  The expression \(\overline{X}^{i}_{jk}\gamma^{\,j}_{1}\gamma^{k}_{1}\) in  equation (33) represents the inhomogeneous term of the transformation of the 4-acceleration. The inhomogeneity of the transformation law  entails that a 4-acceleration that is zero with respect to one  coordinate system is not zero with respect to another coordinate  system. This means that there does not exist a unique standard of  zero 4-acceleration that is intrinsic to the differential topological structure of spacetime. Moreover, even the difference of the  4-accelerations of two bodies at the same spacetime point has no  absolute meaning, unless their 4-velocities happen to be the same.  This shows that while the differential topological structure of  spacetime gives us sufficient structure to do calculus and to derive  the transformation laws for 4-velocities and 4-accelerations by way  of simple differentiation, it does not provide sufficient structure  with which to determine a standard of zero 4-acceleration. Therefore, as Weyl repeatedly emphasized, no solution to the problem of motion  is possible, unless “we dare to acknowledge the  inertial structure as a real thing that not only exerts  effects upon matter but in turn suffers such effects”. In  other words there must exist a structure in addition to the  differential topological structure in the form of a geometric  structure field, or in Weyl’s words, geometrisches  Strukturfeld, which constitutes the inertial structure of  spacetime, and which provides the standard of zero 4-acceleration.  Since this field provides the standard of zero 4-acceleration we can  call it a geodesic 4-acceleration field, or simply, geodesic  acceleration field. A particle in free motion is one that is exclusively governed by this geodesic acceleration field. 