On the Electromagnetic Energy Flow Along Geodesics

Abstract

We present a field-theoretic framework for modeling electromagnetic energy propagation in heterogeneous media by introducing the concept of electromagnetic geodesics. Unlike traditional ray optics, which assumes either a straight-line propagation or a simple bending in refractive media, our approach formulates wave propagation as geodetic motion in a curved spatial geometry induced by variations in refractive index. Building on earlier work, we move beyond scalar refractive index analogies and instead construct a local Riemannian metric characterized by an orthogonal geometric tensor derived from the Helmholtz representation. This tensor encodes spatial anisotropy and curvature, enabling a rigorous description of energy flow through complex media. We derive the electromagnetic geodesics by formulating and solving a Lagrangian system, yielding equations of motion for wavefront trajectories, group velocity, and intensity distribution. The concept of refractive tension—the vector displacement between Euclidean and transformed positions—plays a central role in defining the transformation matrix and associated metric. Numerical simulations for a spherical inhomogeneity embedded in vacuum demonstrate the emergence of curved geodesics and localized energy redistribution, illustrating the model's potential for interpreting interstellar electromagnetic phenomena and refractive effects in astrophysical environments. In particular, it shows the spatial dispersion of a the energy flow in the vicinity of the spherical inhomogeneity.

Plain Language Summary
We propose a new way to describe how light and electromagnetic energy move through complex materials. Instead of straight or bent paths, waves follow curved “geodesics” shaped by refractive index changes. This geometric approach explains energy flow, predicts intensity patterns, and helps interpret astrophysical and interstellar phenomena.