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Is Weyl unified theory wrong or incomplete?

In 1918, H. Weyl proposed a unified theory of gravity and electromagnetism based on a generalization of Riemannian geometry. In spite of its elegance and beauty, a serious objection was raised by Einstein, who argued that Weyl's theory was not suitable as a physical theory . According to Einstein, the theory led to the prediction of a "second clock effect", which is not observed by experiments. We briefly revisit this point and argue that a preliminary discussion on the very notion of proper time is needed in order to consider Einstein's critical point of view.