to many-dimensional space. The writer has keen recollections of a period of much perplexity, when he had not realised that a four-dimensional space with "no curvature" is not the same as a "flat" space! Three-dimensional geometry does not prepare us for these surprises.
Picturing the space-time in the gravitational field round the earth as a pucker, we notice that we cannot locate the pucker at a point; it is "somewhere round" the point. At any special point the pucker can be pressed out flat, and the irregularity runs off somewhere else. That is what the inhabitants of Jules Verne's projectile did; they flattened out the pucker inside the projectile so that they could not detect any field of force there; but this only made things worse somewhere else, and they would find an increased field of force (relative to them) on the other side of the earth.
What determines the existence of the pucker is not the values of the 's at any point, or, what comes to the same thing, the field of force there. 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; but there is this
