Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation

Abstract

We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the nonrelativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.

Fantasia of a Superfluid Universe — In Memory of Kerson Huang

“…(20). Based on the numerical computations in refs., 29,30 we find that the first order Coriolis term is the most important. We show that it leads to vortex formation, and therefore that quantum vortices can be generated by the frame-dragging effect of black holes.…”

“…Huang had been looking for this solution for a long time as he suggested that the so-called "non-thermal filaments" observed near the center of the Milky Way (where there are super massive black holes) could be parts of vortex rings surrounding the blackholes, as pointed out in ref. 1,9,29,30 In fact the current-current interaction (7) mimics rotation terms with local (space-dependent) angular velocities as shown above in the Kerr metric case, so it is possible to have a vortex-ring lattice as a solution to the NLKG in the Kerr metric. For the gravitational collapse problem of rotating blackholes, the spacetime outside is described by the Kerr metric while the spacetime inside should be the generalization of the Robertson-Walker metric with angular momentum.…”

“…In ref. 29 we have shown that all these effects can be traced to a universal mechanism characterized by operator terms in the NLKG that can be associated with the Coriolis and the centrifugal force:…”

“…We have also generalized Feynman's relation to the relativistic regime. 29 Consider N vortices in rotating bucket of radius R and angular velocity Ω. At the wall of the bucket, the superfluid velocity is v s = ΩR.…”

“…In ref. 29 we solve numerically the NLKG with various inertial or interaction terms in (2+1)-and (3+1)-dimensional Minkowski spacetime. In Fig.…”