Page:The principle of relativity (1920).djvu/209

 of motion with reference to K_{1} follow easily from the following consideration. With reference to K_{0}, 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 K_{1}; If we put

(45) Γ_{μυ}^{τ} = - {^{μυ} _{τ}}

we get the motion of the point with reference to K_{1}, given by

(46) d^2 x_{τ}/ds^2 = Γ_{μυ}^{τ} (dx_{μ}/ds) (dx_{υ}/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 K_{0}, 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_{μυ}, among which there is no relation in the special case when K_{0} 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.