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

Abstract

A magnetic field above the Schwinger critical value $B_{\rm crit} = 10^9$ Tesla is much higher than any magnetic field known by now in the interstellar bulk except in the vicinity of observed magnetars with magnetic fields between $10^9$ and $10^{11}~$Tesla. Above the critical magnetic field, calculated by Schwinger in the lowest order perturbation in quantum electrodynamics (QED), one reaches the threshold for electron-positron pair creation, 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 a term with a higher power, $B^4$. In this case, in the region $B<B_{\rm crit}$, we show that interesting effects appear, among them the Cherenkov radiation and the reduction in the speed of light. The latter effects appear because of 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 $\tau$ of gamma-ray bursts (GRB) having traveled through the entire cosmological distances in an average strong magnetic field such as $10^6~$Tesla, reaches an experimentally considerable value of $\tau = 2.4$ hours. 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 give an 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).