Physical measurements in general relativity and requirements on a physical riemannian space-time

The spatial invariant line element, which is regarded as the local distance in Einstein's theory of measurement, is actually a space contraction measured from a local Minkowski space, resting but at free fall. It is shown that Einstein's theory of measurement is actually based on only invalid applications of special relativity; and is not an integral part of general relativity. In addition to theoretical difficulties pointed out by Whitehead, Einstein's approach seems to necessitate his covariance principle that creates further problems. It is shown that a physical Riemannian space-time must have a frame of reference with the Euclidean-like structure that is compatible with physical measurements. Thus, the invariant line element provides the space contractions while the Euclidean-like structure provides the measurement of local distance. Furthermore, the notion of gauge invariant measurable quantity is proven an illusion in mathematics only. For instance, in the light deflection, the impact parameter and the shortest distance from the sun cannot be gauge invariant when measurements are involved.