If the whole universe were suddenly plunged in a mass of gelatine which has set, and one were to squeeze or alter the shape in any way of this gelatinous mass, there would still be something unchanged in the coagulated stuff. What is this something? And what is the calculus to use for it? The answer to these questions was the last stage for Einstein to cover in order to establish the equations of gravitation and General Relativity.
Here it was the penetrating genius of Henri Poincaré that indicated the path. It is very necessary to insist on this, as justice has not been done in the matter to the great French mathematician.
If all the bodies in the universe were to be simultaneously dilated, and to an identical extent, we should have no means of knowing it. Our instruments and our own bodies being similarly dilated, we should not perceive this formidable historical and cosmic event. It would not distract us for a moment from the trivialities of the hour.
What is more, not only will it be unrecognisable if worlds are modified in such a fashion as to alter the scale of lengths and time, but it would be impossible to distinguish between two worlds, if one single point of the first corresponds to each point of the second; if to each object or event of the one world there corresponds one of the same character, placed exactly in the same position, in the other. Now the successive and diverse deformations which we impose upon the gelatinous mass in which we metaphorically enclosed our entire universe in an earlier paragraph give us precisely indistinguishable worlds from this point of view. Poincaré has the distinction of first calling our attention to this and proving that the relativity of things must be understood in this very broad sense.
The amorphous and plastic continuum in which we place the universe has a certain number of properties which are exempt from all idea of measurement. The study of these properties is the work of a special geometry, a qualitative geometry. The theorems of this geometry have this peculiarity, that they would still be true even if the figures were copied by a clumsy draughtsman who made gross errors in the proportions and substituted irregular and wavy lines for straight lines.
This is the geometry which, as Poincaré ably indicated, must be used for the four-dimensional and, according to its regions, more or less Euclidean continuum which is the Einsteinian universe. It is precisely this geometry which states what there is in common between the forms of objects seen by the drunken man and those seen by the water-drinker.
It is along this route, or a route analogous to this, that Einstein at last reached success. The universe being a more or less warped continuum, he proposed to apply to it the geometry created by Gauss for the study of surfaces of variable curvature: a geometry generalised by Riemann. It is by means of this special geometry that we express the fact that the “Interval” of events is an invariant.
Here is an illustration which will, I think, lead us to the heart of the problem of gravitation and to the solution of it.