Recursive Penrose processes in electrically charged black hole spacetimes: Backreaction and energy extraction

Abstract

We study a recursive Penrose process and the energy extraction for the decay of electrically charged particles in a Reissner-Nordström black hole spacetime with anti-de Sitter (AdS) asymptotics, incorporating the backreaction on the black hole's mass and charge. A recursive process requires that the decay products are confined in a finite region so that the emitted particles bounce back for further decay. In AdS spacetimes, the confinement arises naturally. Outgoing particles encounter a turning point and are reflected. One may impose a mirror at finite radius, but in AdS, backreaction makes these two confinement methods equivalent. Let $Q_n$ be the black hole charge after $n$ decays, and define $n_{\rm c}$ as the index for which the black hole's charge is zero, $Q_{n_{\rm c}}=0$. For $n_{\rm c}$ integer the black hole's charge decreases and reaches exactly zero after a finite number of decays, terminating the process. However, the last particle turns back, and encountering zero charge, falls into the hole. The final state is a charged black hole whose charge equals the sum of the original black hole and the initial particle charges. For $n_{\rm c}$ noninteger, the black hole charge decreases and can be arbitrarily small, but is never zero. The last allowed decay occurs at $n=n_{c}^-$, where $n=n_{c}^-$ is the greatest integer less than $n_{\rm c}$. Any further decay invalidates the approximations, the particles would carry a charge comparable to the black hole mass, transforming the problem into a two-body problem. The would-be subsequent decay would violate cosmic censorship and the process terminates before any inconsistency arises. In the integer and noninteger cases, the system yields a finite energy gain. Backreaction ensures that the process extracts a finite amount of energy. No black hole bomb occurs, the system works at most as an energy factory.

Figures (13)

• Figure 1: It is plotted the function $g(r) \equiv - \frac{r^4}{M^2 \, l^2} + \left(\frac{E^2}{m^2} - 1\right) \frac{r^2}{M^2} + 2 \left(1 - \frac{EeQ}{m^2M}\right) \,\frac{r}{M} + \left(\frac{e^2}{m^2} - 1\right) \frac{Q^2}{M^2}$ defined from Eq. \ref{['eq:turning_points_penrose']} as a function of the radius$r$. The turning points correspond to $\dot r=0$, i.e., $g(r)=0$. They are the radius $r_{\rm v}$ (black dot) in the vicinity of the event horizon and of no interest to us, the internal turning point with radius $r_{\rm i}$ (first vertical black line), and the external turning point with radius $r_{\rm o}$ (second vertical black line). The quantities used are unitless, with the black hole mass $M$ serving as the rescaling quantity. The rescaled quantities are $\bar{Q}=\frac{Q}{M}$, $\bar{E} =\frac{E}{M}$, $\bar{e} =\frac{e}{M}$, $\bar{m} =\frac{m}{M}$, $\bar{l} =\frac{l}{M}$, and $\bar{r} =\frac{r}{M}$, The values used are $\bar{Q}=0.78$, $\bar{E} = 0.00018$, $\bar{e} = 0.00068$, $\bar{m} =0.0001$, and $\bar{l} =15.3$. For these values of the rescaled quantities, the event horizon is at radius $\bar{r}_+ =1.60283$, and the rescaled turning point radii have values $\bar{r}_{\rm v} = 2.29101$, $\bar{r}_{\rm i} = 5.9899$, and $\bar{r}_{\rm o} = 17.9198$. We are interested in the motion of particles between $\bar{r}_{\rm i}$ and $\bar{r}_{\rm o}$.
• Figure 2: Left: It is plotted the electric charge $e_{2n+1}$ of the odd particles as a function of the number $n$ of the decay. Right: It is plotted the electric charge of the even particles as a function of the number $n$ of the decay. Both plots are for a recursive Penrose process occurring in a Reissner-Nordström-AdS black hole spacetime when the index $n_{\rm c}$ is an integer, where $n_{\rm c}$ is the number for which the charge of the black hole reaches a zero value. The plots are for $0 \leq n \leq n_{\rm c}$. The quantities used are unitless, with the initial black hole mass $M_0$ serving as the rescaling quantity. So the rescaled electric charges $\bar{e}_{2n+1}=\frac{e_{2n+1}}{M_0}$ and $\bar{e}_{2n+2}=\frac{e_{2n+2}}{M_0}$ are the quantities plotted as a function of $n$. The other rescaled quantities are $\bar{Q}_0=\frac{Q_0}{M}$ and $\bar{e}_0=\frac{e_0}{M_0}$. The values used are $\bar{Q_0}=0.78$, $\bar{e_0}=0.00068$, and $\beta_2 = 1.223$. For these values one has that the value of $n$ when the black hole electric charge is zero is $n_{\rm c} = 35$, for which the process naturally comes to a halt.
• Figure 3: It is plotted the electric charge of the black hole $Q_n$, as a function of the number $n$ of the decay, for a recursive Penrose process occurring in a Reissner-Nordström-AdS black hole spacetime when the index $n_{\rm c}$ is an integer, where $n_{\rm c}$ is the number for which the charge of the black hole reaches a zero value. The plot is for $0 \leq n \leq n_{\rm c}$. The final black hole charge $Q_{\rm f}$ is also displayed. The quantities used are unitless, with the initial black hole mass $M_0$ serving as the rescaling quantity. So the rescaled black hole charge $\bar{Q}_n=\frac{Q_n}{M_0}$ is the quantity plotted as a function of $n$. The rescaled final black hole charge $\bar{Q}_{\rm f} =\frac{Q_{\rm f}}{M_0}$ is also displayed. The other rescaled quantity is $\bar{e}_0=\frac{e_0}{M_0}$. The values used are $\bar{Q_0}=0.78$, $\bar{e_0}=0.00068$, and $\beta_2 = 1.223$. For these values one has that the value of $n$ when the black hole electric charge is zero is $n_{\rm c} = 35$, for which the process naturally comes to a halt.
• Figure 4: Left: It is plotted the energy of odd particles $E_{2n+1}$ as a function of the number $n$ of the decay. Right: It is plotted the energy of even particles $E_{2n+2}$ as a function of the number $n$ of the decay. Both plots are for a recursive Penrose process occurring in a Reissner-Nordström-AdS black hole spacetime when the index $n_{\rm c}$ is an integer, where $n_{\rm c}$ is the number for which the charge of the black hole reaches a zero value. The plots are for $0 \leq n \leq n_{\rm c}$. The quantities used in the plots are unitless, with the initial black hole mass $M_0$ serving as the rescaling quantity. The rescaled quantities are then $\bar{E}_{2n+1}=\frac{E_{2n+1}}{M_0}$, $\bar{E}_{2n+2}=\frac{E_{2n+2}}{M_0}$, $\bar{Q}_0=\frac{Q_0}{M_0}$, $\bar{e}_0=\frac{e_0}{M_0}$, $\bar{l}=\frac{l}{M_0}$ and $\bar{r}_{\rm i}= \frac{r_{\rm i}}{M_0}$. The values used are $\bar{Q_0}=0.78$, $\bar{e_0}=0.00068$, $\bar{m}_0 = 0.0001$, $\bar{l} = 15.3$, $\bar{r}_{\rm i}= 5.9899$, $\alpha_2 = 0.3$, and $\beta_2 = 1.223$. For these values one has that the value of $n$ when the black hole electric charge is zero is $n_{\rm c} = 35$, for which the process naturally comes to a halt.
• Figure 5: It is plotted the mass of the black hole $M_n$ as a function of the number $n$ of the decay, for a recursive Penrose process occurring in a Reissner-Nordström-AdS black hole spacetime when the index $n_{\rm c}$ is an integer, where $n_{\rm c}$ is the number for which the charge of the black hole reaches a zero value. The plot is for $0 \leq n \leq n_{\rm c}$. The quantities used in the plot are unitless, with the initial black hole mass $M_0$ serving as the rescaling quantity. So, the rescaled quantities are $\bar{M}_n=\frac{M_n}{M_0}$, $\bar{E}_{2n+1}=\frac{E_{2n+1}}{M_0}$, $\bar{E}_{2n+2}=\frac{E_{2n+2}}{M_0}$, $\bar{Q}_0=\frac{Q_0}{M_0}$, $\bar{e}_0=\frac{e_0}{M_0}$, $\bar{l}=\frac{l}{M_0}$ and $\bar{r}_{\rm i}= \frac{r_{\rm i}}{M_0}$. The values used are $\bar{Q_0}=0.78$, $\bar{e_0}=0.00068$, $\bar{m}_0 = 0.0001$, $\bar{l} = 15.3$, $\bar{r}_{\rm i}= 5.9899$, $\alpha_2 = 0.3$, and $\beta_2 = 1.223$. For these values one has that the value of $n$ when the black hole electric charge is zero is $n_{\rm c} = 35$, for which the process naturally comes to a halt.