9 6 2 A P P E N D I X F of squares of differentials that is, it corresponds to a four-dimensional Euclidean manifold, which is the case in special relativity. Following that, he spoke of the “reduced Riemann- Christoffel tensor,” , which is—he said—the only second-order tensor dependent on gravitational potentials that is linear with respect to its second-order derivatives and which has no derivatives of a higher order. (In matter-free space, gravitational equations are the ten that result when this last tensor equals zero. Such equations are chosen to define the g values, and therefore metrics and gravitation, because since they come from the equation of a tensor, they are covariant—it would not be so accurate to call them invariant—and, because they possess the above-men- tioned properties and with them the interval , they take on the general form of a non- Euclidean space. To establish those equations as a definition of the g values is merely a hypothesis: there are ten equations that are reduced to six by four others that relate them to one another and which come from the fact that natural laws, according to the principle of general relativity, must be covariant for all systems of Gaussian coordinates. Their definitive acceptance, as in the case of the law of gravity, depends on how well they coincide with experimental re- sults derived. Supporting these equations is the fact that they satisfy Laplace’s equation, which happens, as we know, with Newtonian potential. Moreover, we will see later in a more general case that, at first attempt, they are reduced to Newtonian equations.) Geodesic Equation in Riemannian Space (Equation of the Movement of a Point) Now, Dr. Einstein said, let us look for a line similar to the straight line in a Riemannian space, that is, the shortest line between two points: lines that possess that property are called geodesic. (The importance of these lines stems from the fact that in general relativity, be- cause of the principle of equivalence, the force of gravity ceases to be an agent and is re- placed by metric characters of a four-dimensional manifold. A point set in motion in space moves along a certain path, which is a geodesic line, in the same way that a point forced to remain stationary, on a sphere, for example, and set in motion upon it moves along the same kind of line. The movement will therefore be inertial movement.) Dr. Einstein worked out the equation of those lines, saying that they represented simul- taneously the equations of movement of a point that is free from the actions of other forces in a gravitational field. He pointed out that, according to those equations, one cannot dis- tinguish between inertial and gravitational forces this is only a repetition of the principle of equivalence, which is the basis of all these considerations. General Equations of Gravitation Now, Dr. Einstein went on, we have to familiarize ourselves with the law of gravitation itself. (In this case laws within a space where matter is found.) The equation of Newtonian mechanics, he said, comes from Poisson, according to which the Laplacian of the gravitational potential is equal to the gravitational constant k times the density of the matter. Rμν ds2