Superconductor Meissner effects for gravito-electromagnetic fields in harmonic coordinates due to non-relativistic gravitational sources

It is well known that a covariant Lagrangian for relativistic charged particles can lead to a vanishing Hamiltonian. Alternatively, it is shown that using a "space+time" Lagrangian leads to a new canonical momentum and minimal coupling rule that describes the coupling of both electromagnetic and gravitational fields to a relativistic charged particle. Discrepancies between Hamiltonians obtained by various authors are resolved. The canonical momentum leads to a new form of the London equations and London gauge. Using the linearized Einstein field equation in harmonic coordinates, and a non-relativistic ideal fluid, leads to gravito-electromagnetic field equations. These are used to obtain new penetration depths for both the magnetic and gravito-magnetic fields. A key result is that the gravito-magnetic field is expelled from a superconductor only when a magnetic field is also present. The flux quantum in the body of a superconductor, and the quantized supercurrent in a superconducting ring are 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.