Confined Penrose process with charged particles

We show that kinematics of charged particles allows us to model the growth of particles’ energy by consecutive particle splits, once a spherical mirror as a perfectly reflective boundary is placed outside a charged black hole. We consider a charged version of the Penrose process, in which a charged particle decays into two fragments, one of them has negative energy and the other has positive energy that is larger than that of the parent particle. The confinement system with the mirror makes the particles’ energy amplified each time a split of the parent particle occurs. Thus, the energy is a monotonically increasing function of time. However, the energy does not increase unboundedly, but rather asymptotes to a certain finite value, implying no instability of the system in this respect.

Figure 1

Schematic picture for charged Penrose process. The positively charged particle, particle 0, splits into two oppositely charged particles, particles 1 and 2, within a generalized ergoregion (gray region) of the BH at the center (black region).

Figure 2

Schematic figure of $g_i$. The horizontal dotted lines denote values of the energy for particle $i$. (a) $g_1(x)$ for particle 1 with negative charge. Fragment’s energy must be negative if the split occurs inside the generalized ergoregion (shaded range), $x_H<x_s<x_E$. Since particle 1 moves in the range $x\le x_s$, it cannot escape to infinity, but falls into the BH. (b) $g_2$ for particle 2 with positive charge. Particle’s energy must be positive and escapes outwardly.

Figure 3

Minimum of $q_{\min }$ as a function of the background charge $Q$. The value of a negative charge $q_1$ that is supposed to be dropped into BHs must be below these lines in order to obtain negative $E_1$.

Figure 4

(a) The energy amplification factor $\eta$ for the first 20 decays under the parameters of $\mathrm{\Delta }=0.5$, $\delta =0.5$, $E_0=1.1$, $Q=0.9$, and ${\alpha }_2=0.5$, 0.7, and 0.9. (b) Same as (a) but for $\mathrm{\Delta }=0.5$, $\delta =0.5$, $E_0=1.1$, ${\alpha }_2=0.5$ and $Q=0.1$, 0.5, and 0.9. (c) Same as (a) but for $\delta =0.5$, $E_0=1.1$, $Q=0.5$ and $\mathrm{\Delta }=0.1$, 1, and 10. (d) Same as (a) but for $\mathrm{\Delta }=0.5$, $E_0=1.1$, $Q=0.5$ and $\delta =0.1$, 0.5, and 1.

Figure 5

Potentials of outgoing positive energy particles after the $n$-h decay. $n=0$ denotes the potential for particle 0. Particles splits at the same radius $x_s$, denoted as a vertical solid line. (a) Potentials under the parameter set of $Q=0.9$, $E_0=1.2$, $\mathrm{\Delta }=0.1$, $\delta =0.1$, ${\alpha }_2=0.5$ for (a) $n=0$, 1, 2, 3, 5 and (b) $n=5$, 10, 15.