8 8 2 A P P E N D I X H passes a curve of each system, thus two numbers correspond to each point. These points are called its curvilinear coordinates, which lacks any physical meaning, because it is necessary to know what the curves are. Nevertheless, all the properties can be described with these numbers. In spite of the fact that Gaussian coordinates lack physical meaning, they give the relations among real things when they are eliminated. The entire geometry of the surface is known if we know the distances between points infinitely close to one another in any region of the surface. We are going to see how this condition can be transformed into formulas. Taking a portion of an infinitely small surface, one can consider it a plane and apply Eu- clidean geometry to it, in particular the Pythagorean theorem. In local coordinates, the dis- tance between two nearby points would be . If we take Gaussian coordinates and , the local coordinates will, for each point, be functions of Gaussians and the distance will take the general form , where the g’s are functions of the x’s and depend on the form of the surface and of the net- work of coordinate curves selected. The g’s characterize the surface and cannot be taken as functions given beforehand. In the general relativistic problem, something completely analogous happens. We have seen, first, that one cannot attribute, a priori, any physical meaning to the coordinates. Sec- ond, for a small part of our universe, the Euclidean coordinates are valid, for example on the Earth for a freely falling observer. Free bodies have no acceleration, and therefore there is no gravitation. For an infinitely small portion of our universe one can find a system of coordinates with respect to which special relativity is valid. In special relativity the quantity , which refers to two events infinitely close to each other in time and space, does not depend on the inertial system of coordinates. This statement is equivalent to the Lorentz transfor- mation. turns out to be an invariant under transformations of the Lorentz group and can be measured with rulers and clocks, because the dx’s etc. are directly observable differen- tials. We have seen that in our space of four dimensions we can take small parts without grav- ity and then, for these, invariance persists for the local coordinates. If we take Gaussian co- ordinates (four arbitrary numbers), which fulfill only the condition of continuity, we can describe the universe in the same way as Gauss describes the surface. The coordinates alone mean nothing, they only give the form we mentioned before. The g’s, of which now there are ten, represent directly measurable quantities and give a metric relationship or also define a gravitational field. In effect, let us consider an inertial system without solid bodies then the g’s will assume the values 1, 1, 1, –1, which are constant this is the only special case known up to now if there is a gravitational field, the g’s are no longer constant. If we change the coordinate system, for example, by taking a rotating one, the g’s appear as vari- able coefficients, and we may say that the variability of the g’s give the gravitational field. ds2 dX2 dX1 2 += x1 x2 ds2 gikd xidxk = (i, k 1, 2) = ds2 dx2 dy2 dz2 c2dt2 – + + = ds2 ds2