[Paper Review] Maxwell's equations in 4-dimensional Euclidean space

José B. Almeida|ArXiv.org|Mar 9, 2004

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper formulates Maxwell’s equations in 4-dimensional Euclidean space using geometric algebra, embedding the electromagnetic vector potential in the frame vector $g_0$. It derives relativistic electrodynamics from spatial curvature, showing that solutions to Maxwell’s equations describe vector potential rotation on planes, constrained to hyperspherical surfaces, thus explaining photon confinement to great circles on a 4D hypersphere model of the Universe.

ABSTRACT

The paper formulates Maxwell's equations in 4-dimensional Euclidean space by embedding the electromagnetic vector potential in the frame vector $g_0$. Relativistic electrodynamics is the first problem tackled; in spite of using a geometry radically different from that of special relativity, the paper derives relativistic electrodynamics from space curvature. Maxwell's equations are then formulated and solved for free space providing solutions which rotate the vector potential on a plane; these solutions are shown equivalent to the usual spacetime formulation and are then discussed in terms of the hypersphere model of the Universe recently proposed by the author.

Motivation & Objective

• To reformulate Maxwell’s equations in 4-dimensional Euclidean space using geometric algebra, providing a new geometric foundation for electromagnetism.
• To resolve the inconsistency in prior work where photons were constrained to great circles rather than following geodesics in a 4D hyperspherical model of the Universe.
• To demonstrate that relativistic electrodynamics can emerge from curved 4D Euclidean geometry, not Minkowski spacetime.
• To show that solutions to Maxwell’s equations naturally lead to vector potential rotation, explaining electromagnetic wave behavior in this framework.
• To address the evanescence problem in the $x^0$ direction by linking it to the artificial flattening of the 4D hypersphere geometry.

Proposed method

• Uses geometric (Clifford) algebra in 4D Euclidean space, with orthonormal frame vectors $\sigma_\mu$ and pseudoscalar $I = \sigma_0\sigma_1\sigma_2\sigma_3$, to express Maxwell’s equations compactly.
• Embeds the electromagnetic vector potential $A$ within the frame vector $g_0$, allowing the geometric structure of space to influence electromagnetic dynamics.
• Derives Maxwell’s equations in the form $\nabla^2 A = J$ using geometric algebra, condensing the full set into a single equation.
• Solves the free-space Maxwell equation using exponential solutions involving bivectors, yielding rotating vector potentials in 3D planes.
• Analyzes solutions in terms of hyperspherical geometry, showing that constant $x^0$ surfaces correspond to 3D hyperspheres where $A$ rotates with frequency $\omega$.
• Identifies the evanescence of solutions in the positive $x^0$ direction as a consequence of approximating the curved 4D hypersphere as flat, suggesting a need for resonant mode solutions in full curved geometry.

Experimental results

Research questions

• RQ1 Can Maxwell’s equations be consistently formulated in 4-dimensional Euclidean space using geometric algebra?
• RQ2 How does relativistic electrodynamics emerge from a 4D Euclidean geometry that differs fundamentally from Minkowski spacetime?
• RQ3 Why are photons constrained to great circles on a 4D hypersphere in the author’s cosmological model, and can this be explained by the solutions to Maxwell’s equations?
• RQ4 What is the origin of the evanescent behavior in the $x^0$ direction, and how does it relate to the curvature of the 4D hyperspherical Universe?
• RQ5 Can the apparent inconsistency in the $x^0$ dependence be resolved by considering the full curved geometry rather than a flat approximation?

Key findings

• Maxwell’s equations in 4D Euclidean space are successfully formulated using geometric algebra, condensed into the equation $\nabla^2 A = J$.
• Solutions to the free-space Maxwell equation describe a vector potential that rotates in a 3D plane with angular frequency $\omega$, progressing along the direction normal to the plane.
• The vector potential rotation is equivalent to a circularly polarized electromagnetic wave in the spacetime formulation, with $A = \alpha' \sigma_1 e^{-i\sigma_3 \omega t}$ when $x^3$ is interpreted as time.
• Solutions are constrained to hypersurfaces of constant $x^0$, which correspond to 3D hyperspheres, explaining why photons follow great circles rather than geodesics.
• The evanescent behavior in the positive $x^0$ direction and exponential growth in the negative direction are artifacts of flattening the 4D hypersphere geometry.
• The inconsistency in the $x^0$ dependence is attributed to the approximation of the curved 4D space as flat, suggesting that full solutions must account for resonant modes on the hypersphere.