The two shadows of a single black hole: Vacuum birefringence phenomena within Einstein-Nonlinear-Electrodynamics

Abstract

One of the main features of nonlinear electrodynamics (NED) is the existence of an effective geometry that describes the geodesic motion of photons. A detailed analysis of the properties of effective geometry is of utmost importance for a better understanding of NED theories and their possible imprints on physics, especially in the context of black holes (BHs). We consider a NED model that depends on the two electromagnetic scalar invariants and obtain that the motion of photons in NED exhibits \textit{vacuum birefringence}, i.e., photons can propagate along two distinct paths, depending on their polarization. As a consequence of this phenomenon, we show that static black hole solutions sourced by NED can admit two distinct unstable light rings, leading to the formation of two distinct shadows. Moreover, to explore the potential astrophysical relevance of our results, we also compare them with the astrophysical observations for the shadow radius of Sagittarius A*. We place upper limits on the charge-to-mass ratio of the NED-sourced black hole. We also show that the motion of photons in this context can be interpreted as nongeodesic curves subjected to a four-force term from the perspective of an observer in the spacetime metric, generalizing previous results in the literature for NED models that depend on a single electromagnetic scalar invariant.

Figures (15)

• Figure 1: Metric function of the EH BH solution, considering $\mu = 0.02M^{2}$ and distinct values of $Q/M$, as a function of $r/M$. For this case, $r_{\rm{ext}} = 0.98222 M$ and $Q_{\rm{ext}} = 1.0042M$. We also exhibit the Schwarzschild case, $Q = 0$, for comparison purposes.
• Figure 2: Comparison between the location of the event horizon with that of the signature radii, considering $Q = M$, as functions of $\mu/M^{2}$. For this case, $\mu_{\rm{ext}} = 40M^{2}/729$ and $Q_{\rm{ext}} = 1.0129M$.
• Figure 3: Comparison between the LRs of EH and RN BH geometries, as functions of $Q/M$. For the EH case, we consider the LRs of both polarizations. Here, we set $\mu = 0.05$, and $Q_{\rm{ext}} = 1.0115M$. The inset zoomed the LRs for extreme charge-to-mass BH values.
• Figure 4: The comparison between the numerical results of Subsec. \ref{['Subsec:VA']} and the analytical expressions for the radii of the LRs in the EH spacetime with $\mu=0.05$. In the top panel, we present the comparison for the $\mathcal{P}^+$ polarization, whereas the bottom panel shows the corresponding result for the $\mathcal{P}^-$ polarization.
• Figure 5: Ratio between the shadows radius of the effective geometry, $r_{s}^{\rm{SG}}$, and the effective metrics, $r_{s}^{\pm}$, as functions of $Q/M$ (top panel) and $\mu/M^{2}$ (bottom panel). In the top panel, we set $\mu/M^{2} = 0.05$, with $Q_{\rm{ext}} = 1.0115M$, while in the bottom panel, we fixed $Q = M$, with $\mu_{\rm{ext}} = 40M^{2}/729$.