3+1 formulation of light modes in nonlinear electrodynamics

Abstract

We present a 3+1 formulation of the light modes in nonlinear electrodynamics described by Plebanski-type Lagrangians, which include post-Maxwellian, Born-Infeld, ModMax, and Heisenberg-Euler-Schwinger QED Lagrangians. In nonlinear electrodynamics, strong electromagnetic fields modify the vacuum such that it acquires optical properties. Such a field-modified vacuum can possess electric permittivity, magnetic permeability, and a magneto-electric response, inducing novel phenomena such as vacuum birefringence. By exploiting the mathematical structures of Plebanski-type Lagrangians, we establish a streamlined procedure and explicit formulas to determine light modes, i.e., refractive indices and polarization vectors for a given propagation direction. We also work out the light modes of the various Lagrangians for an arbitrarily strong magnetic field. The 3+1 formulation advanced in this paper has direct applications to the current vacuum birefringence research: terrestrial experiments using permanent magnets/ultra-intense lasers for the subcritical regime and astrophysical observation of X-rays from highly magnetized neutron stars for the near-critical and supercritical regimes. © 2025 Author(s).