Gravitoelectromagnetic knot fields

TL;DR

This paper constructs knot solutions in linearized gravity's gravitoelectromagnetic equations, analyzing their geometric properties, geodesics, and invariants, inspired by analogous electromagnetic knot fields.

Contribution

It introduces a novel class of gravitoelectromagnetic knot solutions in vacuum, extending the concept of electromagnetic knots to gravitational perturbations.

Findings

Derived dual metric tensors for GEM knot solutions.

Calculated geodesic equations for these solutions.

Computed scalar invariants and energy-momentum pseudo-tensors.

Abstract

We construct a class of knot solutions of the gravitoelectromagnetic (GEM) equations in vacuum in the linearized gravity approximation by analogy with the Ra\~{n}ada-Hopf fields. For these solutions, the dual metric tensors of the bi-metric geometry of the gravitational vacuum with knot perturbations are given and the geodesic equation as a function of two complex parameters of the GEM knots are calculated. Finally, the Landau--Lifshitz pseudo-tensor and a scalar invariant of the GEM knots are computed.