Abstract
This work discusses about the interplay of gravitation and superfluids. We focus our attention on the specific type of compact binary systems consisting of one neutron star and one black hole. Here the neutron star is regarded as a huge block of superfluid neutrons while the black hole is to provide the desired gravitational fields. To avoid complexity of the highly nonlinear Einstein field equation, we reformulate it in weak-field approximation in which the Einstein equation reduces to a set of four equations similar to that of Maxwell in electrodynamics. In this simplification, in addition to the usual Newtonian gravitational field, we also have its dual field, the so-called gravitomagnetic field whose role in gravity is much like to magnetic field in electrodynamics. Even so, there exist many essential differences between these two dynamics, among which the most interesting is the anti-Meissner effect, contrary to that in type-I superconductor. We found that even the gravitomagnetic field oscillates, rather than exponentially decays, inside superfluids, the compression of gravitomagnetic field lines also provides a repulsive force similar to superconducting levitation. But the situations are different from and more complicated than those in superconductivity. This gravitational levitation combined with the effect due to gravitational radiation results in quasi-stable balance states of the two stars and small variations appear on the orbital period of the binary system. Furthermore, the left-handed nature of gravitomagnetic induction together with the instability of high-level superfluid vortices inside the neutron star causes the star rotation to decelerate and occasionally show mini-glitches during the binary evolution. However, these phenomena are extremely tiny and may not be easily discovered by astronomical observations. It should be noticed that the consideration and calculation in this work are based on the weak-field approximation for Einstein equation, beyond which the nonlinearity of spacetime is supposed to govern the whole scenario of binary evolution. Fortunately, it can be seen that the obtained results all happen and therefore behave well within the limits.