Particle motion in the Einstein-Euler-Heisenberg rotating black hole spacetime

We study the motion of charged test particles and photons in the gravitational field of the rotating electrically charged Einstein-Euler-Heisenberg black hole solution. This describes a nonlinear electromagnetic generalization of the Kerr-Newman solution, which endows the vacuum with an effective dielectric constant. The orbits of photons are analyzed by means of the effective Plebański pseudometric related to the geometrical metric and to the electromagnetic energy-momentum tensor. The QED induced modifications of the shape of the shadow are presented and discussed.

Figure 1

The potentials $V_{\mathrm{eff}}^{\pm }$ as a function of $r/M$ for the KN case (dashed line) and the rotating EEH one (continuous line). The figure on the rhs shows the enlargement of the corresponding region of the figure on the lhs. For the figures on the top, $\tilde{q}$ is the screened value at the radius of the minimum of the KN potential, while for those on the bottom, $\tilde{q}$ is the screened value at the maximum. The EH effect at the maximum is more visible since it lays at a smaller value of $r$. The other parameters are $M=1\times {10}^4{M}_{\odot }$, $Q=0.9M$, $a=0.3M$, $\mu =1$, $L=0.8M$, $K=4M^2$, and $q=0.5$.

Figure 2

The potentials $V_{\mathrm{eff}}^{\pm }$ as a function of $r/M$ for the KN case (dashed line) and the EEH one (continuous line), with $\tilde{q}$ the screened value at the minimum of the KN potential. For the figure on the lhs, $M=2\times {10}^3{M}_{\odot }$, while for that on the rhs, $M=1\times {10}^3{M}_{\odot }$. Due to the screening effect on the charges, there is no maxima and minima. The other parameters are $Q=0.9M$, $a=0.3M$, $\mu =1$, $L=0.8M$, $K=4M^2$, and $q=0.5$.

Figure 3

The potentials $V_{\mathrm{eff}}^{\pm }$ as a function of $r/M$ for the KN case (dashed line) and the EEH one (continuous line), with $\tilde{q}$ the screened value at the minimum of the KN potential and for a bigger value of the test charge $q=5$. For the figure on the lhs, $M=4\times {10}^4{M}_{\odot }$, while for that on the rhs, $M=2\times {10}^4{M}_{\odot }$. The other parameters are $Q=0.9M$, $a=0.3M$, $\mu =1$, $L=0.8M$, and $K=4M^2$.

Figure 4

The shadow of the EEH rotating black hole (continuous line) and that of KN (dashed line), for different values of $Q$. As $Q$ increases, the EEH nonlinear effect becomes more visible. The parameters are $M=1\times {10}^4{M}_{\odot }$, $a=0.3M$, and ${\theta }_0=\pi /2$.

Figure 5

The shadow of the EEH rotating black hole (continuous line) and that of KN (dashed line), for different values of $M$. As $M$ decreases, the EEH nonlinear effect becomes more relevant. The parameters are $Q=0.8M$, $a=0.6M$, and ${\theta }_0=\pi /2$.

Figure 6

The shadow of the EEH rotating black hole (continuous line) and that of KN (dashed line), for nonvanishing values of $a$. Even for a small fixed value of $Q=5\times {10}^{-4}M$ the EEH nonlinear effect is still visible, due to the nonlinear contribution depending on the angular momentum $a$, (62). The mass corresponds to that for a stellar black hole, $M=5M_{\odot }$. The parameters satisfy the condition $a^2+{\tilde{Q}}^2\le M^2$. The angle ${\theta }_0=\pi /2$.