Influence of the vacuum polarization effect on the motion of charged particles in the magnetic field around a Schwarzschild black hole

Petar Pavlović, Andrey Saveliev, and Marko Sossich

Phys. Rev. D 100, 084033 – Published 15 October 2019

Abstract

The consequences of the vacuum polarization effect in magnetic fields around a Schwarzschild black hole on the motion of charged particles are investigated in this work. Using the weak electromagnetic field approximation, we discuss the nonminimal coupling between magnetic fields and gravity caused by the vacuum polarization and study the equations of motion for the case of a magnetic field configuration which asymptotically approaches a dipole magnetic field. It is shown that the presence of nonminimal coupling can significantly influence the motion of charged particles around black holes. In particular, the vacuum polarization effect, leading to strong amplification or suppression of the magnetic field strength around the event horizon (depending on the sign of the coupling parameter), can affect the scattering angle and minimal distance for the electrons moving in the gravitational field of the black hole as well as the dependence of these parameters on the asymptotic magnetic field strength, initial distance, and the black hole mass. It is further demonstrated that the nonminimal coupling between gravity and astrophysical magnetic fields, caused by the vacuum polarization, can cause significant changes of the parameter space corresponding to bound trajectories around the black hole. In certain cases, the bounded or unbounded character of a trajectory is determined solely by the presence of nonminimal coupling and its strength. These effects could in principle be used as observational signatures of the vacuum polarization effect and also to constrain the value of the coupling parameter.

Received 21 August 2019

DOI:https://doi.org/10.1103/PhysRevD.100.084033

Physics Subject Headings (PhySH)

Alternative gravity theories Classical black holes Fluids & classical fields in curved spacetime General relativity General relativity equations & solutions

Accretion disk & black-hole plasma

Astrophysical electromagnetic fields

Astrophysical & cosmological simulations

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Petar Pavlović^*

Department of Physics, Ramashna Mission Vivekananda Educational and Research Institute, Belur Math 711202, West Bengal, India

Andrey Saveliev ^†

Institute of Physics, Mathematics and Information Technology, Immanuel Kant Baltic Federal University, Ul. Aleksandra Nevskogo 14, 236016 Kaliningrad, Russia and Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, Leninskiye Gory 1-52, 119991 Moscow, Russia

Marko Sossich^‡

Faculty of Electrical Engineering and Computing, Department of Physics, University of Zagreb, Unska 3, 10 000 Zagreb, Croatia

^*petar.pavlovic@desy.de
^†andrey.saveliev@desy.de
^‡marko.sossich@fer.hr

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Vol. 100, Iss. 8 — 15 October 2019

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Images

Figure 1
Results for the trajectories of electrons for different magnetic field strengths/black hole masses. Top panel: Sample trajectories for different magnetic field strengths with $M = 10^{6} M_{⊙}$ (left legend at the bottom)/different Black hole masses with $B_{rad, 0} = 10^{- 17} T$ (right legend at the bottom), at $\tilde{q} = 0.01$ (left) and $\tilde{q} = 0.8$ (right), where the former depicts how the deflection angle $δ$ is measured. Bottom panel: Minimum distance from the center of the black hole (left) and deflection angle (right), where the vertical dashed lines indicate the values of $\tilde{q}$ used in the top panel. In all cases the default initial conditions $r_{0} = 20 \times (2 M)$ and $v_{r, 0} = - 0.9 c$ have been used. The black line indicates the fiducial scenario defined in Eq. (27).
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Figure 2
Results for the trajectories of electrons for different initial distances from the black hole. Top panel: Sample trajectories for different values of $r_{0}$ , at $\tilde{q} = - 0.5$ (left) and $\tilde{q} = 0.5$ (right). Bottom panel: Minimum distance from the center of the black hole (left) and deflection angle (right), where the vertical dashed lines indicate the values of $\tilde{q}$ used in the top panel. The black line indicates the fiducial scenario defined in Eq. (27). The dashed blue line indicates that the trajectory is bound, such that in that case a maximum distance (blue dotted line) can be shown.
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Figure 3
Results for the trajectories of electrons for different initial velocities. Top panel: Sample trajectories for different values of $v_{r, 0}$ , at $\tilde{q} = - 0.5$ (left) and $\tilde{q} = 0.5$ (right). Bottom panel: Minimum distance from the center of the black hole (left) and deflection angle (right), where the vertical dashed lines indicate the values of $\tilde{q}$ used in the top panel. The black line indicates the fiducial scenario defined in Eq. (27). The dashed lines indicates that the trajectory is bound, such that in that case a maximum distance (corresponding dotted line) can be shown.
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Figure 4
“Phase diagram” for the condition of the trajectory of a charged particle at a SBH in the $r_{0}$ vs $v_{r, 0}$ parameter space. Here, the gray region denotes an unbound state, i.e., $r \to \infty$ for $τ \to \infty$ , while the turquoise region shows the parameter combination for which $r$ does not go to infinity for $τ \to \infty$ , i.e., either a bound trajectory around the black hole or the case where the particle falls into it. For the black hole the default values of $M = 10^{6} M_{⊙}$ and $B_{rad, 100} = 10^{- 17} T$ have been used. Note that $v_{r, 0} < 0$ depicts the situation where initially the particle velocity points radially away from the black hole, while for $v_{r, 0} > 0$ it points toward it. The thick black line in each panel shows the “phase transition” of the case without a magnetic field. Top left: The case without a magnetic field. Top right: The case with a dipole magnetic field. Bottom: Phase diagram including the effect of vacuum polarization for $\tilde{q} = - 1.0$ (left) and $\tilde{q} = 1.0$ (right).
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