$e^{-}-e^{+}$ pair creation by vacuum polarization around electromagnetic black holes

The concept of “dyadotorus” was recently introduced to identify in the Kerr-Newman geometry the region where vacuum polarization processes may occur, leading to the creation of $e^{-}-e^{+}$ pairs. This concept generalizes the original concept of “dyadosphere” initially introduced for Reissner-Nordström geometries. The topology of the axially symmetric dyadotorus is studied for selected values of the electric field and its electromagnetic energy is estimated by using three different methods all giving the same result. It is shown by a specific example the difference between a dyadotorus and a dyadosphere. The comparison is made for a Kerr-Newman black hole with the same total mass energy and the same charge to mass ratio of a Reissner-Nordström black hole. It turns out that the Kerr-Newman black hole leads to larger values of the electromagnetic field and energy when compared to the electric field and energy of the Reissner-Nordström one. The significance of these theoretical results for the realistic description of the process of gravitational collapse leading to black hole formation as well as the energy source of gamma ray bursts are also discussed.

Figure 1

The behavior of the electromagnetic energy (7) in solar mass units is shown as a function of the mass parameter $\mu$ for selected values of the charge parameter $\lambda =[0.1,0.5,1]$, from bottom to top. The straight lines (dashed) correspond to the maximum energy extractable from a Reissner-Nordström black hole given by $Q^2/(2r_{+})$.

Figure 2

The total energy of pairs (9) is plotted as a function of the two mass and charge parameters $\mu$ and $\lambda$. The different curves correspond to selected values of the energy (in ergs). Only the solutions below the solid line are physically relevant. The configurations above the solid line correspond instead to unphysical solutions with $r_{\mathrm{ds}}<r_{+}$. The plot is reproduced from 33 with the kind permission of the authors.

Figure 3

The space of parameters $(\lambda ,\mu )$ is shown for different values of the rotation parameter $\alpha =a/M=[0,0.4,0.6,0.8,0.9,0.99]$ and fixed value of the polar angle $\theta =\pi /3$. The region below each curve represents the allowed region for the existence of the dyadoregion with fixed $\alpha$. The configurations above each line correspond to unphysical solutions where $r_{\pm }^d<r_{+}$ for the selected set of parameters. The value of the parameter $k$ has been set equal to one.

Figure 4

The projection of the dyadotorus on the $X-Z$ plane ($X=r\sin \theta$, $Z=r\cos \theta$ are Cartesian-like coordinates built up simply using the Boyer-Lindquist radial and angular coordinates) is shown for an extreme Kerr-Newman black hole with $\mu =10$, $\lambda =1.49\times {10}^{-4}$, and different values of the parameter $k$: (a) $k=0.9$ (orange), (b) $k=1.0$ (red), (c) $k=1.1$ (light blue), (d) $k=1.5$ (blue). The boundary of the dyadoregion becomes a toruslike surface for $k\approx 0.998$, according to Eq. (22). The black disk represents the black hole horizon.

Figure 5

The projections of the dyadotorus on the $X-Z$ plane corresponding to different values of the ratio $|\mathbf{E}|/E_c\equiv k$ are shown in (a) for $\mu =10$ and $\lambda =1.49\times {10}^{-4}$. The corresponding plot for the dyadosphere with the same mass energy and charge to mass ratio is shown in (b) for comparison.

Figure 6

Three-dimensional images of the dyadotorus are shown using Kerr-Schild coordinates. The parameter choice is the same as in Fig. 4. The surfaces have been cut in half for a better view of the interior. The horizon instead has been shown entirely (black region).

Figure 7

The dyadotorus is shown on an embedding diagram. The choice of the parameters is the same as in Fig. 4 as concerns (a), (c), and (d). In the case of (b) the value of the parameter $k$ has been changed to the critical value $k\approx 0.998$ in order to satisfy the condition (22) with the equality sign, so representing the limiting case in which the dyadotorus still appears as a toruslike surface [as in (c) and (d)] for the chosen values of parameters $\mu$ and $\alpha$. The surfaces have been cut in half for a better view of the interior, where the embedding of the horizon is also shown (the black shaded region is Euclidean, whereas the white regions are Minkowskian). Note that in this case the coordinates $(X,Y,Z)$ are given by Eq. (30).

Figure 8

The projections on the $X-Z$ plane of the embedding diagrams of Fig. 7 are shown. Dashed lines correspond to the Minkowskian part of the embedding of the outer horizon.