Topologically Nontrivial Solution in Einstein-Dirac Gravity on the Hopf Bundle

详细

A topologically nontrivial solution in Einstein-Dirac gravity with a cosmological constant is obtained. The space-time has the Hopf bundle as a spatial section. It is shown that the Hopf invariant is related to the spinor current density. Two Dirac spinors are used for obtaining a diagonal energy-momentum tensor. Solutions to the nongravitating Dirac equation in background Lorentzian space-time with the Hopf bundle as a spatial section are also obtained. Nongravitating solutions of the Dirac equation are characterized by two quantum half-integer numbers m, n.