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

TL;DR

This paper explores how spinor fields and rotating fluids in general relativity can induce vortex fields that violate energy conditions, potentially leading to exotic topologies like wormholes, supported by exact solutions.

Contribution

It introduces a new mechanism for vortex field induction in general relativity and provides exact solutions showing possible nontrivial topologies.

Findings

Vortex fields violate energy conditions.

Exact solutions suggest wormhole-like geometries.

Magnetic fields can also induce similar effects.

Abstract

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: $\omega^i = \frac12\eps^{iklm}e_{(a)k}e^{(a)}_{l;m}$, where $e^{(a)}_k$ are components of the tetrad. The energy-momentum tensor $T_{ik}(\omega)$ 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.