To this fine and profound criticism which M. Painlevé raises may be added that of Wiechert, who has pointed out various other hypotheses introduced by Einstein in the course of his calculations.
In a word, Einstein seems not to have kept entirely clear of the Newtonian premises which he repudiates. He has not the disdain for them that one would suppose, and he does not hesitate to have recourse to them occasionally for the purpose of helping out his calculations. That is rather to pay a little reverence to the idols you have burned.
In reply the Einsteinians will doubtless say that, if they introduce Newtonian axes in the course of their arguments, it is to make the results of calculation comparable to the result of experimental measurements. The axes introduced into their equations have for the Relativists the sole privilege of being those to which experimenters refer their measurements. But we must admit that that is no small privilege.
That is not all. The principle of General Relativity amounts to this: All systems of reference are equivalent for expressing natural laws, and these laws are invariant to any system of reference to which they are related. That means in effect: There are relations between objects of the material world which are independent of the one who observes them, and particularly of his velocity. Thus, when a triangle is drawn on paper, there is something in the triangle which characterises it and which is identical, whether the observer passes very quickly or very slowly, or at any speed and in any direction whatever, beside the paper.
M. Painlevé observes, with some reason, that in this form the principle is a sort of truism. It is a severe verdict, yet it expresses a certain fact. The real relations of external objects cannot be altered by the standpoint of the observer.
Einstein replies that it is at all events something to have provided a sieve by which we may sift the laws and formulæ which serve to represent the phenomena that have been empirically observed: a criterion which they must pass before they are recognised as correct. This is true. Newton’s law, in its classic form, did not meet this criterion. This proves that it was not quite so obvious. A truth that was unknown yesterday has become to-day a truism. So much the better.
In expressing one of the conditions which must be satisfied by natural laws the theory of Relativity at least has what is called in philosophical jargon a “heuristic” value. But it is none the less true, as M. Painlevé points out with great force and clearness, that the principle of General Relativity, considered in this light, would be unable to provide precise laws. It would be quite consistent with a law of gravity in which the attraction would be in inverse proportion, not to the square, but to the seventeenth or hundredth power, or any power whatever, of the distance.
In order to extract the correct law of gravitation from the principle of General Relativity we have to add to it the Einsteinian interpretation of the result of the Michelson experiment—to wit, that relatively to any observer whatsoever light travels locally with the same velocity in every direction. We have also to add various hypotheses which M. Painlevé regards as Newtonian.
To the critical discussion of Relativity which he so brilliantly presented at the Academy of Sciences M. Paul Painlevé added a valuable mathematical contribution of which the chief result is the following: It is possible to excogitate other laws of gravitation than that offered by Einstein, and all of them will fulfil the Einsteinian conditions.