Relativistic potential of Weyl: a gateway to quantum physics in the presence of gravitation?

The well-noted correspondence between gravitation and electrodynamics emphasizes the importance of the Lanczos tensor—the potential of the Weyl tensor—which is an inherent structural element of any metric theory of gravity formulated in a 4-dimensional pseudo Riemannian spacetime. However, this ingenious discovery has gone largely unnoticed. We elucidate this important quantity by deriving its expressions in some particularly chosen spacetimes and try to find out what it represents actually. We find out that the Lanczos potential tensor does not represent the potential of the gravitational field, as is ascertained by various evidences. Rather, it is enriched with various signatures of quantum character, which provides a novel insight into the heart of a geometric embodiment of gravity. (paper)