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Spin-spin interaction in general relativity and induced geometries with nontrivial topology

We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. It is shown that both these matter distributions can induce a vortex field described by the curl 4-vector of a tetrad: $ω^i = \frac12\eps^{iklm}e_{(a)k}e^{(a)}_{l;m}$, where $e^{(a)}_k$ are components of the tetrad. The energy-momentum tensor $T_{ik}(ω)$ of this field has been found and shown to violate the strong and weak energy conditions which leads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exact solutions to the equations of general relativity have been found. It is also shown that other vortex fields, e.g., the magnetic field, can also possess such properties.