Gauge-covariant Raychaudhuri dynamics for spin-nondegenerate Lorentz-violating congruences

Abstract

We investigate the Raychaudhuri dynamics of charged spin--nondegenerate Lorentz--violating particle congruences under minimal electromagnetic coupling. The coupling is introduced through the gauge--covariant momentum $P_{\mu}=\pi_{\mu}-qA_{\mu}$, so that the branch dispersion relation keeps its free functional form, while the electromagnetic field enters through the evolution of $P_{\mu}$. For a generic branch $\mathcal D^{(\pm)}(P)$, the tangent $k^{\mu}_{(\pm)}$ and the momentum Hessian $M^{\mu\nu}_{(\pm)}$ determine the covariant acceleration, $a^{\mu}_{(\pm)}=-qM^{\mu\nu}_{(\pm)}F_{\nu\rho}k^{\rho}_{(\pm)}$. As a consequence, the Raychaudhuri equation acquires the branch-dependent electromagnetic source $-q\nabla_{\mu}\!\left(M^{\mu\nu}_{(\pm)}F_{\nu\rho}k^{\rho}_{(\pm)}\right)$. We apply this construction to the $b_{\mu}$, $H_{\mu\nu}$, and $d_{\mu\nu}$ sectors, obtaining the corresponding branch tangents, Hessians, accelerations, and focusing equations. In flat spacetime, the electromagnetic field modifies the expansion through the divergence of the effective branch force. Therefore, uniform fields may bend the trajectories, whereas local focusing requires field gradients or, in the magnetic case, a coupling to an already deformed congruence. We also develop the analogous description for semiclassical quasiparticle beams, where the band Hessian plays the role of an effective electromagnetic response tensor. For anisotropic parabolic, Dirac--like, and Weyl--type dispersions, the same geometric structure relates electromagnetic textures to beam focusing. In two-branch systems, the opposite Hessians of the branches can produce focusing in one congruence and defocusing in the other, giving a quasiparticle realization of branch--dependent birefringence.