Symmetry Analysis of Proper-Time Maxwell's Equations: Invariant Solutions and Physical Implications

Gill and Zachary's proper-time reformulation of Maxwell's equations introduces a source-dependent speed b = sqrt(c^2 + u^2), distinct from standard electrodynamics. This study applies Lie symmetry analysis to the resulting wave equations, yielding new invariant solutions, conservation laws, and physical interpretations. In the one-dimensional case, we derive E(x, tau) = A x / tau + B + k * integral of rho(x) dx, revealing radiative behavior linked to b. Conservation laws via self-adjointness, singularity analysis near tau = 0, and simulations of accelerating charges extend Gill and Zachary's framework. The findings suggest new insights into propagation and energy dynamics, clarified as coordinate effects rather than violations of causality, offering fresh perspectives on relativistic electrodynamics.