Below the Schwinger critical magnetic field value, quantum vacuum and gamma-ray bursts delay

Abstract

A magnetic field above the Schwinger critical value $B_{\mathrm{crit}}={10}^9$ Tesla is much higher than any magnetic field known by now in the interstellar bulk except in the vicinity of observed magnetars having magnetic fields between 10⁹ and 10¹¹ Tesla. Above the critical magnetic field limit, calculated by Schwinger in the lowest order perturbation in quantum electrodynamics (QED), one reaches the threshold for electron-positron pair creation (through the intermediate electric field, as known also from standard electrodynamics), which has interesting consequences. Therefore, finding out whether one could encounter some consequences of interest also for the values of the magnetic field below the Schwinger critical point, we invoke the next higher-order effect in QED, which is emerging from the Quantum Vacuum Effect. The latter is equivalent to the use of the Euler-Heisenberg effective theory in nonlinear electrodynamics, where the Lagrangian has now a term with a higher power, $B^4$. In this case, in the region $B<B_{\mathrm{crit}}$, we show that interesting effects appear, among them the Cherenkov radiation and the reduction in the speed of light. The latter effects appear due to the quantum vacuum mimicking a medium. We also present quantitative arguments for such a close analogy. As a rough estimate, we show that the time delay τ of gamma-ray bursts (GRB) having traveled through the entire cosmological distances in an average strong magnetic field such as 10⁶ Tesla, reaches an experimentally considerable value of $\tau =2.4$ hours. Of course in the vicinity of magnetars, the magnetic field is much stronger, of the order of ${10}^9-{10}^{11}$ Tesla. However, in this case the linear scale of GRB trajectory through such regions would be much smaller. For the latter, we also give a corresponding estimate for the number of the magnetars along the trajectory and also for the delay. Finally, we shall dwell on the recently raised issue in the literature, namely the Lorentz invariance violation (LIV).