Gravitoelectromagnetic Field Equations III: A Vector Calculus Approach

Electromagnetism, as unified by James Clerk Maxwell, brought together electricity and magnetism into a single coherent framework that remains foundational to modern physics. The mathematical structure of Maxwell’s equationshas since inspired attempts to describe gravitational phenomena in analogous terms. Early efforts in gravitomagnetism[3], as well as studies of gravitomagnetic effects [5], have revealed deep structural connections between gravitational and electromagnetic forces. In particular, gravitomagnetism describes how moving masses—especially rotatingones—interact with spacetime in a manner that mathematically mirrors how moving electric charges generate magnetic fields. These effects have been confirmed experimentally by Gravity Probe B [2] and by satellite laser-rangingexperiments using LARES and LAGEOS [1].In this preprint, we continue constructing a formulation of gravitational magnetism using vector calculus, providing a more direct and accessible framework compared to existing gravitomagnetic treatments in the literature [3, 5].We present the first three elements of what will be a set of governing equations: a static gravitational Gauss’s law, acurvature-modified Maxwell–Faraday analogue, and a gravitational-charge reformulation of the Lorentz force. Subsequent work will extend this to a full gravitomagnetic Ampere-law analogue and further time-dependent and rotational `effects