A 3+1 formalism for quantum electrodynamical corrections to Maxwell equations in general relativity

摘要

Magnetized neutron stars constitute a special class of compact objectsharbouring gravitational fields that deviate strongly from the Newtonian weakfield limit. Moreover strong electromagnetic fields anchored into the star giverise to non-linear corrections to Maxwell equations described by quantumelectrodynamics (QED). Electromagnetic fields close to or above the criticalvalue of $\BQ=4.4\times10^9$~T are probably present in some pulsars and formost of the magnetars. To account properly for emission emanating from theneutron star surface like for instance thermal radiation and its polarizationproperties, it is important to include general relativistic (GR) effectssimultaneously with non-linear electrodynamics. This can be achieved through a3+1 formalism known in general relativity and that incorporates QEDperturbations to Maxwell equations. Starting from the lowest order correctionsto the Lagrangian for the electromagnetic field, as given for instance byBorn-Infeld or Euler-Heisenberg theory, we derive the non-linear Maxwellequations in general relativity including quantum vacuum effects. We alsoderive a prescription for the force-free limit and show that these equationscan be solved with classical finite volume methods for hyperbolic conservationlaws. It is therefore straightforward to include general relativity and quantumelectrodynamics in the description of neutron star magnetospheres by usingstandard classical numerical techniques borrowed from Maxwell and Newtontheory. As an application, we show that spin-down luminosity correctionsassociated to QED effects are negligible with respect to GR corrections.