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Vacuum breakdown around a Kerr black hole surrounded by a magnetic field

C. Cherubini, R. Moradi, J. A. Rueda, R. Ruffini

Abstract

We present the invariant characterization of the region where vacuum breakdown into electron-positron ($e^+e^-$) pairs occurs due to an overcritical electric field, the dyadoregion, in the case of a Kerr black hole (BH) in the presence of an external, asymptotically uniform test magnetic field aligned with the BH rotation axis, using the Wald solution. We calculate the dyadoregion morphology, the electromagnetic energy available to the pairs, the pair-creation rate, the pair number density, the average energy per pair, and the pair energy density and pressure. These results provide initial conditions for simulating the subsequent dynamics of the pair-produced plasma and astrophysical applications in the context of high-energy transients involving BHs in strong electromagnetic fields.

Vacuum breakdown around a Kerr black hole surrounded by a magnetic field

Abstract

We present the invariant characterization of the region where vacuum breakdown into electron-positron () pairs occurs due to an overcritical electric field, the dyadoregion, in the case of a Kerr black hole (BH) in the presence of an external, asymptotically uniform test magnetic field aligned with the BH rotation axis, using the Wald solution. We calculate the dyadoregion morphology, the electromagnetic energy available to the pairs, the pair-creation rate, the pair number density, the average energy per pair, and the pair energy density and pressure. These results provide initial conditions for simulating the subsequent dynamics of the pair-produced plasma and astrophysical applications in the context of high-energy transients involving BHs in strong electromagnetic fields.

Paper Structure

This paper contains 9 sections, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Defining and characterizing the dyadoregion
  3. Minimum magnetic field for pair creation
  4. Electromagnetic energy in the dyadoregion
  5. Pair-plasma thermodynamical properties
  6. Discussion and conclusions
  7. Electromagnetic field of the Wald solution in the locally non-rotating frame
  8. Electromagnetic energy budget
  9. Parallel electric and magnetic fields of the Wald solution

Figures (3)

  • Figure 1: Contour of constant electric field intensity $\tilde{E} = E_c$ (solid blue curve), in the $x$-$z$ plane of Kerr-Schild, Cartesian coordinates. The black-filled disk is the Kerr BH horizon. In this example, the BH spin parameter is $\xi = 0.5$ and magnetic field strength $\beta = 200$, which corresponds to $B_0 = 8.8\times 10^{15}$ G. The dashed gray lines show the ends of the polar lobes which have a semi-aperture spherical polar angle $\theta_p \approx \arccos{(\sqrt{3}/3)}\approx 55^\circ$.
  • Figure 2: Dyadoregion electromagnetic energy given by Eq. (\ref{['eq:energyfinal']}), as a function of the magnetic field strength in the range $B_0=(50,400)B_c= (0.22,1.76)\times 10^{16}$ G, for selected values of the BH spin parameter, $a/M = 0.3$ (blue), $0.5$ (red), $0.7$ (green), $0.9$ (orange), and mass $M = 3 M_\odot$.
  • Figure 3: Upper left: plasma temperature $k_B T/(m_e c^2)$ around the Kerr BH of spin parameter $\xi = 0.5$ and magnetic field strength parameter $\beta = 400$. The dark-gray dashed contour is the dyadoregion radius given by the condition $\tilde{E}=E_c$. Upper right: plasma parameter $P/P_{\rm mag}$ for the same parameters as the upper right plot. Lower left: plasma temperature at the horizon, $k_B T_+/(m_e c^2)$, as a function of the BH spin $\xi=a/M$, for selected values of the magnetic field $\beta=50$ (blue), $200$ (red), and $400$ (green). Lower right: Plasma parameter at the horizon, $P_+/P_{\rm mag}$, for the same parameter as the lower left plot. The temperature and pressure of the plasma are given in Eqs. (\ref{['eq:tpairs']}) and (\ref{['eq:plasmaEOSP']}), respectively.