Non-Null Torus Knotted Gravitational Waves from Gravitoelectromagnetism

TL;DR

This work presents a novel non-null torus-knotted gravitational monochromatic wave (NNTKGMW) as a full perturbative solution to the linearized Einstein equations in vacuum within an extended gravitoelectromagnetism (GEM) framework. By fixing gauge and extending the GEM sector to include $h_{ij}$, the authors encode torus-knotted topology into both GEM potentials $A_i$ and spatial perturbations $\gamma_{ij}$, yielding a real line element $ds^2(k)$ with curvature sourced by knot data. They compute the Riemann and Ricci tensors, show $R=0$, derive geodesic equations, and analyze dual GEM geometry and helicities, revealing that the dual geometry mirrors the original and that spatial helicities vanish while GEM helicities persist in the field sector. The results demonstrate that linearized vacuum gravity can carry nontrivial topological information linked to torus knots, with implications for topology–gravity interplay and future explorations of nonlinear effects, dual interpretations, and knotted gravitational-wave configurations.

Abstract

In this paper, we construct a non-null torus-knotted gravitational monochromatic wave solution of the linearized Einstein equations in vacuum, employing the gravitoelectromagnetic (GEM) framework by analogy with classical electrodynamics. We derive the geometric objects, including the line element, the Riemann tensor, the Ricci tensor, the Ricci scalar, and the geodesic equation for this background. Also, we investigate two properties inherent to this solution due to its GEM origin: the dual GEM potential and GEM helicity.