# Energy extraction in electrostatic extreme binary black holes

A<EMAIL_ADDRESS>Nora Breton<EMAIL_ADDRESS>and I. Cabrera-Munguia<EMAIL_ADDRESS>1 Departamento de
Física, Centro de Investigación y de Estudios Avanzados del I. P. N. Apdo.
Postal 14-740, Mexico City, Mexico
2Departamento de Física y Matemáticas, Universidad Autónoma de Ciudad Juárez,
32310 Ciudad Juárez, Chihuahua, México

###### Abstract

Relying on the Penrose process mechanism, we study the possibility of energy
extraction from a binary system composed of two extreme electrostatic black
holes (BHs) oppositely charged, separated by a strut described by Bonnor’s
metric (BM). We determined and plotted the generalized ergosphere that
surrounds only one of the BH. We demonstrate the existence of non closed
orbits of negative energy outside the event horizon; these orbits allow the
possibility of energy extraction by particle disintegration from a system
described by the BM. Besides we prove that the extraction process can occur
when a charged test particle and the BH have opposite charges; also, we
analyzed the efficiency of the process.

##  I Introduction

The Kerr metric is a stationary solution of Einstein’s field equation that
gives the more general description of a rotating BH Kerr1963 ; Visser2007 ;
Teukolsky2015 ; this metric possesses an interesting region called the
ergosphere, delimited by the stationary limit surface and the outer event
horizon. Within the ergosphere the timelike Killing vector
$(\partial_{t})^{\mu}$ becomes spacelike, this special feature allows that
particles inside the ergosphere can have negative energy. However, the
particle can yet avoid enter the event horizon and can escape back to
infinity.

The Penrose process is a mechanism proposed by Penrose and Floyd PenFloyd1971
for extracting energy from a rotating BH taking advantage of the fact that
test particles inside the ergosphere can have negative energy states. It
consists of a particle that reaches the ergosphere and at some point it
disintegrates into two fragments, one of the fragments is trapped within the
ergosphere with negative energy and the other one escapes back to infinity
with more energy than the one of the incident particle; by conservation of
energy then rotating energy through angular momentum has been extracted.
However, the ergosphere is a characteristic region of stationary solutions,
then by means of the Penrose process it is, in principle, impossible to
extract energy from a static BH. However, for electrostatic BHs it is possible
to define a region where charged test particles can have negative energy
DenardoRuffini1972 ; Dadich1980 ; Christo1971 .

Although the Penrose process applied to a single particle might seem
unfeasible to carry out, it is possible to establish relations between the
Penrose process and some astrophysical observations. For instance, the
collisional Penrose process might eventually eliminate dark energy particles
in the vicinity of a supermassive BH once the multiple particles that scatter
inside the ergosphere achieve an arbitrarily high center of mass energy
Schnittman2018 . On the other hand, the influence of an external magnetic
field surrounding a rotating BH can form accretion disks of charged ionized
matter Kolos2021  and this may be related to high-frequency oscillations
noticed in microquasars, galactic nuclei, or even the magnetic Penrose process
itself Dadhich2018  where ultra high-energy particles around rotating
magnetized BHs are created Kolos2017 ; Kolos2015 . Moreover, the radiative
Penrose process is connected to synchrotron radiation of charged particles
moving within the ergosphere of a magnetized BH, where such a process
considers a special type of radiated photons having negative energy relative
to a distant observer Stuchlik2021 ; Kolos2018 . In addition, recent numerical
studies on plasmas and jets suggest the main role of negative energy particles
and the Penrose process in the total flux coming from the BH jets Parfrey2019
. Finally, the electromagnetic Penrose process Bhat1985 ; Tursunov2021 ;
Parthasarathy1986 ; Wagh1985 ; Wagh1989 ; Nucamendi2022  allows events of high
energy emission in contrast with the efficiency of 20.7$\%$ of the well-known
Penrose process due only to the rotation of a Kerr BH.

The present paper aims to investigate energy extraction via the Penrose
process in a BH binary system. In Richartz2021  the energy extraction is
analyzed for the Majumdar-Papapetrou (MP) BH binary metric Majumdar ;
Papapetrou ; Hartle1972 , which is an exact solution of the Einstein-Maxwell
equations describing two static charged BHs whose charges equal their masses,
$|Q_{i}|=M_{i}$, $i=1,2$. Therefore, in the MP binary metric, the BHs remain
in equilibrium since their mutual gravitational attraction compensates their
mutual electric repulsion no matter how far apart the sources are. Our purpose
is to use the method developed in DenardoRuffini1972 ; Christo1971  to
determine negative energy states for charged test particles and prove that by
means of the Penrose process is possible the energy extraction in a binary
system composed of two electrostatic oppositely charged BHs described by
Bonnor’s metric (BM) Bonnor1979 ; Cabrera2011 , where now the gravitational
attraction does not counterbalance the electrical (attractive) force, and
therefore, a conical singularity arises BachW ; Israel  in between the
sources. In particular we study the generalized ergosphere, its dependence on
charged test particle and how the energy extraction efficiency is affected by
the presence of a BH companion with opposite charge.

Our paper is organized as follows. In Sec. II.1, we introduce the spacetime
described by the BM and derive the motion equations for charged massive test
particles in a static and axisymmetric spacetime. In Sec. II.2 the generalized
ergosphere and the existence of negative orbits is analyzed. In Sec. III the
Penrose process is described and the conservation equations are presented
(mass, charge, energy, linear momentum and angular momentum). In Sec. IV the
constrictions over the parameters and the maximum efficiency of the process
are presented and in the last section conclusions are given.

##  II Bonnor’s Binary BH

The stationary axisymmetric line element in Weyl’s cylindrical coordinates
$(t,\rho,z,\phi)$ is the Papapetrou metric Papapetrou  given by

$ds^{2}=f^{-1}\big{[}e^{2\gamma}(d\rho^{2}+dz^{2})+\rho^{2}d\phi^{2}\big{]}-f(dt-\omega
d\phi)^{2},$ (1)

where the condition $\omega=0$ defines the double Reissner-Nordström (DRN)
Manko2007  solution of Einstein-Maxwell equations that describes a two-body
system composed of two electrostatic BHs; to keep the two BHs apart, a line
source should be introduced, the strut BachW ; Israel . As a consequence there
arise an angle deficit in $\phi$ that is a conical singularity. The
interacting force associated to the strut between the BHs is given by
Manko2007

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{M_{1}M_{2}-(Q_{1}-\delta)(Q_{2}+\delta)}{R^{2}-(M_{1}+M_{2})^{2}+(Q_{1}+Q_{2})^{2}},$
$\displaystyle\delta$
$\displaystyle=\frac{M_{2}Q_{1}-M_{1}Q_{2}}{R+M_{1}+M_{2}},$ (2)

where $M_{i}$ and $Q_{i}$, for $i=1,2,$ are the masses and electric charges,
respectively, while $R$ defines an arbitrary separation distance among the BH
centers. Regarding the last point, the half-length BH horizons $\sigma_{i}$
assume the next simple formulas VCH2002 ; Alekseev2007

$\displaystyle\sigma_{i}$
$\displaystyle=\sqrt{M_{i}^{2}-Q_{i}^{2}-2(-1)^{i}Q_{i}\frac{M_{2}Q_{1}-M_{1}Q_{2}}{R+M_{1}+M_{2}}},$
$\displaystyle i$ $\displaystyle=1,2,$ (3)

and thus the DRN spacetime contains a total, of five independent parameters
within the set $\\{M_{i},Q_{i},R\\}$. The reader should note that if one
exchanges the physical properties of the sources; i.e., $M_{1}\leftrightarrow
M_{2}$ and $Q_{1}\leftrightarrow Q_{2}$, where $\delta\to-\delta$, the
interaction force remains invariant. Also, it is possible to observe that
$\sigma_{1}\leftrightarrow\sigma_{2}$ under this physical interchange.

The extremal condition $\sigma_{i}=0$ leads to two possible scenarios. The
first one is the MP case, where $|Q_{i}|=M_{i}$, and there is no need of
introducing the strut since the gravitational attraction balances the electric
repulsion and the net force is null, $\mathcal{F}=0$.

The second scenario corresponds to the Bonnor’s metric (BM); in this case the
electric charges, $Q_{i}$, are opposite in sign and related to the masses and
distance according to Cabrera2011

$\displaystyle
Q_{1}=M_{1}\sqrt{\frac{(R+M_{2})^{2}-M_{1}^{2}}{(R-M_{2})^{2}-M_{1}^{2}}},$
(4) $\displaystyle
Q_{2}=-M_{2}\sqrt{\frac{(R+M_{1})^{2}-M_{2}^{2}}{(R-M_{1})^{2}-M_{2}^{2}}},$

where the electric charges and masses fulfill the condition $|Q_{i}|>M_{i}$,
$i=1,2$. The metric functions in the BM are given by

$\displaystyle f$
$\displaystyle=\bigg{(}\frac{1}{1+g_{1}}+\frac{1}{1+g_{2}}-1\bigg{)}^{2},\qquad
e^{2\gamma}=\bigg{(}\frac{(1-g_{1}g_{2})g_{+}g_{-}}{4d^{2}}\bigg{)}^{4},$
$\displaystyle g_{1}$
$\displaystyle=\frac{(1+d)M_{1}-(1-d)M_{2}}{(1+d)r_{-}-(1-d)r_{+}},\quad
g_{2}=\frac{(1+d)M_{2}-(1-d)M_{1}}{(1+d)r_{+}-(1-d)r_{-}},\qquad
d=\sqrt{\frac{R^{2}-(M_{1}-M_{2})^{2}}{R^{2}-(M_{1}+M_{2})^{2}}}$
$\displaystyle g_{+}$ $\displaystyle=1+d-(1-d)\frac{r_{+}}{r_{-}},\qquad
g_{-}=1+d-(1-d)\frac{r_{-}}{r_{+}},\qquad r_{\pm}=\sqrt{\rho^{2}+(z\pm
R/2)^{2}}.$ (5)

In Weyl’s cylindrical coordinates the BHs and their horizons are represented
by points at the $z$-axis $(\rho=0,z=\pm R/2)$. The independent parameters of
the metric functions in the BM (5) are $M_{1}$, $M_{2}$, and $R$, while the
electric potential $A_{\mu}$ is given by

$A_{\mu}=(A_{t},A_{\rho},A_{\phi},A_{z})=\left(\frac{1}{1+g_{1}}-\frac{1}{1+g_{2}},0,0,0\right).$
(6)

Contrary to the MP scenario, in the BM the electrical force is attractive
rather than repulsive, which means that the gravitational attraction will
never balance the electromagnetic force. After replacing (4) in (2) it is
possible to show that the interaction force associated to the strut between
the BHs is given by Cabrera2014

$\mathcal{F}=\frac{2M_{1}M_{2}}{R^{2}-\left(M_{1}+M_{2}\right)^{2}}\left(1+\frac{2M_{1}M_{2}}{R^{2}-\left(M_{1}+M_{2}\right)^{2}}\right),$
(7)

and because the masses and distance satisfy the inequality $(M_{1}+M_{2})<R$,
only attractive scenarios will be allowed corresponding to a positive force of
the strut, $\mathcal{F}>0$. On the other hand, since the BM is static, the
spacetime does not possess an ergosphere in the usual sense where the timelike
Killing vector becomes spacelike, and consequently the energy associated to
geodesic motion of neutral particles is always positive, i.e., the energy
extraction is not possible. However, charged particles can interact with
charged BHs via Lorentz forces. According to the ideas proposed by Denardo-
Ruffini DenardoRuffini1972  and Dadhich Dadich1980  for a single charged BH as
well as Sanches-Richartz for a BH binary Richartz2021 , we can define a
particular region where negative energy trajectories and energy extraction
are, in principle, possible. In what follows we study the motion of charged
particles in the BH binary BM and derive the energy extraction.

###  II.1 Motion of charged particles

The motion equations for a test particle with charge-mass ratio $\mu$ in a
spacetime characterized by the metric $g_{\mu\nu}$ and interacting with the
electric potential $A_{\mu}$, can be obtained from the Euler-Lagrange
equations with the Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\mu
A_{\alpha}\dot{x}^{\alpha},$ (8)

where the dot means derivative with respect to an affine parameter. In terms
of the metric coefficients in Eq. (1), the Lagrangian is

$\mathcal{L}=\frac{1}{2}\left(f^{-1}\big{(}e^{2\gamma}(\dot{\rho}^{2}+\dot{z}^{2})+\rho^{2}\dot{\phi}^{2}\big{)}-f\dot{t}^{2}\right)+\mu
A_{t}\dot{t},$ (9)

which does not depend explicitly on the coordinates $t$ and $\phi$. Then we
can identify two motion constants of the test particle: its energy $E$ and
angular momentum $L$ per unit mass, as measured by observers at infinity,
given by

$\displaystyle E$
$\displaystyle=-\frac{\partial\mathcal{L}}{\partial\dot{t}}=f\dot{t}-\mu
A_{t},$ (10) $\displaystyle L$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=f^{-1}\rho^{2}\dot{\phi},$

where after solving for $\dot{\phi}$ and $\dot{t}$, and substituting the
result into Eq. (9), we obtain the motion equations for $\rho$ and $z$
coordinates, via the Euler-Lagrange equations, as

$\displaystyle\ddot{\rho}-\frac{fe^{-2\gamma}}{2}\frac{\partial}{\partial\rho}\left(\frac{(E+\mu
A_{t})^{2}}{f}-\frac{fL^{2}}{\rho^{2}}\right)+\left(\frac{\dot{z}^{2}-\dot{\rho}^{2}}{2}\right)\frac{\partial}{\partial\rho}\Big{(}\ln\left(fe^{-2\gamma}\right)\Big{)}-\dot{\rho}\dot{z}\frac{\partial}{\partial
z}\bigg{(}\ln\left(fe^{-2\gamma}\right)\bigg{)}=0,$ (11)
$\displaystyle\ddot{z}-\frac{fe^{-2\gamma}}{2}\frac{\partial}{\partial
z}\left(\frac{(E+\mu
A_{t})^{2}}{f}-\frac{fL^{2}}{\rho^{2}}\right)+\left(\frac{\dot{\rho}^{2}-\dot{z}^{2}}{2}\right)\frac{\partial}{\partial
z}\bigg{(}\ln\left(fe^{-2\gamma}\right)\bigg{)}-\dot{\rho}\dot{z}\frac{\partial}{\partial\rho}\bigg{(}\ln\left(fe^{-2\gamma}\right)\bigg{)}=0.$

It is worth mentioning that the motion equations for $\rho$ and $z$, in terms
of the metric (1) are given in Dubeibe2016  for neutral test particles. On the
other hand, an explicit expression for the energy $E$ is obtained by plugging
the constants of motion (10) into the normalization condition of the four
velocity $\dot{x}^{\mu}\dot{x}_{\mu}=-1$,

$E=-\mu
A_{t}\pm\sqrt{e^{2\gamma}\left(\dot{\rho}^{2}+\dot{z}^{2}\right)+\frac{L^{2}f^{2}}{\rho^{2}}+f},$
(12)

where the positive root is taken for a positive energy at infinity when
$\mu=0$ Christo1971 . After some straightforward algebra, Eq. (12) can be
expressed as

$\dot{\rho}^{2}+\dot{z}^{2}=E_{\rm{eff}}^{2}(\rho,z)-V_{\rm{eff}}(\rho,z),$
(13)

where

$\displaystyle E_{\rm{eff}}^{2}(\rho,z)$ $\displaystyle=\frac{\left(E+\mu
A_{t}\right)^{2}}{e^{2\gamma}},$ $\displaystyle V_{\rm{eff}}(\rho,z)$
$\displaystyle=\frac{1}{e^{2\gamma}}\left(\frac{L^{2}f^{2}}{\rho^{2}}+f\right).$
(14)

These expressions are subject to the constraints:

$E_{\rm{eff}}(\rho,z)\geq 0,\qquad E_{\rm{eff}}^{2}(\rho,z)\geq
V_{\rm{eff}}(\rho,z).$ (15)

The system of equations (11) fully describes the motion of a charged test
particle in a static and axisymmetric spacetime; this set of equations can be
numerically solved once the appropriate initial conditions are chosen. In
order to solve this system of equations, the values for the energy $E$,
angular momentum $L$, and initial values for $\rho$, $z$, and $\dot{z}$ should
be given; the initial value for $\dot{\rho}$ is determined from Eq. (12). With
these initial values Eqs. (11) can be solved for $\rho(\lambda)$ and
$z(\lambda)$ with $\lambda$ being the affine parameter; the full description
of the motion of a test particle is obtained when the set of Eqs. (10) is
solved using $\rho(\lambda)$ and $z(\lambda)$.

###  II.2 Generalized ergosphere

From the expression (12), we know that the energy is determined by the angular
momentum $L$, charge-mass ratio $\mu$, electric potential $A_{t}$ and the
coordinates and velocities: $\rho$, $z$, $\dot{\rho}$ and $\dot{z}$ at a
specific time. In order to know if there are test particles with negative
energy we consider the minimum possible energy with a fixed charge $\mu$ and
position $(\rho,z)$; this is the energy associated with test particles at
rest. Replacing $\dot{\rho}=0$, $\dot{z}=0$, and $L=0$ in (12), we get

$E_{\rm{min}}=-\mu A_{t}+\sqrt{f},$ (16)

since $\sqrt{f}\geq 0$, then the existence of test particles with negative
energies is defined by the term $-\mu A_{t}$. In order that $E_{\rm{min}}<0$,
$\mu A_{t}>0$. Hence we need to determine the sign of $A_{t}$ given by,

$A_{t}=\frac{1}{1+g_{1}}-\frac{1}{1+g_{2}}.$ (17)

Using the explicit form of $g_{1}$ and $g_{2}$ in Eq. (5) one can easily
verify that the sign of $A_{t}$ is different if $(\rho,z)$ are inside or
outside the circle delimited by

$\bar{\rho}^{2}+\bigg{(}\bar{z}+\frac{1}{2}\left(\frac{1+M_{R}^{2}}{1-M_{R}^{2}}\right)\bigg{)}^{2}=\left(\frac{M_{R}}{1-M_{R}^{2}}\right)^{2},$
(18)

where $\bar{\rho}=\rho/R$, $\bar{z}=z/R$ and $M_{R}=M_{2}/M_{1}$. Note that
this region encircles the smaller mass. We distinguish two scenarios:
i) For $M_{1}>M_{2}$ the circle represented by (18) surrounds the BH with mass
$M_{2}$ and electric charge $Q_{2}<0$. Whether $(\rho,z)$ is located inside
(outside) this circle, then, it follows that $A_{t}>0(A_{t}<0)$. ii) For
$M_{1}<M_{2}$, the circle depicted by (18) surrounds the BH with mass $M_{1}$
and electric charge $Q_{1}>0$. If $(\rho,z)$ is localized inside (outside) the
circular region, then $A_{t}>0(A_{t}<0)$.

It should be stressed that the region (18) never includes both sources. For
the case $M_{1}>M_{2}$, the circle remains on the semi-plane $z<0$ while for
$M_{2}>M_{1}$ it is located on the semi-plane $z>0$. Hence, for a given value
$\mu>0$, the condition $\mu A_{t}>0$ cannot be simultaneously fulfilled; in
this case the ergosphere encircles the lower BH with electric charge
$Q_{2}<0$. While if $\mu<0$, the condition $\mu A_{t}>0$ is fulfilled in the
region encircling the upper BH with electric charge $Q_{1}>0$.

Figure 1:  The sign of the electric potential, $A_{t}$ depends on the chosen
$(\rho,z)$; (a) If $M_{1}>M_{2}$, $0<M_{R}<1$ then $A_{t}$ is positive inside
the circle that surrounds the bottom source, and negative outside. (b) On the
other hand, if $M_{1}<M_{2}$, $M_{R}>1$ then $A_{t}$ is negative inside the
circle that surrounds the upper source, and positive outside.

Figure 2:  It is shown the $\phi=0$ (meridional) plane section of the
generalized ergosphere (shaded surface) for different values of $\mu$ and
fixed values of $M_{1}=0.5$, $M_{2}=0.2$, and $R=1$. Recalling that negative
energy states are achieved via the condition $\mu A_{t}>0$, where $A_{t}$
depends on $M_{1}$, $M_{2}$, and $R$. The black circumference represents Eq.
(18) and the dots symbolize the BHs. The shadow regions in (a)-(e) represent
the ergosphere for $\mu<0$, where the condition $\mu A_{t}>0$ is satisfied
outside the black circumference, thus, the ergosphere surrounds only the upper
positively charged source and excludes the region delimited by the black
circumference. On the other hand, the shadow regions in (f)-(h) represent the
ergosphere for $\mu>0$ that is surrounding the bottom (negatively charged)
source, where now $\mu A_{t}>0$ is fulfilled inside the black circumference.
For an arbitrarily large value of $|\mu|$ the ergosphere tends to occupy the
space inside the black circumference. The reader must be aware that the
ergosphere always surrounds the BH carrying on an opposite charge to the test
particle.

Figure 3:  Energy levels of the BM generalized ergosphere are illustrated with
$R=1$, $M_{1}=0.5$, $M_{2}=0.2$ and selected values of $\mu$. The color bar
represents $E_{min}$. The dots indicate the location of the BHs. The
horizontal and vertical axes are $\rho$ and $z$, respectively.

When the values of $M_{1}$ and $M_{2}$ are exchanged the graphic in Fig. 1, is
rotated about the $z=0$ plane; then, without loss of generality, we shall
consider only the case $M_{1}>M_{2}$.

In order for $E_{min}<0$ the sign of $\mu$ should be the same than the sign of
$A_{t}$. In contrast with the MP metric, analyzed in Richartz2021 , where the
two BH are positively charged and then energy can be extracted only by
negatively charged test particles, in the BM energy can be extracted by
negative or positive charged test particles.

The condition $E_{min}<0$, Eq. (16) can be written as

$\bar{\rho}^{2}+\bar{z}^{2}+\frac{2\mu\left(\bar{M}_{1}\bar{r}_{+}-\bar{M}_{2}\bar{r}_{-}\right)\bar{d}}{\bar{d}^{2}-1}+\frac{\bar{d}^{2}+1}{\bar{d}^{2}-1}\bar{r}_{+}\bar{r}_{-}<\frac{1}{4},$
(19)

where $\bar{r}_{\pm}=r_{\pm}/R$, $\bar{M}_{1,2}=M_{1,2}/R$ and
$\bar{d}=\sqrt{\left(1-(\bar{M}_{1}-\bar{M}_{2})^{2}\right)/\left(1-(\bar{M}_{1}+\bar{M}_{2})^{2}\right)}$
are dimensionless quantities restricted by
$\left(\bar{M}_{1}+\bar{M}_{2}\right)<1$. Recall that $(\bar{\rho},\bar{z})$
are the coordinates of the initial particle, then (19) restricts the position
of the initially at rest particle. Note that this reparametrization is
equivalent to fix $R=1$. The inequality (19) determines the generalized
ergosphere on the BM. The sketches of the ergosphere for $\mu>0$ and $\mu<0$
are shown in Fig. 2; the charged test particle can have negative energy in the
region encircling the BH with opposite charge. For particles with $\mu<0$ the
generalized ergosphere surrounds the upper source Fig. 2.a-2.e. If $|\mu|$ is
sufficiently large the generalized ergosphere would surround the region
delimited by (18) without including it. On the other hand, for particles
$\mu>0$ the generalized ergosphere surrounds the bottom source, Fig. 2.f-2.h;
and if $|\mu|$ is sufficiently large the generalized ergosphere is delimited
by (18).

From Fig. 2 we see that the generalized ergosphere depends on the parameters
$\mu$, $M_{1}$ and $M_{2}$, i.e., not only on the spacetime geometry but also
on the charge of the test particle via $\mu$. The minimum energy per unit mass
at a given point inside the ergosphere also depends on $\mu$ and one the
masses of the BHs. To illustrate this, the energy levels of the ergosphere are
shown in Fig. 3 for different values of $M_{1}$, $M_{2}$ for charge-mass ratio
$\mu$ positive and negative. One can see that in general the shape of the
ergosphere does not change and the magnitude of the energy levels is larger as
the test particle gets closer to the BH oppositely charged. The ergospheres
then surround one of the BHs, the one with charge opposite to the test
particle and then never merge to include both BHs, in this sense resembling
the double–Kerr for counterrotating BHs Herdeiro2009 .

###  II.3 Negative energy trajectories

The negative energy trajectories of the charged test particles in the BM are
confined inside the ergosphere defined by (19); in the case illustrated the
particle falls into one of the BHs.

We shall describe two classes of orbits around the BH binary described by BM.
The first class of trajectories are orbits with angular momentum $L=0$; this
condition implies that the trajectories are confined to a meridional plane,
i.e., a plane with $\phi$ constant; for simplicity we take $\phi=0$. Fixing
the parameters $M_{1}$, $M_{2}$, $\mu$ and $R$, and given a set of initial
conditions $\rho(0)$, $z(0)$ and $\dot{z}(0)$ we can calculate
$\dot{\rho}(\lambda)$ and solve the motion equations (11). In Fig. 4 we show
some examples of trajectories of particles with negative energy that are
confined to the generalized ergosphere. Fig. 4.a exhibits three trajectories
for $\mu=-5$ and Fig. 4.b exhibits three trajectories for $\mu=20$. Note in
these trajectories one of them ends at one of the BHs.

Figure 4:  (a) Examples of trajectories for particle with charge-mass ratio
$\mu=-5$; the other parameters are specified for each trajectory. $T_{(0)}$:
$E=-0.1$, $\rho(0)=-1/2$, $z(0)=1$, $\dot{\rho}(0)=-0.73758$ and
$\dot{z}(0)=0$. $T_{(1)}$: $E=-0.00001$, $\rho(0)=0$, $z(0)=0$,
$\dot{\rho}(0)=1.32396$ and $\dot{z}(0)=0$. $T_{(2)}$: $E=-0.0001$,
$\rho(0)=1/2$, $z(0)=3/2$, $\dot{\rho}(0)=0.250857$ and $\dot{z}(0)=0$. (b)
Examples of trajectories for particle with charge-mass ratio $\mu=20$; the
other parameters are specified for each trajectory. $T_{(0)}$: $E=-0.002$,
$\rho(0)=0$, $z(0)=-1$, $\dot{\rho}(0)=0.998144$ and $\dot{z}(0)=0$.
$T_{(1)}$: $E=-0.002$, $\rho(0)=-1/4$, $z(0)=-1/4$, $\dot{\rho}(0)=0.423686$
and $\dot{z}(0)=0$. $T_{(2)}$: $E=-0.002$, $\rho(0)=1/4$, $z(0)=-1/4$,
$\dot{\rho}(0)=0.423686$ and $\dot{z}(0)=0$. The parameters are $M_{1}=0.4$,
$M_{2}=0.2$, and $R=1$.

The second class of orbits is a projection of geodesics in the plane $z=0$; in
general, a particle that initially is located at $z=0$ will not remain in this
plane. Since the generalized ergosphere for positive $\mu$ is always contained
in the circle depicted by Eq. (18) (see Figs. 2.f-2.h), then the ergosphere
does not reach the plane $z=0$; then we only show the movement for negative
$\mu$. Setting $z=0$ in motion Eqs. (11) and energy condition Eq. (13), the
motion in the $z=0$ plane is constrained to the region
$E^{2}_{\rm{eff}}(\rho,0)\geq V_{\rm{eff}}(\rho,0)$ where the equality is
satisfied for circular orbits. Fig. 5 shows the energy and effective potential
for BM at the $z=0$ plane. The corresponding trajectory is also shown in Fig.
5.

Figure 5: Left: Effective energy (black curve) and effective potential (red
dashed curve) for $L=0.71$, $E=-0.15$, $M_{1}=0.5$, $M_{2}=0.2$ and $R=1$.
Right: example of a trajectory of negative energy at the $z=0$ plane with
initial conditions $\rho(0)=2.19858$, $\phi(0)=0$ and $\dot{\rho}(0)=1.22395$.
The blue circle corresponds to the generalized ergosphere.

##  III Penrose process

Now we investigate the possibility of energy extraction from the binary BM.
Considering the Penrose process developed in PenFloyd1971 , extended for RN
BHs DenardoRuffini1972  and lately applied to MP binary BH Richartz2021 , we
addressed the Penrose process for the BM. It consists in sending a charged
particle towards the binary BH; at some point, once inside the generalized
ergosphere, the particle breaks up into two fragments, one of them escapes to
infinity with more energy than the initial one, while the other remains inside
the ergosphere until it falls into one of the BHs. We denote the initial
particle with subscript $0$, the particle that falls into the BH with
subscript $1$ and the particle that escapes with subscript $2$. We consider
that the incident particle follows the trajectory $T_{(0)}$, which starts
outside the ergosphere and ends inside it, at the break-up point
$(\rho_{*},\phi_{*},z_{*})$. From the break-up point emerge two particles with
trajectories labeled $T_{(1)}$ for the particle with negative energy
($E_{(1)}<0$) that remains inside the ergosphere; and $T_{(2)}$ that
corresponds to the particle escaping to infinity. The $T_{(i)}$ trajectories
are timelike paths $x^{\mu}_{i}(\lambda)$ parametrized by the proper time
$\lambda$. We denote with $m_{(i)}$, $\mu_{(i)}$, $E_{(i)}$, $L_{(i)}$, and
$P^{\mu}_{(i)}$ to the mass, charge-mass ratio, energy per unit mass, angular
momentum per unit mass (with respect to the $z$ axis) and four momentum of the
$i$-particle, respectively. The quantities that characterize each particle
should fulfill charge, energy and momentum conservation equations. From the
charge conservation we have,

$\mu_{(0)}m_{(0)}=\mu_{(1)}m_{(1)}+\mu_{(2)}m_{(2)}.$ (20)

On the other hand, if we consider that at the break-up point the four momentum
is conserved,

$P^{\nu}_{(0)}=P^{\nu}_{(1)}+P^{\nu}_{(2)}.$ (21)

From the temporal component in Eq. (21) we have the conservation of the total
energy, i.e.

$E_{(0)}m_{(0)}=E_{(1)}m_{(1)}+E_{(2)}m_{(2)},$ (22)

while the spatial components of Eq. (21) are the conservations of linear
momenta in each component, i.e.,

$\displaystyle
m_{(0)}\dot{\rho}_{(0)}=m_{(1)}\dot{\rho}_{(1)}+m_{(2)}\dot{\rho}_{(2)},$ (23)
$\displaystyle
m_{(0)}\dot{z}_{(0)}=m_{(1)}\dot{z}_{(1)}+m_{(2)}\dot{z}_{(2)},$

where the derivatives $\dot{\rho}_{(i)}$ and $\dot{z}_{(i)}$ should be
evaluated at the break-up point. Besides, the conservation of angular momentum
is

$m_{(0)}L_{(0)}=m_{(1)}L_{(1)}+m_{(2)}L_{(2)}.$ (24)

Finally there is an additional restriction on the masses $m_{(i)}$; squaring
the four momentum (21) and using the condition $P^{\mu}_{(1)}P_{\mu(2)}$
(future-pointing timelike vectors)Richartz2021 ; Bhat1985 , we have,

$m^{2}_{(1)}+m^{2}_{(2)}<m^{2}_{(0)}.$ (25)

##  IV ENERGY EXTRACTION EFFICIENCY

The efficiency $\eta$ of the Penrose process can be defined as the ratio
between the output energy (energy of the outgoing particle) and the input
energy (energy of the incident particle). From (22), we have

$\eta=\frac{E_{(2)}m_{(2)}-E_{(0)}m_{(0)}}{E_{(0)}m_{(0)}}=-\frac{E_{(1)}m_{(1)}}{E_{(0)}m_{(0)}}.$
(26)

In order to maximize the efficiency of the process we need to make $E_{(1)}$
as large as possible and $E_{(0)}$ as small as possible. On the other hand the
mass of the negative energy fragment $m_{(1)}$ should be as massive as
possible in comparison with $m_{(0)}$. In order to deduce the values of the
parameters that maximizes the efficiency we choose particular values for the
BM parameters $M_{1}$, $M_{2}$, $R$, the break-up point coordinates
$(\rho_{*},z_{*})$, and the charge-mass ratio $\mu_{(1)}$. With these
considerations we analyze how much energy can be extracted from the BM binary
BH.

###  IV.1 Maximum efficiency

The minimum energy of the incident particle, that comes from infinity,
according to Eq. (12) is $E_{(0)}=1$ and it corresponds to the particle having
zero kinetic energy at infinity. On the other hand, with the purpose of
maximizing the efficiency, the absolute value of the energy $E_{(1)}$ should
be as large as possible, this occurs when the particle $m_{(1)}$ is initially
at rest. Recalling that $E_{(1)}<0$, at the break-up point we set

$\dot{\rho}_{(1)}=\dot{z}_{(1)}=\dot{\phi}_{(1)}=0,$ (27)

hence, the angular momentum per unit mass and energy per unit mass are,
respectively, $L_{(1)}=0$ and

$E_{(1)}(\rho_{*},z_{*})=E^{min}_{(1)}(\rho_{*},z_{*})=-\mu_{(1)}A_{t*}+\sqrt{f_{*}},$
(28)

where $A_{t*}$ and $f_{*}$ are evaluated at the break-up point
$(\rho_{*},z_{*})$. Now, we determine the restrictions over the masses
$m_{(0)}$, $m_{(1)}$ and $m_{(2)}$. From the linear momentum conservation Eqs.
(23), we have

$\displaystyle m^{2}_{(2)}=$ $\displaystyle
m^{2}_{(0)}\frac{\dot{\rho}^{2}_{(0)}+\dot{z}^{2}_{(0)}}{\dot{\rho}^{2}_{(2)}+\dot{z}^{2}_{(2)}}+m^{2}_{(1)}\frac{\dot{\rho}^{2}_{(1)}+\dot{z}^{2}_{(1)}}{\dot{\rho}^{2}_{(2)}+\dot{z}^{2}_{(2)}}$
(29)
$\displaystyle-2m_{(0)}m_{(1)}\frac{\dot{\rho}_{(0)}\dot{\rho}_{(1)}+\dot{z}_{(0)}\dot{z}_{(1)}}{\dot{\rho}^{2}_{(2)}+\dot{z}^{2}_{(2)}}.$

If we consider the condition given by Eq. (27), and using
$\left(\dot{\rho}^{2}_{(i)}+\dot{z}^{2}_{(i)}\right)$ from Eq. (13),
substituting in Eq. (29), yields

$m_{(2)}=\sqrt{m^{2}_{(0)}-2m_{(0)}m_{(1)}\alpha_{(0)}+m^{2}_{(1)}},$ (30)

where

$\alpha_{(0)}=\frac{1+A_{t*}\mu_{(0)}}{\sqrt{f_{*}}},$ (31)

where $f_{*}$ and $A_{t*}$ are evaluated at the break-up point. $\alpha_{(0)}$
can be written as $\alpha_{(0)}=E_{\rm{eff(0)}}/\sqrt{V_{\rm{eff(0)}}}$ using
Eqs. (14) for an initially at rest particle at infinity ($E_{(0)}=1$, $L=0$),
hence $\alpha_{(0)}$ is always positive. The MP case is recovered with
$f_{*}\rightarrow 1/U^{2}_{*}$ and $A_{t*}\rightarrow 1/U_{*}-1$ where $U_{*}$
is the interaction potential for MP binary BH Richartz2021 .

From the inequality (25), squaring (30), and using the fact that the masses
are positive, we obtain

$0<m_{(1)}<m_{(0)}\left(\alpha_{(0)}-\sqrt{\alpha^{2}_{(0)}-1}\right).$ (32)

Where a necessary condition for the masses to have real values is
$\alpha^{2}_{(0)}\geq 1$, namely,

$\alpha_{(0)}=\frac{1+A_{t*}\mu_{(0)}}{\sqrt{f_{*}}}\geq 1,$ (33)

then the values of $\mu_{(0)}$ depend on the location of the break-up point,
since $A_{t*}$ is positive (negative), depending if it is evaluated inside
(outside) of the region defined by Eq. (18). For the case where the break-up
point occurs inside Eq. (18), $\mu_{0}$ is constrained by

$\mu_{(0)}\geq\frac{\sqrt{f_{*}}-1}{A_{t*}}.$ (34)

Have in mind that the extraction process occurs in this region if $\mu_{(0)}$
is positive and the BH is negatively charged. On the other hand, if the break-
up point occurs outside (18), the values which $\mu_{(0)}$ can take are
constrained by

$\mu_{(0)}\leq\frac{\sqrt{f_{*}}-1}{A_{t*}},$ (35)

and the extraction process takes place if $\mu_{(0)}$ is negative. Note that
the change of sign in the inequality (35) is because $A_{t*}<0$. Then, to
maximize the range of $m_{(1)}$ the inequality (33) must be saturated; this
occurs when the inequalities (34) or (35) are saturated, i.e., when
$\mu_{(0)}\rightarrow(\sqrt{f_{*}}-1)/A_{t*}$ to the left or to the right
according to the sign of $\mu_{(0)}$. In this case, one can choose
$m_{(1)}\rightarrow m_{(0)}$ thus maximizing the ratio $m_{(1)}/m_{(0)}$ that
appears in (26). Then, the efficiency $\eta$ of the Penrose process in the BM
is bounded by

$\eta<\eta^{b}=-E_{(1)}^{min}(\rho_{*},z_{*}).$ (36)

The efficiency upper bound denoted by $\eta^{b}$ is a function of $\mu_{(1)}$,
the break-up point coordinates $(\rho_{*},z_{*})$ and the BM parameters
$M_{1}$, $M_{2}$, $R$ according to Eq. (28). The effect of varying these
parameters is analyzed in the next subsection.

Figure 6:  It is illustrated the upper bound efficiency $\eta^{b}$ as a
function of $z_{*}$ for fixed $\rho_{*}$ and $M_{1}=0.5$, $M_{2}=0.2$, $R=1$;
in (a) $\mu_{(1)}=-1.7$, while in (b) $\mu_{(1)}=3$. In the region where the
Penrose process takes place, note that as the break-up point gets closer to
the point where the BH is located, the maximum efficiency $\eta^{b}$ increases
up to a maximum [see Eqs. (38)-(39)]; conversely, if the break-up point moves
away the BH then $\eta^{b}\rightarrow 0$. Negative $\eta^{b}$ occurs if the
break-up points are outside the generalized ergosphere.

Figure 7: It is shown in (a) the upper bound efficiency $\eta^{b}$ as a
function of $M_{T}=M_{1}+M_{2}$ for $M_{R}=M_{2}/M_{1}=0.25$, $R=1$ and
$\mu=-1.7$ for selected break-up points, denoted by $X$, $Y$ and $Z$. The
values of $M_{Tcrit}$ are $M_{Tcrit}=0.658999$, $M_{Tcrit}=0.645854$ and
$M_{Tcrit}=0.823573$ for the break-up points $X$, $Y$ and $Z$, respectively.
(b) Generalized ergospheres for selected values of $M_{T}$. The black square
markers are the locations of the break-up points $(\rho_{*},z_{*})=(0,0.8)$,
$(\rho_{*},z_{*})=(0,0.3)$ and $(\rho_{*},z_{*})=(0,0)$ that are labeled as
$X$, $Y$ and $Z$, respectively. Each curve in (a) corresponds to one of the
break-up points in (b). The dots represent the BHs.

###  IV.2 Dependence of the maximum efficiency on the parameters.

From Eqs. (28) and (36) the explicit expression of the efficiency upper bound
$\eta^{b}$ is given by

$\eta^{b}=\mu_{(1)}A_{t*}-\sqrt{f_{*}}.$ (37)

The efficiency bound $\eta^{b}$ has a linear dependence respect to the charge-
mass ratio $\mu_{(1)}$. The dependence with respect to the break-up point
coordinates ($\rho_{*}$, $z_{*}$) can be understood with the help of the
energy levels shown in Fig. 3; the efficiency upper bound increases as the
break-up point approaches one of the BHs, i.e., the Penrose process is more
efficient if the break-up point is located near one of the horizons. If this
is the case, using Eq. (4) the upper bound efficiency for $\mu_{(1)}>0$ is

$\eta^{b}=\mu_{(1)}\sqrt{\frac{(R-M_{1})^{2}-M^{2}_{2}}{(R+M_{1})^{2}-M^{2}_{2}}}=-\mu_{(1)}\frac{M_{2}}{Q_{2}},$
(38)

and for $\mu_{(1)}<0$ the upper bound is

$\eta^{b}=-\mu_{(1)}\sqrt{\frac{(R-M_{2})^{2}-M^{2}_{1}}{(R+M_{2})^{2}-M_{1}^{2}}}=-\mu_{(1)}\frac{M_{1}}{Q_{1}}.$
(39)

A relevant characteristic of the efficiency bound, Eqs. (38) and (39), is that
it depends on the charge-mass ratio $\mu_{(1)}$, and it can be chosen
arbitrarily large, even such that the resulting efficiency is greater than
one, $\eta^{b}>1$, similarly to the case of one single charged BH Bhat1985 ;
Parthasarathy1986 ; Nucamendi2022  interacting with charged particles.

Moreover, for a given $M_{1}$, the ratio $|M_{2}/Q_{2}|$ in Eq. (38) decreases
monotonically in the interval $0<M_{2}<R-M_{1}$; however, as $M_{2}$
approaches $(R-M_{1})$ the ratio $|M_{2}/Q_{2}|\to 0$ and consequently
$\eta^{b}\to 0$. i.e. as the value of the total mass approaches the BH
separation distance $R$, $M_{1}+M_{2}\to R$, the maximum efficiency decreases
up to zero. The same behavior occurs for a fixed $M_{2}$ and varying $M_{1}$,
see Eq. (39).

From (37) we can identify two scenarios according to the sign of the charge-
mass ratio $\mu$, while the sign of $A_{t}$ is defined by the break-up point
location.

How $\eta^{b}$ depends on the the break-up location, ($\rho_{*}$, $z_{*}$), is
illustrated in Fig. 6.a for several values of $\rho_{*}$ and fixed $M_{1}$,
$M_{2}$, $R$ and $\mu_{(1)}<0$. On the other hand, the dependence of the
break-up coordinates of $\mu_{(1)}>0$ is illustrated in Fig. 6.b. The
efficiency is negative when the particle that escapes to infinity carries less
energy than the incident particle. Note that the extraction process occurs
only when the test particle and the BH are of opposite charges.

Since the upper bound efficiency given by Eqs. (38) and (39) can be written in
terms of the BH charges that are such that $|Q_{i}|>M_{i}$, then the upper
bound efficiency is less than $(-\mu_{(1)})$ for arbitrary $M_{1}$, $M_{2}$,
and $R$; this is in contrast with the MP case, analyzed in Richartz2021 ,
where the upper bound is $(-\mu_{(1)})$. In the case that one of the BH masses
is zero, the extraction process occurs if $\mu_{(1)}$ and the BH are
oppositely charged and in that case the maximum efficiency is
$\eta^{b}=\pm\mu_{(1)}$, for a negative or positively charged BH,
respectively.

To determine how the efficiency depends on the BH masses, we analyze (37) in
terms of the mass ratio $M_{R}=M_{2}/M_{1}$ with $M_{2}<M_{1}$, total mass
$M_{T}=M_{1}+M_{2}$ and fixed $\mu_{(1)}$ and $R=1$, such that $0<M_{R}<1$ and
$0<M_{T}<1$. The dependence on the total mass $M_{T}$ for a fixed mass ratio
$M_{R}$, $\mu_{(1)}$ with different break-up points is shown in Fig. 7. In
Fig. 7.a initially the efficiency is negative because the break-up points are
located outside the ergosphere and the total mass is small (therefore the
ergosphere is small as well); for a fixed $M_{R}$ the generalized ergosphere
gets bigger as the total mass $M_{T}$ increases until a critical value
$M_{Tcrit}$, where the generalized ergosphere has its largest size and the
upper bound efficiency is maximum.

When $M_{Tcrit}<M_{T}<R$, the efficiency $\eta^{b}$ behaves similarly than in
Eqs. (38)-(39); i.e. it decreases monotonically, even reaching negative
values; the reason is that as $M_{T}\to R$ the ergosphere shrinks.

The critical value of the total mass $M_{Tcrit}$ is given by

$\frac{d}{dM_{T}}\left(\eta^{b}\right)=0,\quad\frac{d^{2}}{dM_{T}^{2}}\left(\eta^{b}\right)<0,$
(40)

where $\eta^{b}$ is the upper bound efficiency in Eq. (37), where $A_{t*}$ and
$f_{*}$ have been rewritten in terms of the total mass $M_{T}$ and mass ratio
$M_{R}$ using $M_{1}=M_{T}/(M_{R}+1)$ and $M_{2}=M_{R}M_{T}/(M_{R}+1)$.

For $\mu_{(1)}>0$ the ergosphere and upper bound efficiency exhibit the same
qualitative behaviour as shown in Fig. 8; the critical mass can be calculated
with Eqs. (40).

The mass ratio $M_{R}$ for $M_{T}=1/2$ and $\mu_{(1)}<0$, for different break-
up points is shown in Fig. 9. In this case the ergosphere and upper bound
$\eta^{b}$ decrease as $M_{R}$ increases, i.e., the ergosphere and upper bound
efficiency decrease when the masses $M_{1}$ and $M_{2}$ tend to the same
value. The maximum of the upper bound efficiency $\eta^{b}$ occurs when
$M_{R}\to 0$, i.e., when $M_{2}\to 0$ but less or equal to $-\mu_{(1)}$. In
Fig. 10 is shown $\eta^{b}$ as a function of $M_{R}$ for $\mu_{(1)}>0$. In
this case the upper bound efficiency $\eta^{b}$ and the ergosphere increase as
$M_{R}$ increases and the maximum $\eta^{b}$ for an arbitrary break-up point
occurs when $M_{R}\to 1$.

Figure 8: It is shown the (a) upper bound efficiency $\eta^{b}$ as a function
of $M_{T}$ for $M_{R}=0.75$, $R=1$ and $\mu=3$ for different break-up points;
$M_{Tcrit}=0.809496$, $M_{Tcrit}=0.326936$ and $M_{Tcrit}=0.611923$ correspond
to the break-up points $X$, $Y$ and $Z$, respectively. (b) Generalized
ergospheres for different $M_{T}$. The square markers are the locations of the
break-up points $(\rho_{*},z_{*})=(0,-0.2)$, $(\rho_{*},z_{*})=(0,-0.55)$ and
$(\rho_{*},z_{*})=(0,-0.75)$ that are labeled $X$, $Y$ and $Z$ respectively.
The dots represent the BHs. Each curve in (a) corresponds to one of the break-
up points in (b).

Figure 9: It is plotted the (a) upper bound efficiency $\eta^{b}$ as a
function of $M_{R}$ for $M_{T}=0.5$, $R=1$ and $\mu=-1.7$ for selected break-
up points. (b) Generalized ergospheres for different $M_{R}$. The square
markers are the locations of the break-up points $(\rho_{*},z_{*})=(0,0.88)$,
$(\rho_{*},z_{*})=(0,0.3)$ and $(\rho_{*},z_{*})=(0,0)$labeled as $X$, $Y$ and
$Z$, respectively. The dots represent the BHs. Each curve in (a) corresponds
to one of the break-up points in (b).

Figure 10: It is illustrated in (a) the upper bound efficiency $\eta^{b}$ as a
function of $M_{R}$ for $M_{T}=0.5$, $R=1$ and $\mu=3$ for selected break-up
points. (b) Generalized ergospheres for different $M_{R}$. The square markers
are the locations of the break-up points $(\rho_{*},z_{*})=(0,-0.24)$,
$(\rho_{*},z_{*})=(0,-0.6)$ and $(\rho_{*},z_{*})=(0,-0.7)$ with labels $X$,
$Y$ and $Z$ respectively. The dots represent the BHs. Each curve in (a)
corresponds to one of the break-up points in (b).

###  IV.3 Examples of the Penrose process

Once we have described the efficiency $\eta^{b}$, in this section some
concrete examples of the Penrose process in the BM are given. In these
examples the efficiency approaches the theoretical maximum described by Eq.
(38) for test particles with $\mu_{(1)}>0$ and Eq. (39) for test particles
with $\mu_{(1)}<0$. First we fix $M_{1}$, $M_{2}$, and $R$, and the charge
mass ratio $\mu_{(1)}$ for a particular break-up point
$(\rho_{*},z_{*},\phi_{*})$ located inside the generalized ergosphere.
According to the analysis in Sec. IV.1 we set $m_{(0)}=1$,
$L_{(0)}=L_{(1)}=L_{(2)}=0$, which means that the trajectories are confined to
the meridional plane $\phi=\phi_{*}$. The energy $E_{(1)}$ is determined by
Eq. (28) and we set $E_{(0)}=1$. The two scenarios for the process correspond
to negative and positive charged test particle $\mu_{(1)}$; the charge mass
ratio $\mu_{(0)}$ is bounded by two different limits depending where the
break-up point ($\rho_{*}$, $z_{*}$) occurs according to Eqs. (34) and (35).

If the break-up point is located outside the region bounded by (18), then
according to Eqs. (32) and (35), we choose

$\mu_{(0)}=\frac{\sqrt{f_{*}}-1}{A_{t*}}-\epsilon_{1},$ (41)
$m_{(1)}=1-\nu_{1},$ (42)

Figure 11: Examples of Penrose process with the maximum efficiency in a BM.
The efficiencies of the processes are 90$\%$ of the theoretical maximum. In
both panels the incoming trajectory $T_{(0)}$ (black curve) splits (at the
black point) into the negative energy trajectory $T_{(1)}$ (red curve) and the
trajectory of the particle that escapes $T_{(2)}$. The Bonnor parameters in
both cases are fixed as $M_{1}=0.5$, $M_{2}=0.2$, $R=1$, the parameter $\nu$
and $\epsilon$ are fixed as $\nu_{1,2}=10^{-2}$, $\epsilon_{1,2}=10^{-5}$, the
charge-mass ratio, the break-up point and the angle $\theta_{(}0)$ are fixed,
respectively, as: (a) $\mu_{(1)}=-5$, $(\rho_{*},z_{*})=(1/4,1/4)$ and
$\theta_{(0)}=0.0872665$; (b) $\mu_{(1)}=20$, $(\rho_{*},z_{*})=(0.3,-0.8)$
and $\theta_{(0)}=0.296706$. The initial values of the parameters that
generate these trajectories are shown in Tables 1 and 2. The respective upper
bound efficiency is $\eta^{b}=1.20854$ (left panel) and $\eta^{b}=0.208203$
(right panel).

where $\epsilon_{1}$ and $\nu_{1}$ are small and positive. Substituting Eqs.
(41) and (42) into Eq. (32), we find that $\epsilon_{1}$, and $\nu_{1}$,
satisfy

$\nu_{1}>\frac{A_{t*}\epsilon_{1}}{\sqrt{f_{*}}}\left(\sqrt{1-\frac{2\sqrt{f_{*}}}{A_{t*}\epsilon_{1}}}+1\right).$
(43)

On the other hand, if the break-up point occurs inside of the circle (18),
then according to (32) and (34), we choose

$\mu_{(0)}=\frac{\sqrt{f_{*}}-1}{A_{t*}}+\epsilon_{2},$ (44)

$m_{(1)}=1-\nu_{2},$ (45)

Substituting these expressions into (32), we find that $\epsilon_{2}$ and
$\nu_{2}$ satisfy

$\nu_{2}>\frac{A_{t*}\epsilon_{2}}{\sqrt{f_{*}}}\left(\sqrt{1+\frac{2\sqrt{f_{*}}}{A_{t*}\epsilon_{2}}}-1\right),$
(46)

the charge mass ratio $\mu_{(2)}$ and energy per unit mass $E_{(2)}$ can be
determined from Eqs. (20) and (22).

For a given angle between the velocities,
$\theta_{(0)}=\arg(\dot{\rho}_{(0)}+i\dot{z}_{(0)}$), Eqs. (47) and (48) can
be used to determine $\dot{\rho}_{(0)}$ and $\dot{z}_{(0)}$ at the break-up
point. According to (27), we have $\dot{\rho}_{(1)}=\dot{z}_{(1)}=0$ for the
negative energy fragment. For the incident particle, if the break-up point is
outside the circular region (18) and $E_{(0)}=1$ then

$\dot{\rho}^{2}_{(0)}+\dot{z}^{2}_{(0)}=\frac{\big{(}1+A_{t*}(1-\epsilon_{1})\big{)}^{2}}{e^{2\gamma_{*}}}-\frac{f_{*}}{e^{2\gamma_{*}}},$
(47)

while if the break-up point is inside (18) we have,

$\dot{\rho}^{2}_{(0)}+\dot{z}^{2}_{(0)}=\frac{\big{(}1+A_{t*}(1+\epsilon_{2})\big{)}^{2}}{e^{2\gamma_{*}}}-\frac{f_{*}}{e^{2\gamma_{*}}}.$
(48)

While from linear momentum conservation (23) $\dot{\rho}_{(2)}$ and
$\dot{z}_{(2)}$ are calculated.

With this setting of parameters the trajectories $T_{(0)}$, $T_{(1)}$ and
$T_{(2)}$ can be completely determined and the efficiency of the Penrose
process is given by $\eta_{1,2}=(1-\nu_{1,2})\eta^{b}$. In order to maximize
the efficiency we choose $\nu_{1,2}$ as small as possible. However,
$\epsilon_{1,2}$ and $\nu_{1,2}$ should not be zero because the inequalities
(43) or (46) cannot be saturated; but it is possible, in principle, to set
infinitesimally small values for $\epsilon_{1,2}$ and $\nu_{1,2}$. According
to these considerations we explore two specific examples where the efficiency
corresponds to $90\%$ for a break-up point located outside and inside of the
region bounded by (18) resulting that the extraction process occurs with the
BH with positive or negative charge, respectively.

Fig. 11.a shows the Penrose extraction from a BH with negative charge. The BM
parameters $M_{1}$, $M_{2}$, and $R$ are fixed. It corresponds to the
extraction process where the break-up point occurs outside (18) and the
charge-mass ratio of the negative energy fragment (red line) is negative
$\mu_{(1)}$ (e. g. Fig. 2.c), the setting of these parameters enables us to
determine the trajectories $T_{(1)}$, having in mind that the energy $E_{(1)}$
associated to the trajectory $T_{(1)}$, Eq. (16), is the minimum possible and
the trajectories $T_{(0)}$ and $T_{(2)}$ are fully described once we fix
$\nu_{1,2}$, $\epsilon_{1,2}$, and $\theta_{(0)}$. On the other hand, in Fig.
11.b it is shown the Penrose process where the break-up point occurs inside
the region bounded by (18) for a negatively charged particle. We can highlight
two different features of the Penrose process in the BM that are in contrast
with the MP binary: first, the process of extraction can occur for positive or
negative charged test particles inside the generalized ergosphere that
surrounds the BH with charge of opposite sign to the one of the test particle.
So, any observer can recover information of the energy extraction only when
the particle that gains energy escapes back to infinity. Second, in some
scenarios the particle escaping with more energy can be trapped by the other
BH, as shown in Fig. 11.b; in this case the observer at infinity will not
receive any information of the energy extraction process. As far as we know
the second phenomenon has not been reported in the MP binary BH. The
parameters that generate these examples satisfy the conservations Eqs.
(20)-(25) and are given in Tables 1 and 2.

Table 1: Initial values that generate the trajectories $T_{(0)}$, $T_{(1)}$ and $T_{(2)}$ in the Penrose process shown in Fig. 11.a. The derivatives $\dot{\rho}_{(i)}$ and $\dot{z}_{(i)}$ are evaluated at the break-up point. $i$ | $m_{(i)}$ | $\mu_{(i)}$ | $E_{(i)}$ | $L_{(i)}$ | $\dot{\rho}_{(i)}$ | $\dot{z}_{(i)}$ |
---|---|---|---|---|---|---|---
0 | 1 | 2.70371 | 1 | 0 | 1.45015 | 0.126872 |
1 | 0.9 | -5 | -1.20854 | 0 | 0 | 0 |
2 | 0.0998852 | 72.1199 | 20.9008 | 0 | 14.5182 | 1.27018 |
Table 2: Initial values that generate the trajectories $T_{(0)}$, $T_{(1)}$ and $T_{(2)}$ in the Penrose process shown in Fig. 11.b. The derivatives $\dot{\rho}_{(i)}$ and $\dot{z}_{(i)}$ are evaluated at the break-up point. $i$ | $m_{(i)}$ | $\mu_{(i)}$ | $E_{(i)}$ | $L_{(i)}$ | $\dot{\rho}_{(i)}$ | $\dot{z}_{(i)}$ |
---|---|---|---|---|---|---|---
0 | 1 | 2.70371 | 1 | 0 | 1.45015 | 0.126872 |
1 | 0.9 | -5 | -1.20854 | 0 | 0 | 0 |
2 | 0.0998852 | 72.1199 | 20.9008 | 0 | 14.5182 | 1.27018 |

###  IV.4 Stability of the Bonnor metric

In Bernard2019  it is studied the effect of perturbing a black hole binary
(BHB) with a massless scalar field and it is shown that initially the dominant
mode is monopolar and once this initial mode dies away there arises an
exponentially damped sinusoid mode. It is as well find out that BHB possesses
global quasinormal modes whose ringdown parameters do not depend on the
initial conditions but only on the masses and the separation, and that
relaxation time scale increases with the separation. Moreover simulations
indicate that the perturbations die away in time. Besides, Wong2019  agrees in
the exponential decay of a massless scalar perturbation, however indicating
that for confined BHB there might be mechanisms that could trigger
instabilities. More analysis is required for other kind of perturbations to
elucidate general conditions under which BHB are stable systems.

Regarding stability of the binary BM against electromagnetic perturbations and
specifically how energy extraction affects stability the following
considerations point to the stability of the BM: since the generalized
ergosphere never encircles both BH, then the perturbation affects only one of
the BH, the one with charge opposite to the test particle; therefore we can
apply stability criteria valid for one single charged BH, i.e. a Reissner-
Nordström BH, that we know is stable; and then we guess that the BM remains
stable regarding energy extraction. The situation would be very different if
the system be under some kind of confinement that could stimulate
superradiance effects; in any case a deeper analysis is required to elucidate
the BM stability.

##  V Conclusions

We have analyzed in detail the possibility of energy extraction from the
Bonnor BH binary (BM), that describes two oppositely charged BH kept apart by
a strut Bonnor1979 ; Cabrera2011  that prevents the two BH collide; this is in
contrast to the Majumdar-Papapetrou (MP) binary where the gravitational and
electromagnetic forces are balanced.

We determined a generalized ergosphere that depends on the parameters of the
BM and the charge-mass ratio $\mu$ of the test particle and we showed that
energy extraction is possible; the sign of the electric potential $A_{t}$ is
defined by the point $(\rho,z)$ where it is evaluated. A first difference with
the MP case is that the generalized ergosphere exists for positive and
negatively charged test particles $\mu$. Another remarkable difference with
respect to the MP case is that the ergosphere cannot include both BHs, but
only one; the ergosphere encloses the BH with charge opposite to the one of
the test particle. We found that for some initial conditions the particle that
escapes with more energy is trapped by the other BH and in this case the
observer at infinity would not receive information of the energy extraction.

We studied the conditions that optimize the efficiency of the process. The
efficiency is enhanced if the break-up point is located near the horizon of
the BH charged oppositely to the test particle. We determined the total mass
$M_{T}=M_{1}+M_{2}$ and mass ratio $M_{R}=M_{2}/M_{1}$ that renders the
highest efficiency $\eta^{b}$. The behavior of $\eta^{b}$ as a function of
$M_{R}$ depends on the sign of $\mu$. If $\mu<0$ then $\eta^{b}$ decreases
when $M_{R}\to 1$; while if $\mu>0$ then $\eta^{b}$ increases when $M_{R}$
approaches $1$. Moreover the maximum efficiency $\eta^{b}$ does not increase
monotonically as $M_{T}$ increases, but there is a certain $M_{Tcrit}$ such
that for $M_{T}>M_{Tcrit}$, $\eta^{b}$ decreases, and even can reach negative
values. The upper bound efficiency in the BM is always smaller than the MP one
Richartz2021 .

Due to the vast recent observations reported by the LIGO-Virgo Collaboration,
so far it has been able to identify multiple candidates for compact binary
systems. We believe that the study of the Penrose process in BM contributes to
the understanding of actual BHBs as it extends the analysis carried out for a
MP BH in Richartz2021 . Further analysis of BHB would give relevant
information regarding other astrophysical aspects Misner1972 ; Bekenstein1973
; Brito2015 , for instance there are proposals that link compact binary
systems with superradiance Rosa2015 ; Wong2019 . Moreover, the magnetic
variant of the Penrose process that takes into account the combined influence
of external magnetic field and the rotation of a BH seems to be connected to
the origin of accretion disks where the energy extraction into jets can
befall, or even the generation of ultra-high energy cosmic rays Kolos2021 ;
Kolos2020 . In this direction we aim to develop further research of the
magnetic Penrose process in BHB.

##  VI Acknowledgments

NB acknowledges partial support by CONACyT Project No. 284498. ICM
acknowledges financial support of SNI-CONACyT, México, grant CVU No. 173252.
AB acknowledges financial support by CONACyT, México, through the PhD
Scholarship with CVU No. 933515.

## References

  * (1) R. P. Kerr, Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics, Phys. Rev. Lett. 11, 237 (1963).
  * (2) M. Visser, The Kerr spacetime: A Brief introduction, in Kerr Fest: Black Holes in Astrophysics, General Relativity and Quantum Gravity (2007) arXiv:0706.0622.
  * (3) S. A. Teukolsky, The Kerr metric, Class. Quantum Grav. 32, 124006 (2015).
  * (4) R. Penrose and R. M. Floyd, Extraction of Rotational Energy from a Black Hole, Nat. Phys. Sci. 229, 177 (1971).
  * (5) D. Christodoulou, Reversible Transformations of a Charged Black Hole, Phys. Rev. D 4, 3552 (1971).
  * (6) G. Denardo and R. Ruffini, On the energetics of Reissner Nordström geometries, Phys. Lett. B 45, 259 (1973).
  * (7) N. Dadhich, The Penrose Process of Energy Extraction in Electrodynamics, ICTP Preprint, IC-80/98 (1980).
  * (8) J. Schnittman, The collisional Penrose process, Gen. Relativ. Gravit. 50, 77 (2018).
  * (9) Z. Stuchlík, M. Kološ, and A. Tursunov, Penrose process: Its variants and astrophysical applications, Universe 7 (2021).
  * (10) N. Dadhich, A. Tursunov, B. Ahmedov, and Z. Stuchlík, The distinguishing signature of magnetic Penrose process, Mon. Not. Roy. Astron. Soc. 478, L89 (2018).
  * (11) Z. Stuchlík, M. Kološ, and A. Tursunov, Possible signature of the magnetic fields related to quasi-periodic oscillations observed in microquasars, Eur. Phys. J. C 77, 860 (2017).
  * (12) M. Kološ, Z. Stuchlík, and A. Tursunov, Quasiharmonic oscillatory motion of charged particles around a Schwarzschild black hole immersed in a uniform magnetic field, Classical and Quantum Gravity 32, 165009 (2015).
  * (13) M. Kološ, Z. Stuchlík, and A. Tursunov, Radiative penrose process: Energy gain by a single radiating charged particle in the ergosphere of rotating black hole, Phys. Rev. D 103, 024021 (2021).
  * (14) M. Kološ, Z. Stuchlík, and A. Tursunov, Light escape cones in local reference frames of kerr-de sitter black hole spacetimes and related black hole shadows, Eur. Phys. J. C 78, 180 (2018).
  * (15) K. Parfrey, A. Philippov, and B. Cerutti, First-Principles Plasma Simulations of Black-Hole Jet Launching, Phys. Rev. Lett. 122, 035101 (2019).
  * (16) M. Bhat, S. Dhurandhar, and N. Dadhich, Energetics of the Kerr-Newman black hole by the penrose process, J. Astrophys. Astron. 6, 85 (1985).
  * (17) A. Tursunov, B. Juraev, Z. Stuchlík, and M. Kološ, Electric Penrose process: High-energy acceleration of ionized particles by nonrotating weakly charged black hole, Phys. Rev. D 104, 084099 (2021).
  * (18) S. Parthasarathy, S. M. Wagh, S. V. Dhurandhar, and N. Dadhich, High Efficiency of the Penrose Process of Energy Extraction from Rotating Black Holes Immersed in Electromagnetic Fields, Astrophys. J. 307, 38 (1986).
  * (19) S. M. Wagh, S. V. Dhurandhar, and N. Dadhich, Revival of the Penrose Process for Astrophysical Applications, Astrophys. J. 290, 12 (1985).
  * (20) S. M. Wagh and N. Dadhich, The energetics of black holes in electromagnetic fields by the penrose process, Phys. Rept. 183, 137 (1989).
  * (21) S. Shaymatov, P. Sheoran, R. Becerril, U. Nucamendi, and B. Ahmedov, Efficiency of Penrose process in spacetime of axially symmetric magnetized Reissner-Nordström black hole, Phys. Rev. D 106, 024039 (2022).
  * (22) L. T. Sanches and M. Richartz, Energy extraction from non-coalescing black hole binaries, Phys. Rev. D 104, 124025 (2021).
  * (23) S. D. Majumdar, A class of exact solutions of Einstein’s field equations, Phys. Rev. 72, 390 (1947).
  * (24) A. Papapetrou, A static solution of the equations of the gravitational field for an arbitrary charge distribution, Proc. R. Irish Acad., Sect. A 51, 191 (1947).
  * (25) J. B. Hartle and S. W. Hawking, Solutions of the Einstein-Maxwell equations with many black holes, Commun. Math. Phys. 26, 87 (1972).
  * (26) W. B. Bonnor, A three-parameter solution of the static Einstein-Maxwell equations, J. Phys. A Math Gen. 12, 853 (1979).
  * (27) I. Cabrera-Munguia, V. S. Manko, and E. Ruiz, A combined Majumdar-Papapetrou-Bonnor field as extreme limit of the double-Reissner-Nordström solution, Gen. Relativ. Gravit. 43, 1593 (2011).
  * (28) R. Bach and H. Weyl, Neue Lösungen der Einsteinschen Gravitationsleichungen, Math. Z. 13, 134 (1922).
  * (29) W. Israel, Line sources in general relativity, Phys. Rev. D 15, 935 (1977).
  * (30) V. S. Manko, Double-Reissner-Nordström solution and the interaction force between two spherical charged masses in general relativity, Phys. Rev. D 76, 124032 (2007).
  * (31) G. G. Varzugin and A. S. Chystiakov, Charged rotating black holes in equilibrium, Classical Quantum Gravity 19, 4553 (2002).
  * (32) G. A. Alekseev and V. A. Belinski, Equilibrium configurations of two charged masses in general relativity, Phys. Rev. D 76, 021501(R) (2007).
  * (33) I. Cabrera-Munguia and A. Macías, On naked singularities in the extreme double Reissner-Nordström solution, in AIP Conf. Proc., Vol. 1577 (AIP Publishing LLC, 2014).
  * (34) F. Dubeibe and José D. Sanabria-Gómez, Geodesic motion in a stationary dihole spacetime, Phys. Rev. D 94, 044058 (2016).
  * (35) M. S. Costa, C. A. R. Herdeiro, and C. Rebelo, Dynamical and Thermodynamical Aspects of Interacting Kerr Black Holes, Phys. Rev. D 79, 123508 (2009).
  * (36) L. Bernard, V. Cardoso, T. Ikeda, and M. Zilhaõ, Physics of black hole binaries: Geodesics, relaxation modes and energy extraction, Phys. Rev. D 100, 044002 (2019).
  * (37) L. K. Wong, Superradiant scattering by a black hole binary, Phys. Rev. D 100, 044051 (2019).
  * (38) C. Misner, Stability of Kerr black holes against scalar perturbations, Bull. Amer. Phys. Soc. 17, 472 (1972).
  * (39) Jacob D. Bekenstein, Extraction of Energy and Charge from a Black Hole, Phys. Rev. D 7, 949–953 (1973).
  * (40) R. Brito, V. Cardoso and P. Pani, Superradiance, Lect. Notes Phys. 906, 1 (2015).
  * (41) J. G. Rosa, Testing black hole superradiance with pulsar companions, Phys. Lett. B 749, 226 (2015).
  * (42) A. Tursunov, Z. Stuchlík, M. Kološ, N. Dadhich and B. Ahmedov, Supermassive Black Holes as Possible Sources of Ultrahigh-energy Cosmic Rays, Astrophys. J. 985, 14 (2020).