Straightest Geodetic Lines. A line may be constructed in such a way that its successive elements arise from each other by parallel displacements. This is the natural generalization of the straight line of the Euclidean geometry. For such a line, we have [Pg 82] The left-hand side is to be replaced by ,[17] so that we have We get the same line if we find the line which gives a stationary value to the integral between two points (geodetic line). [17]The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element (), from the direction vector of each point considered. [Pg 83] LECTURE IV THE GENERAL THEORY OF RELATIVITY (continued) WE are now in possession of the mathematical apparatus which is necessary to formulate the laws of the general theory of relativity. No attempt will be made in this presentation at systematic completeness, but single results and possibilities will be developed progressively from what is known and from the results obtained. Such a presentation is most suited to the present provisional state of our knowledge. A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this is a real straight line. The natural, that is, the simplest, generalization of the straight line which is plausible in the system of concepts of Riemann's general theory of invariants is that of the straightest, or geodetic, line. We shall accordingly have to assume, in the sense of the principle of equivalence, that the motion of a material particle, under the action only of inertia and gravitation, is described by the equation, In fact, this equation reduces to that of a straight line if all the components, , of the gravitational field vanish. How are these equations connected with Newton's equations of motion? According to the special theory of relativity, the as well as the , have the values, with respect to an inertial [Pg 84] system (with real time co-ordinate and suitable choice of the sign of ), {\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right\}. \qquad \text{(91)} "> The equations of motion then become We shall call this the "first approximation" to the -field. In considering approximations it is often useful, as in the special theory of relativity, to use an imaginary -co-ordinate, as then the . to the first approximation, assume the values {\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right\}. \qquad \text{(91a)} "> These values may be collected in the relation To the second approximation we must then put where the are to be regarded as small of the first order. [Pg 85]