§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation. A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimensional region, in which the co-ordinates K_{0} can be, and are, so chosen that _g__{μν}’s have special constant values of the expression (4). Let us discuss this motion from the stand-point of any arbitrary co-ordinate-system K₁; it moves with reference to K₁ (as explained in §2) in a gravitational field. The laws of motion with reference to K₁ follow easily from the following consideration. With reference to K₀, the law of motion is a four-dimensional straight line and thus a geodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law of motion for the motion of the material-point with reference to K₁. If we put (45) $$ \Gamma^{\tau}_{\mu\nu} = - \begin{Bmatrix}\mu & & \nu\ \tau&\end{Bmatrix} $$ we get the motion of the point with reference to K₁, given by (46) $$ \frac{d^2 x_{\tau}}{ds^2} = \Gamma^{\tau}_{\mu\nu} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$ We now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field, when there exists no reference-system K₀, with reference to which the special relativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46) contains only the first differentials of _g__{μν}, among which there is no relation in the special case when K₀ exists. If γ_{μν}^{τ}’s vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field. §14. The Field-equation of Gravitation in the absence of matter. In the following, we differentiate gravitation-field from matter in the sense that everything besides the gravitation-field will be signified as matter; therefore the term includes not only matter in the usual sense, but also the electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points. A special case in which the field-equations sought-for are evidently satisfied is that of the special relativity theory in which _g__{μν}’s have certain constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K₀. With reference to this system, all the components B^{ρ}_{μστ} of the Riemann’s Tensor [equation 43] vanish. These vanish then also in the region considered, with reference to every other co-ordinate system.