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 [4], 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 begin constructing a formulation of gravitational magnetism using vector calculus, providinga more direct and accessible framework compared to existing gravitomagnetic treatments in the literature [3, 4]. Wefocus on the first of what will be a set of governing equations, establishing the static gravitational Gauss’s law as thefoundation. Subsequent work will extend this to include time-dependent and rotational effects.

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 equations has 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 rotating ones—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-ranging experiments 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, a curvature-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

Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields In physics, the special theory of relativity, or simply special relativity, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, "On the Electrodynamics of Moving Bodies", the theory is presented as being based on just two postulates: The laws of physics are invariant (identical) in all inertial frames of reference (that is, frames of reference with T…