Gravitational Lensing of Euler-Heisenberg Black Hole Surrounded by Perfect Fluid Dark Matter

TL;DR

We address how PFDM and nonlinear electrodynamics modify gravitational lensing by an Euler–Heisenberg black hole. By deriving and solving geodesic equations in the metric with $f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{aQ^{4}}{20r^{6}}+\frac{\lambda_{\text{DM}}}{r}\ln\frac{r}{|\lambda_{\text{DM}}|}$, the study computes the deflection angle $\alpha$, light-time delay $\Delta T$, bound-orbit precession, and shadow via Ishihara-type finite-distance formalism and optical-geometry techniques. Results show that increasing PFDM parameter $\lambda_{\text{DM}}$ substantially reduces lensing and shadow observables (consistent with an effective negative mass contribution), while the nonlinear electrodynamics parameter $a$ has only tiny effects, and electric charge $Q$ yields modest additional reductions. These findings imply that gravitational lensing measurements around charged black holes could constrain PFDM properties, whereas probing nonlinear QED corrections remains challenging. The work highlights the rich interplay between dark matter and quantum-electrodynamics effects in strong gravity contexts and points to future work on higher-order QED corrections and more realistic dark-matter fluids.

Abstract

In this work, we study the gravitational lensing of Euler-Heisenberg black hole surrounded by perfect fluid dark matter. This kind of black hole solution enables us to investigate the nontrivial interplay between the dark matter effects and nonlinear electrodynamics effects (or quantum electrodynamics effects) on charged black hole systems. The important observables in gravitational lensings are calculated and discussed in our work, including the gravitational deflection angle of light and time delay of light. Additionally, we also explore the massive orbit's bound orbits (and their precession angles) and black hole shadow radius for Euler-Heisenberg black hole in the presence of dark matter. The results indicate that the Euler-Heisenberg black hole with a larger perfect fluid dark matter parameter could greatly reduce the gravitational deflection angle of light, time delay of light, and precession angle of massive object's bound orbit, while the nonlinear electrodynamics effects do not have large influences on these observables. Keywords: Euler Heisenberg Black Hole; Gravitational Lensing; Perfect Fluid Dark Matter; Nonlinear Electrodynamics

Figures (8)

• Figure 1: The gravitational deflection angle of light for finite distance light source and observer. This figure illustrate the gravitational deflection angle $\alpha \equiv \Psi_{\text{O}} - \Psi_{\text{S}} - \Delta \phi_{\text{OS}}$ in the thin lens approximation.
• Figure 2: A schematic plot of the precession of azimuthal angle for a massive object's bound orbit around the central black hole.
• Figure 3: The gravitational deflection angle of light for Euler-Heisenberg black hole surrounded by PFDM. (a) The upper left panel exhibits the influences from black hole electric charge $Q$. (b) The upper right panel shows the influences from nonlinear electrodynamics effects / QED effects. (c) The lower left panel highlights the influences from dark matter in the absence of electric charge and nonlinear electrodynamics effects. (d) The lower right panel highlights the influences from dark matter in the presence of nonlinear electrodynamics effects. In all panels, the horizontal axis labels the impact parameter in photon orbit. The positions of light source and observer are located at a certain distance $r_{\text{O}} = r_{\text{S}} = 200 M$.
• Figure 4: The gravitational time delay of light for Euler-Heisenberg black hole surrounded by PFDM. (a) The upper left panel shows the influences from black hole electric charge $Q$. (b) The upper right panel exhibits the influences from nonlinear electrodynamics effects / QED effects. (c) The lower left panel highlights the influences from dark matter in the absence of black hole charge and nonlinear electrodynamics effects. (d) The lower right panel highlights the influences from dark matter in the presence of nonlinear electrodynamics effects. In all panels, the horizontal axis labels the position of light source, and the vertical axis labels the time delay of light $\Delta T$ measured in unit of black hole mass.
• Figure 5: The precession angles and trajectories of massive particle's bound orbits moving around the Euler-Heisenberg black hole in the presence of PFDM. (a) The upper left panel plots the precession angle of orbits affects by black hole charge. (b) The upper right panel illustrates the trajectories of orbits for different black hole charge values. (c) The middle left panel gives the precession angle of orbits affects by dark mater. (d) The middle right panel illustrates the trajectories of orbits for different PFDM parameter values. (e) The lower left panel shows the precession angle of orbits influenced by nonlinear electrodynamics effects / QED effects. (f) The lower right panel illustrates the trajectories of orbits for different nonlinear electrodynamics parameter values.