Probing nonlinear electrodynamics in slowly rotating spacetimes through neutrino astrophysics

Huge electromagnetic fields are known to be present during the late stages of the dynamics of supernovae. Thus, when dealing with electrodynamics in this context, the possibility may arise to probe nonlinear theories (generalizations of the Maxwellian electromagnetism). We firstly solve Einstein field equations minimally coupled to an arbitrary (current-free) nonlinear Lagrangian of electrodynamics (NLED) in the slow rotation regime $a\ll M$ (black hole’s mass), up to first order in $a/M$. We then make use of the robust and self-contained Born-Infeld Lagrangian in order to compare and contrast the physical properties of such NLED spacetime with its Maxwellian counterpart (a slowly rotating Kerr-Newman spacetime), especially focusing on the astrophysics of both neutrino flavor oscillations (${\nu }_e\to {\nu }_{\mu }$, ${\nu }_{\tau }$) and spin-flip (${\nu }_l\to {\nu }_r$, “$l$” stands for “left” and “$r$” stands for “right”, change of neutrino handedness) mass level crossings, the equivalent to gyroscopic precessions. Such analysis proves that in the spacetime of a slowly rotating nonlinear charged black hole (RNCBH), intrinsically associated with the assumption the electromagnetism is nonlinear, the neutrino dynamics in core-collapse supernovae could be significantly changed. In such an astrophysical environment, a positive enhancement (reduction of the electron fraction $Y_e<0.5$) of the r-process may take place. Consequently, it might result in hyperluminous supernova explosions due to enlargement, in atomic number and amount, of the decaying nuclides. Finally, we envisage some physical scenarios that may lead to short-lived charged black holes with high charge-to-mass ratios (associated with unstable highly magnetized neutron stars) and ways to possibly disentangle theories of the electromagnetism from other black hole observables (by means of light polarization measurements).

Figure 1

Ratio of the off-diagonal term $A(r)$ in Eq. (4) coming from Born-Infeld (BI) Lagrangian and the Maxwell (Ma) Lagrangian for selected values of the parameter $\alpha$ with $bM=0.017$. The value of $bM$ was chosen such that it is in agreement with the condition $bM>{10}^{-4}$, valid for astrophysical bodies with some solar masses, and Eq. (55) is satisfied for all selected $\alpha$. In this case, the associated black holes do have just one horizon and do not have classical counterparts.

Figure 2

Ratio of the polar magnetic fields for the same theories, selected values of $\alpha$, and meaning of the curves as in Fig. 1.

Figure 3

Ratio of radial components of the magnetic field for the same assumptions as Figs. 1 and 2.

Figure 4

Maxwell to Born-Infeld black holes oscillation lengths ratio for selected values of $\alpha$ for a fixed value of $bM$ satisfying $bM>{10}^{-4}$ and Eq. (55).

Figure 5

Spherically symmetric transition probability of neutrino spin-flip, Eqs. (52) and (56), for selected values of $\alpha$ and $bM$, for circular orbits at $u=0.24$. Notice that in this case, $u\le \mathcal{F}[\pi ,1/\sqrt{2}]\sqrt{bM/\alpha }/3$, and so the spin-flip frequency in Born-Infeld theory is larger than its Maxwellian counterpart. We point out that in this case the electromagnetic theories for a given $\alpha$ would differ after $\tau \approx 20M$, which for stars with some solar masses would be equivalent to $({10}^{-4}-{10}^{-3})\mathrm{s}$.

Figure 6

Induced spin-flip frequencies due to slow rotation for Born-Infeld theory when compared to its Maxwellian counterpart for $bM=0.013$ and selected values of $\alpha$ within the context of circular orbits. The Born-Infeld frequency induction due to slow rotation is always larger in magnitude than the classical case.

Figure 7

Born-Infeld to Maxwell ratio of ${\mathrm{\Omega }}^r$ appearing in Eq. (61) for various values of $\alpha$ and $\theta$, with $bM=0.1$.

Figure 8

Born-Infeld to Maxwell ratio of the polar frequency component ${\mathrm{\Omega }}^{\theta }$ appearing in Eq. (61) for various values of $\alpha$ and $\theta$, with $bM=0.1$.

Figure 9

The $Y_e$ for the case $u=0.42$ as a function of the anti-neutrino-to-neutrino local luminosity. For this case, it was chosen $r_{{\nu }_e}^{\nu -sp}=3.5$ Km just to try to simulate the case where the neutrino spheres are close to the horizon of the collapsing system. One sees from this plot that $Y_e^{\text{Schw}}$ is larger than $Y_e^{\mathrm{BI}}$ and $Y_e^{\mathrm{Ma}}$. Further for a given $\alpha$, $Y_e^{\mathrm{BI}}>Y_e^{\mathrm{Ma}}$.