The A B C of Relativity

the earth, this was one of the first discoveries of Galileo. Aristotle thought that heavy bodies fall faster than light ones; Galileo showed that this is not the case, [Pg 141] when the resistance of the air is eliminated. In a vacuum, a feather falls as fast as a lump of lead. As regards the planets, it was Newton who established the corresponding facts. At a given distance from the sun, a comet, which has a very small mass, experiences exactly the same acceleration towards the sun as a planet experiences at the same distance. Thus the way in which gravitation affects a body depends only upon where the body is, and in no degree upon the nature of the body. This suggests that the gravitational effect is a characteristic of the locality, which is what Einstein makes it. As for the gravitational mass in sense (2), i.e., the intensity of the force produced by a body, this is no longer exactly proportional to its inertial mass. The question involves some rather complicated mathematics, and I shall not go into it.[7] We have another indication as to what sort of thing the law of gravitation must be, if it is to be a characteristic of a neighborhood, as we have seen reason to suppose that it is. It must [Pg 142] be expressed in some law which is unchanged when we adopt a different kind of co-ordinates. We saw that we must not, to begin with, regard our co-ordinates as having any physical significance: they are merely systematic ways of naming different parts of space-time. Being conventional, they cannot enter into physical laws. That means to say that, if we have expressed a law correctly in terms of one set of co-ordinates, it must be expressed by the same formula in terms of another set of co-ordinates. Or, more exactly, it must be possible to find a formula which expresses the law, and which is unchanged however we change the co-ordinates. It is the business of the theory of tensors to deal with such formulæ. And the theory of tensors shows that there is one formula which obviously suggests itself as being possibly the law of gravitation. When this possibility is examined, it is found to give the right results; it is here that the empirical confirmations come in. But if Einstein’s law had not been found to agree with experience, we could not have gone back to Newton’s law. We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. [Pg 143] It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example.