8 8 0 A P P E N D I X H Now it is easy to see that this law can also be stated saying that an action can be consid- ered both as inertial and as gravitational. To see it, let us imagine a region of space without a gravitational field, that is, an inertial system k. Let us consider a coordinate system in accelerated motion with respect to k, and consider a free material point that is accelerated with respect to k. But now, who can say that the movement of is accelerated? In the same way, we can say that k is in accelerated motion with respect to and thus the point would be in accelerated motion with respect to it. If, with respect to a system, all bodies are accelerated the same way, we can say there is a gravitational field with respect to the system and nothing prevents us from saying that the bodies are at rest and the system is accelerated. From this point of view the equality of the two masses is completely natural. We will present an example. Let us suppose two systems k and and in the former a mass suspended on a wire whose other end is fixed in . The wire stretches out. Why? There are two ways to view the question. First, let us suppose that is accelerated. The acceleration of the system is transmitted to the mass by the wire, and then the tension measures the inert mass. Second, let us consider that is at rest. Then it is necessary to suppose that gravitation acts on the body and the tension of the wire will measure the gravitational mass. It thus results that the same action can be interpreted as in- ertia or as gravity, and by the general postulate of relativity, the equality of the two masses is absolutely necessary. It is natural to accept this law as valid for any other accelerated motion. Why is this interesting? Because we can produce a gravitational field by merely choos- ing an arbitrary state of motion, and in this manner we have a purely theoretical method for discovering the characteristics of a gravitational field. Let us consider a space that has no gravitational field when referred to a certain system. When, however, it is referred to another system with known motion, there is a gravitational field that can be determined by calculation. It is now necessary to see whether the laws thus obtained for the field coincide with those obtained in the first system, and this is not as easy as it seems, because one must be aware that the fields obtained by this procedure are not the most general ones, that is to say, not all fields can be obtained by a change of coordinates or, what amounts to the same thing, one cannot always make gravitation disappear by changing the system of reference. So, to find the law of a general field, a generalization seems necessary that, we believe, is possible to make. The simplest law of gravitation is that of Newton, which for spaces external to matter corresponds to the mathematical condition expressed by the Laplace-Poisson equation (vanishing of the Laplacian). But there is a great difficulty: the theory of special relativity has changed somewhat the nature of physical time, but it has preserved the geometry. Likewise, it has conserved the theory of time in its main features, that is, by measuring it by identical clocks distributed throughout space, and it was sufficient to give one rule to regulate them. In the general theory this does not work. One can easily see that if the postulate of gen- eral relativity is true, neither Euclidean geometry nor the measure of time by identical clocks can be preserved. Let us take a system k with respect to which there is no gravita- tional field, that is, let k be an inertial system. Let us now take a disk in rotation which con- k′ k′ k′ k′ k′ k′ k′