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The standard of zero 3-acceleration must be provided by a geodesic 3-acceleration field or geodesic directing field, or what Weyl calls the guiding field.
This means that there does not exist a unique standard of zero 4-acceleration that is intrinsic to the differential topological structure of spacetime.
Since this field provides the standard of zero 4-acceleration we can call it a geodesic 4-acceleration field, or simply, geodesic acceleration field.
The differences then transform linearly and homogeneously; consequently, the vanishing or non-vanishing of body accelerations relative to the standard of zero acceleration provided by the geodesic 4-acceleration field (the affine structure), is coordinate independent.
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.
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.
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