Discretization of Maxwell's Equations for Non-inertial Observers Using Space-Time Algebra

TL;DR

This paper introduces a space-time algebra-based discretization method for Maxwell's equations that handles moving geometries without non-relativistic assumptions, enabling accurate modeling of relativistic effects like the Sagnac effect.

Contribution

It develops a 4D mesh construction method from a 3D mesh, extending traditional approaches to relativistic space-time scenarios without simplifying assumptions.

Findings

Efficient modeling of rotating systems using space-time discretization.

Constant material matrices improve computational efficiency in rotating cases.

Successful simulation of Sagnac effect in a relativistic framework.

Abstract

We employ Maxwell's equations formulated in Space-Time Algebra to perform discretization of moving geometries directly in space-time. All the derivations are carried out without any non-relativistic assumptions, thus the application area of the scheme is not restricted to low velocities. The 4D mesh construction is based on a 3D mesh stemming from a conventional 3D mesh generator. The movement of the system is encoded in the 4D mesh geometry, enabling an easy extension of well-known 3D approaches to the space-time setting. As a research example, we study a manifestation of Sagnac's effect in a rotating ring resonator. In case of constant rotation, the space-time approach enhances the efficiency of the scheme, as the material matrices are constant for every time step, without abandoning the relativistic framework.