Superconductor Meissner Effects for Gravito-Electromagnetic Fields in Harmonic Coordinates Due to Non-Relativistic Gravitational Sources

Abstract

There is much discrepancy in the literature concerning the possibility of a superconductor expelling gravito-electromagnetic fields just as it expels electromagnetic fields in the Meissner effect. Contradicting results are found in at least 18 papers written collectively by more than 20 authors and published over the course of more than 55 years (from 1966 to the present year of 2022). The primary purpose of this paper is to carefully explain the reason for the discrepancies, and provide a single conclusive treatment which may bring coherence to the subject. The analysis begins with a covariant Lagrangian for spinless charged particles (Cooper pairs) in the presence of electromagnetic fields in curved space-time. It is known that such a Lagrangian can lead to a vanishing Hamiltonian. Alternatively, it is shown that using a “space + time” Lagrangian leads to an associated Hamiltonian with a canonical momentum and minimal coupling rule. Discrepancies between Hamiltonians obtained by various authors are resolved. The canonical momentum leads to a modified form of the London equations and London gauge that includes the effects of gravity. A key result is that the gravito-magnetic field is expelled from a superconductor with a penetration depth on the order of the London penetration depth only when an appropriate magnetic field is also present. The gravitational flux quantum (fluxoid) in the body of a superconductor, and the quantized supercurrent in a superconducting ring, are also derived. Lastly, the case of a superconducting ring in the presence of a charged rotating mass cylinder is used as an example of applying the formalism developed.