Anisotropic Einstein Universes with a Global Magnetic Field and Sasakian Quasi-Killing Spinors

We consider an Einstein–Dirac–Maxwell system with two charged massless spinors coupled with an electromagnetic field, and construct a family of exact solutions to the system. The solution spacetime is an anisotropic generalization of the static Einstein universe which has a global cosmic magnetic field generated by the current of the spinors. The magnetic field is a force-free field that has played an important role in the study of cosmic magnetic fields. Our exact solution is regarded as a toy model which describes global cosmic magnetic phenomena in the early universe. The spinors are induced from Sasakian quasi-Killing spinors, and the total Dirac current flows along fibers of the Hopf-fibration. The magnetic field is a contact magnetic field.