# Energy extraction from the Reissner-Nordström de Sitter black hole

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 Instituto
Politecnico Nacional; Apdo. Postal 14-740, Mexico City, Mexico
2Departamento de Física y Matemáticas,
Universidad Autónoma de Ciudad Juárez, 32310 Ciudad Juárez, Chihuahua, México

###### Abstract

The energy extraction from an electrostatic black hole by the decay or
splitting of electrically charged particles is analyzed. We determine the
energetic conditions that make the extraction process viable and present a
general expression for the efficiency in terms of the parameters of the
electrostatic black hole and the decaying particles. We also examine the
conditions that optimize the efficiency of the extraction process. We analyze
two particular cases, the first one is the extraction process from a Reissner
Nordström black hole, for charged test particles with nonvanishing angular
momentum; the second one and more interesting corresponds to the energy
extraction from a Reissner Nordström de Sitter black hole. For the latter
there are two regions where the energy extraction is possible, the generalized
ergosphere and a cosmological ergosphere induced by the cosmological horizon.
Under certain conditions the two ergospheres get connected and cover the whole
region between the event horizon and the cosmological horizon, and therefore
the energy extraction is possible at any point in the vicinity of the black
hole. Moreover, the efficiency of the energy extraction can be the same for
different break up points and also there is the possibility of a different
efficiency for the same break up point. The conditions that maximize the
efficiency are determined as well.

##  I Introduction

The Penrose process is a mechanism proposed in PenFloyd1971  to extract energy
from a rotating black hole (BH) like the Kerr spacetime Kerr1963 ; Visser2007
; it considers a decay or splitting process in the region delimited between
the event horizon and the stationary limit surface. When a particle gets in
the ergosphere a timelike trajectory becomes spacelike and the particles can
have negative energy (as measured by an observer at infinity); a particle
inside the ergosphere still can avoid enter into the event horizon and can
escape back to infinity.
Essentially the Penrose process consists of an ingoing particle that falls
freely into the BH; the particle penetrates the ergosphere and then it decays
into two fragments, one them with negative energy that remains confined within
the ergosphere and eventually it falls into the event horizon. Due to energy
conservation, if the second fragment escapes to infinity, it does with a
greater energy than the original particle.
Because the energy extraction process takes place inside the ergosphere, that
is a characteristic region of stationary solutions, in principle, the
extraction process is not possible from static BHs. Nevertheless, as a
variation of Penrose process is the electric Penrose process Christo1971 ;
DenardoRuffini1972 ; Dadich1980  that allows the energy extraction from
electrically charged BH, even for static BHs, via electrostatic interaction
between the charged black hole and charged test particles. Recent studies
about electric Penrose process include the analysis of energy extraction from
a Reissner-Nordström-anti-de Sitter BH Lemos2024  and the possibility to
obtain BH energy factories and BH bombs is studied considering a recursive
Penrose process; other works that carry out the energy extraction from static
BHs are Richartz2021 ; Baez2022 , where systems of equally and oppositely
charged BHs described by Majumdar-Papapetrou Majumdar ; Papapetrou ;
Hartle1972  and Bonnor Bonnor1979 ; Cabrera2011  metric, are analyzed.
There are other variations of the Penrose process that can be associated with
some astrophysical observations. For instance, collision in the 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 ; BSW2009 .
On the other hand, the presence of an external magnetic field surrounding a
rotating BH can give rise to accretion disks comprising charged ionized matter
Kolos2021 , this could potentially be connected to the observation of high-
frequency oscillations observed in microquasars or galactic nuclei where ultra
high-energy particles around rotating magnetized BHs are created Kolos2017 ;
Kolos2015 . The radiative Penrose process is related to synchroton radiation
of charged particles interacting in the ergosphere of a magnetized BH, where
the process could produce a particular type of photons having negative energy
relative to a distant observer Kolos2021 ; Kolos2018 . On the other hand, the
electromagnetic Penrose process Bhat1985 ; Tursunov2021 ; Parthasarathy1986 ;
Wagh1985 ; Wagh1989 ; Nucamendi2022  allows events of high energy emission
that improve significantly the efficiency of 20% of the Penrose process due
solely to the rotation Bardeen1972 ; Wald1974 . It is worth mentioning, that
exist other ways of extracting energy from a BH, for instance, the
superradiance that considers waves instead particles. In superradiance a wave
field is sent into the ergoregion, and under the appropriate conditions an
amplified wave returns to infinity. The superradiant effects are exhibited by
Kerr BH Starobinsky1973 ; Teu1974 . We also point out that the process here
analyzed is different from the energy extraction by Hawking radiation, mainly
because it does not require a time varying background geometry, and the whole
treatment is classical Damour75 .

The present paper aims to investigate energy extraction from a black hole via
the electric Penrose process. In DenardoRuffini1972  was analyzed for the
first time the extraction from a Reissner-Nordström BH via electrostatic
interaction considering the decay of a charged particle, however the study was
restricted to zero angular momentum of the decaying particle. In Lemos2024
the recursive Penrose process for a Reissner-Nordström-anti-de Sitter BH is
studied. Even analysis on binary solutions of electrostatic BHs has been
developed Richartz2021 ; Baez2022 . We present the analysis of the electric
Penrose process for a static, spherically symmetric charged BH; we derive a
general expression for the efficiency of the energy extraction and determine
the energetic conditions that optimize the efficiency. We extend the study of
the RN BH presented in DenardoRuffini1972  to arbitrary angular momentum of
the charges involved; additionally, the particle parameters that maximize the
efficiency of the process are determined. The second case that is examined is
the Reissner-Nordström-de Sitter BH; we have determined the regions where
extraction energy is possible and its relationship with the charges of the
test particles.
Our paper is organized as follows. In Sec.II.1, we introduce the general
static spherically symmetric spacetime and derive the motion equations for
charged massive test particles. In Sec. II.2 the generalized ergosphere and a
general expression to determine the efficiency are presented; the test
particle conditions that optimize the process are discussed as well. In Sec.
III it is analyzed the Penrose process for the RN BH, in the general case of
nonvanishing angular momentum of the particle with negative energy, obtaining
in the limit of zero angular momentum the results in DenardoRuffini1972 . In
Sec. IV the analysis of the extraction process (efficiency, generalized
ergosphere and restrictions) for Reissner-Nordström-de Sitter is presented;
the RNdS BH has two ergospheres and there are cases with the same efficiency
for different break up points and also the possibility of a different
efficiency for the same break up point. In the last section Conclusions are
given.

##  II Penrose mechanism for electrostatic black holes

Considering the Penrose process introduced in PenFloyd1971 , we present an
approach to study the efficiency of this process in the vicinity of an
electrostatic BH. It consists in sending a charged particle towards the BH; at
some point, inside the ergosphere, the particle breaks up into two fragments,
one of them escapes to infinity with more energy than the initial one, while
the second fragment remains inside the ergosphere and eventually penetrates
the horizon of the BH. This mechanism requires that the BH possesses an
ergoregion, which, in principle, only stationary BH possess; therefore by
means of the regular Penrose process it is not possible to extract energy from
a static BH. However, for electrostatic BH it is possible to define a
generalized ergosphere where charged test particles can have negative energies
Christo1971 ; DenardoRuffini1972 ; Dadich1980 .

Let us consider the static spherically symmetric line element,

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2},$
(1)

where $(t,r,\theta,\phi)$ are spherical-like coordinates and the components of
the metric $g_{\mu\nu}$ only depend of $r$ coordinate. We use the signature
$(-,+,+,+)$. The electric potential $A_{\mu}$ of the charged BH is

$A_{\mu}=(A_{t},0,0,0),$ (2)

###  II.1 Motion of charged particles

The equation of motion for a test particle with charge $q$, mass $m$, and
charge mass ratio $e=q/m$, can be obtained from the Euler-Lagrange equations
with the Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+eA_{\alpha}\dot{x}^{\alpha},$
(3)

where the dot means derivative with respect to an affine parameter
$\dot{x}^{\mu}=dx^{\mu}/d\lambda$. In the case of a massless particle
$\lambda$ is a properly chosen affine parameter. For a particle of mass $m$
the affine parameter is $\lambda=\tau/m$, where $\tau$ is the proper time. In
terms of the metric coefficients in Eq. (1), the Lagrangian is

$\mathcal{L}=\frac{1}{2}\left(g_{tt}\dot{t}^{2}+g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}+g_{\phi\phi}\dot{\phi}^{2}\right)+eA_{t}\dot{t},$
(4)

that does not depend explicitly on the $(t,\phi)$ coordinates. Then we can
identify two motion constants of the test particle: its energy and angular
momentum per unit mass $\mathcal{E}$ and $l$ respectively, given by

$l=\frac{L}{m}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=g_{\phi\phi}\dot{\phi},\quad\mathcal{E}=\frac{E}{m}=-\frac{\partial\mathcal{L}}{\partial\dot{t}}=-g_{tt}\dot{t}-eA_{t},$
(5)

where $E$ and $L$ are the energy and angular momentum, respectively. The
motion equations are obtained after solving for $\dot{\phi}$ and $\dot{t}$,

$\dot{\phi}=\frac{l}{g_{\phi\phi}},\quad\dot{t}=-\frac{\mathcal{E}+eA_{t}}{g_{tt}}.$
(6)

and plugging them into the contraction of the four moment
$\dot{x}^{\mu}\dot{x}_{\mu}=-\delta$ ($\delta=1$ for massive particles and
$\delta=0$ for massless particles), obtaining

$\displaystyle\dot{r}^{2}=$
$\displaystyle-\frac{1}{g_{rr}}\left(\frac{\left(\mathcal{E}+eA_{t}\right)^{2}}{g_{tt}}+\frac{l^{2}}{g_{\phi\phi}}+\delta\right).$
(7)

Due to the spherical symmetry, without loss of generality, we restrict
ourselves to the motion on the equatorial plane, then $\theta=\pi/2$ and
$\dot{\theta}=0$.

The radial motion has turning points at $\dot{r}=0$, that is a quadratic
equation in $\mathcal{E}$, whose solutions are the effective potentials,
$\mathcal{E}=V_{\pm}$, given by

$V_{\pm}=\pm\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}l^{2}-g_{tt}\delta}-eA_{t}.$ (8)

Note that the effective potential is different for each particle, i.e. the
particle “i” with angular momentum $l_{i}$ and charge mass ratio $e_{i}$ moves
under the effective potential

$V_{\pm i}=\pm\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}l_{i}^{2}-g_{tt}}-e_{i}A_{t}.$
(9)

Along the paper we analyze the potential corresponding to the particle that
penetrates the horizon, the particle 1.

###  II.2 Generalized ergosphere

The energy extraction from an electrostatic BH is possible if negative energy
states (NES) exist for a test particle; the NES are possible if
$\mathcal{E}=V_{+}<0$.

The region that admits NES is called the“generalized ergosphere”, it is the
region delimited by the event horizon $r_{h}$ and the radius $r_{e}$, defined
from the condition $\mathcal{E}=V_{+}=0$,

$\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}l^{2}-g_{tt}\delta}-eA_{t}=0,$ (10)

and such that $r_{h}<r_{e}$. For non charged test particles, $e=0$, the
condition (10) implies $g_{tt}=0$, equivalently, $r_{h}=r_{e}$, and there is
not ergosphere. Therefore a necessary condition for the existence of NES is
that charged test particles interact with the electrostatic BH; i.e. the
extracted energy is to the expense of the electrostatic BH energy.

Additionality, it can be shown that the maximum region admitting NES is
obtained when the test particle angular momentum is zero $l=0$, from Eq. (10)
this leads to the condition,

$g_{tt}+\left(eA_{t}\right)^{2}=0.$ (11)

Let us consider a charged test particle moving along a timelike geodesic
(particle $0$) that reaches one turning point ($\dot{r}=0$); at that point it
breaks up into two pieces, one of them with negative energy (particle $1$) and
the other one with positive energy (particle $2$). The particle with negative
energy is confined inside the generalized ergosphere until it falls into the
BH, while the other one escapes from the generalized ergosphere with more
energy than the initial one. The $T_{i}$ trajectories are timelike paths
$x_{i}^{\mu}(\lambda)$ parametrized by $\lambda=\tau/m$ where $\tau$ is the
proper time; the incident particle follows the trajectory $T_{0}$, which
starts outside the ergosphere and ends inside it at the break-up point
($r_{*}$,$\theta_{*}$,$\phi_{*}$). From the break-up point emerge two
particles with trajectories labeled as $T_{1}$ for the particle with negative
energy ($E_{1}<0$) that remains in the ergosphere; the trajectory $T_{2}$
corresponds to the particle escaping outside the ergosphere. We denote with
$m_{i}$, $e_{i}$, $\mathcal{E}_{i}$, $l_{i}$ and
$p_{i}^{\mu}=dx^{\mu}_{i}/d\lambda$ to the mass, charge mass ratio, energy and
azimuthal angular momentum per unit mass, and 4-momentum of the $i$ particle,
respectively. These quantities should fulfill the charge and 4-momentum
conservation equations,

$m_{0}e_{0}=m_{1}e_{1}+m_{2}e_{2},$ (12)

and

$p_{0}^{\mu}=p_{1}^{\mu}+p_{2}^{\mu}.$ (13)

At the break-up point the energy is conserved (time 4-momentum component
$p^{t}$); while from the spatial components of the 4-momentum it is derived
the conservation of linear momenta, namely,

$\displaystyle
m_{0}\mathcal{E}_{0}=m_{1}\mathcal{E}_{1}+m_{2}\mathcal{E}_{2},$ (14)
$\displaystyle m_{0}l_{0}=m_{1}l_{1}+m_{2}l_{2}.$ (15)

To determine the efficiency of the Penrose process we need to know the
explicit form of the energies $E_{i}$ of each particle at the break-up point
where $\dot{r}=0$ (turning point). From the radial Eq. (7), we obtain the
energy and angular momentum for the particle $i$ at the break-up point, given
by

$\mathcal{E}_{i}=\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}}\sqrt{l_{i}^{2}+g_{\phi\phi}\delta_{i}}-e_{i}A_{t},\quad
i=0,1,2.$ (16)

and

$l_{i}=\pm\sqrt{-\frac{g_{\phi\phi}}{g_{tt}}}\sqrt{\left(\mathcal{E}_{i}+e_{i}A_{t}\right)^{2}+g_{tt}\delta_{i}}\quad
i=0,1,2.$ (17)

According to Eq. (10), $V_{+}$ and then the NES are independent of the sign of
$l$; thus we can choose $l_{i}$ with any sign. Using Eq. (17), considering
$l_{i}>0$ and Eq. (15) we obtain,

$\displaystyle\sqrt{\left(\mathcal{E}_{0}+e_{0}A_{t}\right)^{2}+g_{tt}\delta_{0}}-\sqrt{\left(\mathcal{E}_{1}+e_{1}A_{t}\right)^{2}+g_{tt}\delta_{1}}$
(18)
$\displaystyle\qquad\qquad-\sqrt{\left(\mathcal{E}_{2}+e_{2}A_{t}\right)^{2}+g_{tt}\delta_{2}}=0;$

solving Eq. (18) and considering Eq. (14) we obtain

$\displaystyle\tilde{E}_{1}=\frac{1}{2}\left(\left(1+\frac{\tilde{\delta}_{1}}{\tilde{\delta}_{0}}-\frac{\tilde{\delta}_{2}}{\tilde{\delta}_{0}}\right)\tilde{E}_{0}+\kappa\sqrt{d_{0}}\right),$
(19)
$\displaystyle\tilde{E}_{2}=\frac{1}{2}\left(\left(1-\frac{\tilde{\delta}_{1}}{\tilde{\delta}_{0}}+\frac{\tilde{\delta}_{2}}{\tilde{\delta}_{0}}\right)\tilde{E}_{0}-\kappa\sqrt{d_{0}}\right),$
$\displaystyle\tilde{E}_{i}=E_{i}+q_{i}A_{t},\quad i=1,2,\quad\kappa=\pm 1,$
$\displaystyle d_{0}=$
$\displaystyle\left(1-2\left(\frac{\tilde{\delta}_{1}+\tilde{\delta}_{2}}{\tilde{\delta}_{0}}\right)+\left(\frac{\tilde{\delta}_{1}-\tilde{\delta}_{2}}{\tilde{\delta}_{0}}\right)^{2}\right)\left(\tilde{E}_{0}^{2}+g_{tt}\tilde{\delta}_{0}\right).$

where $E_{i}=m_{i}\mathcal{E}$, $q_{i}=m_{i}e_{i}$ are the energy and charge
of the $i$-particle, and $\tilde{\delta}_{i}=m^{2}_{i}\delta_{i}$,
$\tilde{\delta}_{i}$ is $m^{2}_{i}$ for massive particles and zero for
massless particles. It is important to remark that the parameter that defines
the generalized ergosphere is the charge-mass ratio of the particle that
penetrates the BH, $e_{1}=q_{1}/m_{1}$, while the generalized ergosphere is
independent on the charge and mass of the ingoing and outgoing particles, $0$
and $2$. The results in DenardoRuffini1972  for the RN BH are recovered with
$\kappa=-1$. We just noticed that in Zaslavski2024  the RN BH energy
extraction is analyzed considering an arbitrary break up point, i.e., not
restricted to be a turning point.

###  II.3 Efficiency of the electric Penrose process

The efficiency $\eta$ of the Penrose process can be defined as the ratio
between the gain in energy and the input energy (energy of the incident
particle),

$\eta=\frac{E_{2}-E_{0}}{E_{0}}=-\frac{E_{1}}{E_{0}}.$ (20)

where the energies $E_{2}$ and $E_{1}$ are given in Eq. (19). Note that
$E_{0}$ implicitly depends on the break-up point $r_{*}$ and on the angular
momentum $L_{0}$, that depends on $r_{*}$ as well.

To obtain a general expression for the efficiency, $\eta$, for an arbitrary
break-up point $r_{*}$, we need to know the energies $E_{0}$, $E_{1}$ and
$E_{2}$ in terms of $r_{*}$. The radial equation, Eq. (7), for a turning point
$\dot{r}=0$, leads to the system of equations,

$g_{\phi\phi}\left(\mathcal{E}_{i}+e_{i}A_{t}\right)^{2}+g_{tt}\left(l_{i}^{2}+g_{\phi\phi}\delta_{i}\right)=0,\quad
i=0,1,2.$ (21)

Substituting the conservation of charge, energy and angular momentum of Eqs.
(12), (14) and (15) into Eq. (21) and considering only massive particles
$\delta_{i}=1$ we obtain three conditions that the parameters of the three
particles involved in the decay should fulfill,

$\displaystyle E_{1}$
$\displaystyle=-q_{1}A_{t}+\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}}\sqrt{L_{1}^{2}+m_{1}^{2}g_{\phi\phi}},\quad
E_{2}=-q_{2}A_{t}+\frac{m_{0}^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}^{2}}\left(E_{1}+q_{1}A_{t}\right)-\frac{\epsilon
D_{0}m_{0}^{2}}{2m_{1}^{2}}\sqrt{\left(E_{1}+q_{1}A_{t}\right)^{2}+m_{1}^{2}g_{tt}},$
(22) $\displaystyle L_{2}$
$\displaystyle=\frac{m_{0}^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}^{2}}L_{1}-\frac{\epsilon
D_{0}m_{0}^{2}}{2m_{1}^{2}}\sqrt{L_{1}^{2}+m_{1}^{2}g_{\phi\phi}},\quad
D_{0}=\sqrt{\left(1-\left(\frac{m_{1}+m_{2}}{m_{0}}\right)^{2}\right)\left(1-\left(\frac{m_{1}-m_{2}}{m_{0}}\right)^{2}\right)},\quad\epsilon=\pm
1.$

where $E_{i}=m_{i}\mathcal{E}_{i}$, $L_{i}=m_{i}l_{i}$ and $q_{i}=m_{i}e_{i}$
are the energy, angular momentum and charge, respectively. The energy $E_{0}$,
angular momentum $L_{0}$ and charge $q_{0}$ are given by the conservation
laws,

$E_{0}=E_{1}+E_{2},\quad L_{0}=L_{1}+L_{2},\quad q_{0}=q_{1}+q_{2}.$ (23)

In Eq. (22) there are six free parameters: the charges $q_{1}$ and $q_{2}$,
the angular momentum $L_{1}$, and the masses $m_{0}$, $m_{1}$ and $m_{2}$.
Additionally, from $D_{0}$ the masses are restricted to,

$m_{0}\geq m_{1}+m_{2}.$ (24)

On the other hand, using Eq. (20) and (22) we can obtain the general
expression for the efficiency of the extraction process, $\eta$, as

$\eta=\frac{2m_{1}^{2}E_{1}}{2m_{1}^{2}\left(q_{1}+q_{2}\right)A_{t}-\left(m_{0}^{2}+m_{1}^{2}-m_{2}^{2}\right)\left(E_{1}+q_{1}A_{t}\right)+\epsilon
D_{0}m_{0}^{2}\sqrt{\left(E_{1}+q_{1}A_{t}\right)^{2}+m_{1}^{2}g_{tt}}},$ (25)

The advantage of Eq. (22) lies in the explicit dependence on the parameters of
the particles involved in the extraction process and the break up point
$r_{*}$, and then we can determine the efficiency at any break up point using
Eq. (25), for an arbitrary static spherically symmetric charged BH. In
contrast to Eq. (19) that is valid for massive and massless particles, Eq.
(25) is only valid for massive particles. Moreover, from Eq. (25) the
generalized ergosphere can be determined from the condition $\eta=0$. The
extracted energy comes from the electric interaction between the charged test
particle and the charged BH, such that if the BH is non-charged, $(A_{t}=0)$,
the efficiency becomes negative, therefore the energy extraction is not
possible; the same as with chargeless test particles $q_{i}=0$. Finally, an
interesting limiting case occurs when the break-up point is located at the
event horizon $r_{*}=r_{h}$; for a static spherically symmetric BH this imply
that $g_{tt}=0$, then Eq. (25) yields,

$\eta_{r_{h}}=-\frac{q_{1}}{q_{0}},$ (26)

it is worth to mention that the efficiency in Eq. (26) is valid for any static
spherically symmetric charged BH, and it occurs if a test particle with
negative energy reaches the event horizon. Moreover it is independent of the
rest of the parameters of the particles and of the mass and charge of the BH.
Additional features of the energies in Eq. (22) and the efficiency in Eq. (25)
are discussed in the next Section for the Reissner-Nordström BH.

###  II.4 The twofold efficiency

From Eq. (22) in case $L_{1}\neq 0$, associated to $\epsilon=\pm 1$, there are
two different values of $L_{2}$ and $E_{2}$, therefore there are as well two
possible efficiencies $\eta_{1}$, $\eta_{2}$, for the same set of free
parameters ($m_{0}$, $m_{1}$, $m_{2}$, $q_{1}$, $q_{2}$, $L_{1}$) at a given
breaking point $r_{*}$.

In contrast, in the case $L_{1}=0$ the process reaches its maximum efficiency
for a given break up point $r_{*}$. In the limit that the break up point
approaches the event horizon $r_{*}\to r_{h}$ the efficiencies converge to the
same limit given by Eq. 26, a value that is independent of $L_{i}$, $m_{i}$
and the BH parameters.

###  II.5 Efficiency with zero angular momentum $L_{1}=0$

As we discussed above, if $L_{1}=0$ ($l_{1}=0$) the generalized ergosphere
reaches its maximum radius, the efficiency is unique, and Eqs. (22) lead to,

$\displaystyle
E_{1}=-q_{1}A_{t}+m_{1}\sqrt{-gtt},\quad|L_{0}|=|L_{2}|=\frac{\sqrt{g_{\phi\phi}}m_{0}^{2}}{2m_{1}}D_{0},$
(27) $\displaystyle\qquad
E_{2}=-q_{2}+\frac{\left(m_{0}^{2}-m_{1}^{2}-m_{2}^{2}\right)}{2m_{1}}\sqrt{-gtt}$

where $E_{0}$ and $L_{0}$ are given by Eq. (23), and the efficiency of Eq.
(25) is,

$\eta=\frac{2m_{1}E_{1}}{2m_{1}(q_{1}+q_{2})A_{t}-\left(m^{2}_{0}+m^{2}_{1}-m^{2}_{2}\right)\sqrt{-g_{tt}}}.$
(28)

####  II.5.1 Efficiency with $D_{0}=0$

Another case where the efficiency has a unique value is when $D_{0}=0$, this
imply that $m_{0}=m_{1}+m_{2}$, and Eqs. (22) lead to,

$\displaystyle
E_{1}=-q_{1}A_{t}+\sqrt{-\frac{g_{tt}}{g_{\phi\phi}}}\sqrt{L_{1}^{2}+m_{1}^{2}g_{\phi\phi}},$
(29) $\displaystyle E_{2}$
$\displaystyle=-q_{2}A_{t}+\frac{m_{2}}{m_{1}}\left(E_{1}+q_{1}A_{t}\right),\quad
L_{2}=\frac{m_{2}}{m_{1}}L_{1},$

where $E_{0}$ and $L_{0}$ are given by Eq. (23); while the efficiency, Eq.
(25) takes the form,

$\eta=\frac{m_{1}E_{1}}{m_{1}\left(q_{1}+q_{2}\right)A_{t}-m_{0}\left(E_{1}+q_{1}A_{t}\right)}.$
(30)

Note that in this case the value of $L_{2}$ is independent of the break-up
point $r_{*}$ and is directly proportional to the angular momentum $L_{1}$.
This condition produces a lower efficiency than for the case $L_{1}=0$.

###  II.6 How the efficiency depends on the masses

As we mentioned above the masses of the particles are restricted to the
condition $m_{0}\geq m_{1}+m_{2}$. If $m_{0}=(m_{1}+m_{2})$ the efficiency is
given by Eq. (30); in this case the efficiency $\eta$ may be larger or smaller
than the efficiency for $m_{0}>(m_{1}+m_{2})$. To understand this effect we
consider the efficiency for fixed $r_{*}$, $m_{0}$ and $m_{1}$; then $m_{2}$
is in the range $0\leq m_{2}\leq m_{0}-m_{1}$. Deriving Eq. (25) with respect
to $m_{2}$ we obtain

$\displaystyle\frac{d\eta}{dm_{2}}=\frac{2\epsilon
m_{2}E_{1}\left(L_{1}+L_{2}\right)D_{0}\sqrt{-g_{tt}g_{\phi\phi}}}{m_{0}^{2}\left(\left(q_{1}+q_{2}\right)A_{t}\sqrt{g_{\phi\phi}}+\epsilon\sqrt{-g_{tt}}C_{0}\right)^{2}},$
(31) $\displaystyle
C_{0}=L_{2}\left(1+\frac{m_{1}^{2}-m_{2}^{2}}{m_{0}^{2}}\right)-L_{1}\left(1-\frac{m_{1}^{2}-m_{2}^{2}}{m_{0}^{2}}\right),$

with the break-up point $r_{*}$ inside the generalized ergosphere it is
guaranteed that $E_{1}<0$ and the sign of the derivative in Eq. (31) is
determined by the factor $-\epsilon\left(L_{1}+L_{2}\right)$. Therefore there
are three possible cases if $m_{0}=m_{1}+m_{2}$. The first case is when
$L_{1}=0$ that leads to $d\eta/dm_{2}>0$, this means that the efficiency
$\eta$ is a monotonically increasing function of $m_{2}$, reaching the maximum
when $m_{0}=m_{1}+m_{2}$ [this is illustrated in Fig. 5 for the RN BH].

For $L_{1}\neq 0$ we can choose $\epsilon=\pm 1$. For $\epsilon=-1$,
$d\eta/dm_{2}>0$, and we conclude again that the efficiency is an increasing
function with respect to $m_{2}$. This means that the maximum efficiency for a
given angular momentum $L_{1}$ and break up point $r_{*}$, is reached when
$m_{2}$ takes its highest possible value that occurs when $m_{0}=m_{1}+m_{2}$.
The last case occurs when the angular momentum $L_{1}\neq 0$ and
$\epsilon=+1$, the sign that the derivative takes depends on the angular
momentum $L_{1}$ and the break-up point $r_{*}$; the sign of the derivative is
negative if,

$g_{\phi\phi}\leq\frac{4L_{1}^{2}}{m_{0}^{2}D_{0}^{2}},$ (32)

in this case the efficiency as a function of $m_{2}$, Eq. (25), is a
decreasing function, hence the minimum efficiency corresponds to
$m_{0}=(m_{1}+m_{2})$, and therefore the maximum efficiency occurs when
$m_{2}$ approaches zero. Nevertheless, when Eq. (32) satisfies the equality a
maximum efficiency for $L_{1}\neq 0$ is reached with $m_{2}$ given by,

$m_{2}=\sqrt{m_{0}^{2}+m_{1}^{2}-\frac{2m_{0}\sqrt{g_{\phi\phi}\left(L_{1}^{2}+m_{1}^{2}g_{\phi\phi}\right)}}{g_{\phi\phi}}}.$
(33)

Note that $m_{2}$, in Eq. (33), takes real values only if the angular momentum
fulfils the condition,

$0<L_{1}^{2}\leq\frac{\left(m_{0}^{2}-m_{1}^{2}\right)^{2}g_{\phi\phi}}{4m_{0}^{2}}.$
(34)

On the other hand, the efficiency will be a strictly decreasing function as we
mentioned above.
The described features of the energy extraction become clear in the next
section where the process for the RN BH is analyzed.

##  III Energy extraction of the Reissner-Nordström BH

The Reissner-Nordström (RN) metric Reissner1916 ; Nordstrom1918  describes a
static spherically symmetric charged BH. The metric is given by (1) with
metric functions

$\displaystyle g_{tt}=-\frac{\Delta}{r^{2}},\quad
g_{rr}=\frac{r^{2}}{\Delta},\quad g_{\theta\theta}=r^{2},$ (35) $\displaystyle
g_{\phi\phi}=r^{2}\sin^{2}{\theta},\quad\Delta=r^{2}-2Mr+Q^{2},$

where $Q$ and $M$ are the electric charge and mass of the BH, respectively.
The electric potential is

$A_{t}=-\frac{Q}{r}.$ (36)

The event horizon $r_{h}$ is given by

$r_{h}=M+\sqrt{M^{2}-Q^{2}}.$ (37)

Considering massive test particles $\delta=1$, the effective potential $V_{+}$
is,

$V_{+}=\frac{\sqrt{\Delta(l^{2}+r^{2})}}{r^{2}}+\frac{eQ}{r}.$ (38)

$V_{+}$ has two contributions, the first one from the gravitational
interaction, that is positive definite; the second one is the electrostatic
interaction that can be positive or negative according to the sign of $eQ$.
Therefore, negative energies ($V_{+}=\mathcal{E}<0$) are obtained if $eQ<0$,
but also is required that the electrostatic interaction dominate over the
gravitational one in the range $r_{h}<r<r_{e}$. And $r_{e}$ is defined from
the condition $V_{+}=0$,

$r^{2}\left(\Delta-Q^{2}e^{2}\right)-l^{2}\Delta=0.$ (39)

Deriving respect to the angular momentum $l$, we find that $r_{e}$ is
maximized if $l=0$. Solving for this condition Eq. (39) leads to
DenardoRuffini1972 ,

$r_{e_{\pm}}=M\pm\sqrt{M^{2}-Q^{2}\left(1-e^{2}\right)},$ (40)

where $r_{e_{+}}$ denotes the boundary of the region where NES can exist,
corresponding to $l=0$. On the other hand, $r_{e_{-}}$ is located within the
event horizon then it is not relevant for the extraction process.

For angular momentum $l\neq 0$ the radius of the generalized ergosphere is
delimited by the event horizon $r_{h}$ and $r_{e_{+}}$ that is given by

$\displaystyle r_{e}$
$\displaystyle=\frac{M}{2}+\frac{\sqrt{\alpha}+\sqrt{\alpha-4\beta}}{2},\quad\beta=\frac{a_{1}+\alpha}{2}+\frac{a_{2}}{2\sqrt{\alpha}},$
(41) $\displaystyle\alpha$
$\displaystyle=M^{2}-2a_{3}+2\sqrt{a_{o}}\cos\Bigr{[}\frac{1}{3}\arccos\left(\frac{b_{o}}{a_{o}^{3/2}}\right)\Bigr{]},$
$\displaystyle b_{o}$
$\displaystyle=a_{3}^{3}+l^{2}\left(4\left(M^{2}-Q^{2}\right)a_{3}+2M^{2}e^{2}Q^{2}\right),$
$\displaystyle a_{o}$
$\displaystyle=a_{3}^{2}-\frac{4l^{2}\left(M^{2}-Q^{2}\right)}{3},\quad
a_{1}=-\frac{3M^{2}}{2}+3a_{3},$ $\displaystyle a_{2}$
$\displaystyle=M\left(3a_{3}-M^{2}-l^{2}\right),\quad
a_{3}=\frac{l^{2}-Q^{2}\left(e^{2}-1\right)}{3}.$

When $l=0$, the solution $r_{e_{+}}$ of Eq. (40) is recovered.

In Fig. 1(a) is shown the effective potential for the particle 1, Eq. (36),
with NES for different values of the angular momentum; the region where
$V_{+}<0$ is $r_{h}<r<r_{e_{+}}$. If the angular momentum increases then
$r_{e}$ decreases, and consequently the generalized ergosphere decreases. In
the limit $|l|\to\infty$ the value of $r_{e}$ tends to $r_{h}$, and the
generalized ergosphere eventually disappears. This happens when the
gravitational term in the effective potential Eq. (38) dominates over the
electrostatic interaction, and the effective potential becomes positive. Note
that the sign of the angular momentum is irrelevant in the effective potential
$V_{+}$. In Fig. 1(b) are shown the effective potentials for different values
of the charge mass ratio $e_{1}$; NES exist only if the condition $eQ<0$ is
satisfied, otherwise the energy extraction is not possible.

Figure 1:  For the RN BH are illustrated the effective potentials with
negative energy states (shaded region) the condition $eQ<0$ is fulfilled with
$Q=0.5M$ and $e=-10$ . The region where NES can exists lies between the event
horizon $r_{h}$ and $r_{e_{+}}$, denoted with the black dots. (a) varying $l$
as shown $l=0,\pm 6,\pm 15$ (b) for different values of charge mass ratio $e$.
The outer radius of the ergosphere $r_{e_{+}}$ is larger for larger (negative)
values of the charge mass ratio $e$. Figure 2:  Illustration of the energy
extraction from an extreme RN BH, $Q=M$. The trajectory $T_{0}$ corresponds to
the ingoing particle coming from infinity; $T_{1}$ is the trajectory of the
particle with negative energy that falls into the black hole and $T_{2}$ is
the trajectory of the outgoing particle that escapes with more energy than the
ingoing particle. The break-up point is at $r_{*}=2M$; the shadow region
represents the generalized ergosphere its outer radius being $r_{e_{+}}=3M$.
The event horizon is $r_{h}=M$. The parameters of the test particles are given
in Table 1.

We illustrate other features of the energy extraction with the extreme RN BH,
that is the RN metric Eq. (35) with $M=Q$; we take the parameters for the
extraction process considered in DenardoRuffini1972 , that satisfy Eq. (19).
When the trajectory $T_{0}$ coming from infinity and entering the ergosphere
reaches a turning point $r_{*}=2M$, the test particle splits into two
particles that follow the trajectories $T_{1}$ and $T_{2}$; $T_{1}$ goes
inside the BH and $T_{2}$ goes back to infinity as is shown in Fig. 2. These
parameters are listed in Table 1.

The case analyzed in DenardoRuffini1972  for a RN BH corresponds to $L_{1}=0$.
For the parameters $q_{i}$ and $m_{i}$ in Table 1, the efficiency is shown as
a function of the break-up point. It is clear that energy extraction is only
possible when the break-up point is located inside the generalized ergosphere,
otherwise the efficiency is negative; this means that if the break-up point is
outside the ergosphere the outgoing particle carries less energy than the
incident particle and there is not energy extraction. In Fig. 3, it is shown
that the maximum efficiency is reached when the break-up point $r_{*}$ is the
closest to the event horizon $r_{h}$; in this case the efficiency is given by
Eq. (26). It is worth to mention that the efficiency for $L_{1}=0$ is larger
than for the case $L_{1}\neq 0$, for a given break-up point in the generalized
ergosphere. The extraction process described in Table 1 has an efficiency of
$\eta=0.1111$.

Table 1: The parameters considered to illustrate the energy extraction from an extreme RN BH, with $Q=M$. The break-up point is at $r_{*}=2M$ and the outer radius of the ergosphere is $r_{e_{+}}=3M$. The event horizon is $r_{h}=M$. This set of parameters satisfy Eq. (19). $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.197 | 6.586 | 4.5 | 2 |
1 | 1 | -2 | -0.5 | 0 |
2 | 1 | 8.586 | 5 | 2 |

In Fig. 3 we illustrate the twofold efficiency with the parameters in Table 1
for $L_{1}=0$, and $L_{1}=2$ with $\epsilon=\pm 1$; the radii $r_{e+}$ and
$r_{e_{1}}$ delimit the generalized ergosphere. We consider other two
scenarios for the parameters in Table 2 and 3 with $L_{1}=0,1,2$.

Additionally, notice that a larger efficiency does not imply that the energy
of the outgoing particle is greater, in fact, from Table 2 and 3, for
$\epsilon=+1$ the energies $E_{0}$ and $E_{2}$ are smaller than for
$\epsilon=+1$ but the efficiency is larger.

Table 2: Parameters of the energy extraction from the extreme RN BH, $Q=M$. The break-up point is at $r_{*}=2M$ and the radius of the ergosphere is $r_{e_{1}}=2.581M$. The event horizon is $r_{h}=M$. These parameters fulfill Eq. (22) with $\epsilon=-1$. $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.197 | 6.586 | 5.49896 | 7.6512 |
1 | 1 | -2 | -0.292893 | 2 |
2 | 1 | 8.586 | 5.79185 | 5.6512 |
Table 3: Parameters for the energy extraction from an extreme RN BH, $Q=M$. The break-up point is at $r_{*}=2M$ and the radius of the ergosphere is $r_{e_{1}}=2.581M$; the event horizon is $r_{h}=M$. These parameters satisfy Eqs. (22) with $\epsilon=1$. $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.197 | 6.586 | 4.50011 | 2.00162 |
1 | 1 | -2 | -0.29289 | 2 |
2 | 1 | 8.586 | 4.793 | 0.00162 |

It is important to mention that the two possible efficiencies for a given
value of $L_{1}$ correspond to the two values that the angular momentum
$L_{2}$ can take; for RN BH we can understand this dependence examining Eq.
(22) for $L_{2}$. In the RN BH case $L_{2}$ and break up point $r_{*}$ satisfy
a hyperbola equation,

$\left(L_{2}-\frac{m_{0}^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}^{2}}L_{1}\right)^{2}-\frac{m_{0}^{4}D_{0}^{2}}{4m_{1}^{2}}r_{*}^{2}=\frac{m_{0}^{4}D_{0}^{2}}{4m_{1}^{4}}L_{1}^{2}.$
(42)

In Fig. 4 it is shown that when $L_{1}\neq 0$ there are two values of the
angular momentum $L_{2}$ for a given break-up point $r_{*}$; therefore there
are two associated values of $E_{2}$ and $\eta$. On the other hand, when
$L_{1}=0$, $L_{2}$ (and therefore $L_{0}$) depends linearly on $r_{*}$; also
note that if $L_{1}=0$ although $L_{2}$ (with $L_{1}=0$) can take two values,
both have the same magnitude, namely, there is no split of efficiency for
$L_{1}=0$.

Figure 3:  Efficiency of the extraction process from RN BH via electrostatic
interaction as a function of the break-up point $r_{*}$ for fixed parameters
$Q=M$, $m_{0}=2.197$, $m_{1}=1$, $m_{2}=1$, $q_{0}=6.586$, $q_{1}=-2$ and
$q_{2}=8.586$, the charge mass ratio for each particle is $e_{i}=q_{i}/m_{i}$.
The radii of the event horizon and of the generalized ergosphere are
$r_{h}=M$, and $r_{e+}=3M$ and $r_{e_{1}}=2.581M$ for $L_{1}/M=0$ and
$L_{1}/M=2$, respectively. Figure 4:  The angular momentum of the outgoing
particle, $L_{2}$ as a function of $r_{*}$, in the energy extraction from an
extreme RN BH. For $L_{1}=0$ $L_{2}$ depends linearly on $r_{*}$. On the other
hand if $L_{1}\neq 0$ $L_{2}$ has two values, and two different scenarios
arise with different efficiencies. The fixed parameters are $Q=M$,
$m_{0}=2.197$, $m_{1}=1$, $m_{2}=1$, $q_{0}=6.586$, $q_{1}=-2$ and
$q_{2}=8.586$. The dashed line represents the event horizon $r_{h}=M$. Figure
5:  The efficiency as a function of $m_{2}$ for $L_{1}=0$ in the extreme RN
BH, $Q=M$. The radius of the event horizon is $r_{h}=M$. The parameters of the
test particles are $m_{0}=2.197$, $m_{1}=1$, $q_{1}=-2$ and $q_{2}=8.586$.
$m_{2}$ is in the range $0\leq m_{2}\leq(m_{0}-m_{1})$. The efficiency takes
its maximum value when $m_{0}=m_{1}+m_{2}$.

The dependence of the efficiency on the masses for RN BH is illustrated in
Figs.5 and 6. In Fig. 6 is shown the efficiency for a given break-up point
$r_{*}$ and angular momentum $L_{1}$ for $\epsilon=\pm 1$. In both figures we
can observe that when $\epsilon=-1$ the efficiency is an increasing function
that reaches its maximum value when $m_{0}=m_{1}+m_{2}$. In the case
$\epsilon=+1$, if the angular momentum does not satisfy Eq. (34) the
efficiency is a decreasing function and its minimum occurs for
$m_{0}=m_{1}+m_{2}$, as is shown in Fig. 6a. If the angular momentum $L_{1}$
fulfills Eq. (34) the efficiency reaches its maximum value when $m_{2}$
satisfies Eq. (33). In the case $m_{0}=m_{1}+m_{2}$, the efficiencies for
$\epsilon=\pm 1$ tend to the one given in Eq. (30).

Figure 6:  The efficiency as a function of $m_{2}$ for the extreme RN BH,
$M=Q$. The radius of the event horizon is $r_{h}=M$. The fixed parameters of
the particles are $m_{0}=2.197$, $m_{1}=1$, $q_{1}=-2$ and $q_{2}=8.586$.
While $m_{2}$ is in the range $0\leq m_{2}\leq m_{0}-m_{1}$. (a) For $L_{1}=2$
the efficiency with $\epsilon=+1(\epsilon=-1)$ is a decreasing (increasing)
function and the minimum (maximum) occurs for $m_{0}=m_{1}+m_{2}$, black
(blue) curve. (b) For $L_{1}=1$, the efficiency for $\epsilon=-1$ is an
increasing function (blue line) an its maximum occurs for $m_{0}=m_{1}+m_{2}$,
while for $\epsilon=+1$ the maximum occurs for $m_{2}=0.956121$ (in the zoom).

##  IV Reissner Nordström de Sitter Black Hole, $\Lambda\geq 0$

The spacetime that describes a Reissner Nordström BH with cosmological
constant $\Lambda$ is given by Eq. (1) with metric coefficients in coordinates
($t$, $r$, $\theta$, $\phi$) are Cai2002

$g_{tt}=-\frac{\Delta}{r^{2}},\quad g_{rr}=\frac{r^{2}}{\Delta},\quad
g_{\theta\theta}=r^{2},\quad g_{\phi\phi}=r^{2}\sin^{2}\theta,$ (43)

where

$\Delta=r^{2}-2Mr+Q^{2}-\frac{\Lambda}{3}r^{4},$ (44)

and $M$ and $Q$ are the mass and charge of the BH, and $\Lambda$ is the
cosmological constant. The RN-de Sitter (RNdS) and RN-anti-de Sitter (RN-AdS)
correspond to positive or negative cosmological constant, respectively, and is
non-asymptotically flat. The event horizons are given by the solution of
$g_{rr}^{-1}=0$, equivalently $\Delta=0$, or $g_{tt}=0$. In contrast with the
horizons of RN BH, Eq. (37), the RN-$\Lambda$ BH has the possibility of two or
three horizons depending on the sign of $\Lambda$.

###  IV.1 Horizons of the RNdS BH

From $\Delta=0$ are determined the cosmological, outer and inner horizons,
$r_{c}$, $r_{h}$, $r_{-}$, respectively,

$\Delta=r^{2}-2Mr+Q^{2}-\frac{\Lambda}{3}r^{4}=0,$ (45)

the roots given by

$\displaystyle r_{c}=\alpha+\sqrt{\alpha^{2}-\beta_{+}},\qquad\quad
r_{h}=\alpha-\sqrt{\alpha^{2}-\beta_{+}},$ (46) $\displaystyle
r_{-}=-\alpha+\sqrt{\alpha^{2}-\beta_{-}},\qquad\beta_{\pm}=2\alpha^{2}-3A_{1}\pm\frac{3M}{2\alpha\Lambda},$
$\displaystyle A_{1}=\frac{1}{2\Lambda},\qquad
A_{2}=\frac{1-4Q^{2}\Lambda}{16\Lambda^{2}},\qquad
A_{3}=-\left(\frac{3M}{4\Lambda}\right)^{2},$
$\displaystyle\alpha=\sqrt{A_{1}+2\sqrt{A_{2}}\cos\Bigr{[}\frac{1}{3}\arccos\left(\frac{A_{1}\left(3A_{2}-A_{1}^{2}\right)-A_{3}}{2A_{2}^{3/2}}\right)\Bigr{]}}.$

This roots 444The fourth root is negative,
$r_{--}=-\alpha-\sqrt{\alpha^{2}-\beta_{-}}$. satisfy the relation
$0<r_{-}<r_{h}<r_{c}$. Fig. 7 exhibits the three horizons for RNdS BH for
fixed $M$, $Q$ and positive $\Lambda$, showing also the RN BH case with two
horizons, Eq. (37).

Figure 7:  $\Delta(r)$ for RN BH (dashed line) and RNdS (solid line) BH. The
parameters have been fixed as $Q=0.5M$ and $\Lambda=0.1/M^{2}$. The dots
correspond to the location of the cosmological horizon $r_{c}=3.8749M$, the
event horizon $r_{h}=2.2926M$ and the inner horizon $r_{-}=0.1339M$.

####  IV.1.1 Degeneracy of the horizons

According to Eq.(37), the RN BH has two horizons when $Q\leq M$. If $M=Q$,
there is a degenerate horizon $r_{h}=r_{-}$ and Eq. (35) describes an extreme
BH. When $M<Q$ there are no horizons, and Eq. (35) exhibits a naked
singularity. However, for RNdS BH the introduction of an additional parameter,
$\Lambda>0$, changes this situation and the relative magnitudes of $M$, $Q$
and $\Lambda$ define the number of horizons. This deserves a deeper analysis
that we carry out in what follows. We distinguish three cases of degeneracy
following the notation in Sharmanthie2013 .

####  IV.1.2 Case I: Degeneration of the event horizon and cosmological
horizon $r_{c}=r_{h}$

The first case of degeneracy is the charged Nariai BH that occurs when the
event horizon and cosmological horizon coincide $r_{h}=r_{c}$, in this case
the three horizons of Eq. (46) take the explicit form,

$\displaystyle r_{h}=r_{c}=\sqrt{A_{1}+2\sqrt{A_{2}}},\quad
r_{-}=-r_{h}+2\sqrt{A_{1}-\sqrt{A_{2}}},$ (47) $\displaystyle
A_{1}=\frac{1}{2\Lambda},\quad
A_{2}=\frac{1-4Q^{2}\Lambda}{16\Lambda^{2}},\quad M=\frac{4\Lambda
r_{h}\left(A_{1}-\sqrt{A_{2}}\right)}{3}.$

Taking the mass of the BH $M$ as a free parameter it is possible to establish
the range for the cosmological constant $\Lambda$ and charge $Q^{2}$ for RNdS
BH,

$0\leq\Lambda\leq\frac{6\left(M+\sqrt{9M^{2}-8Q^{2}}\right)}{\left(3M+\sqrt{9M^{2}-8Q^{2}}\right)^{3}},\quad
0\leq Q^{2}\leq\frac{9}{8}M^{2}.$ (48)

####  IV.1.3 Case II: Extreme BH: Coalesced event and inner horizons
$r_{h}=r_{-}$

The second case of degeneracy occurs when the event and inner horizon coincide
$r_{h}=r_{-}$; this represents an extreme BH surrounded by a cosmological
horizon, and the explicit expressions of the horizons are,

$\displaystyle r_{c}=-r_{h}+2\sqrt{A_{1}+\sqrt{A_{2}}},\quad
r_{h}=r_{-}=\sqrt{A_{1}-2\sqrt{A_{2}}},$ (49) $\displaystyle
M=\frac{4}{3}\Lambda r_{h}\left(A_{1}+\sqrt{A_{2}}\right),\quad
A_{1}=\frac{1}{2\Lambda},\quad A_{2}=\frac{1-4Q^{2}\Lambda}{16\Lambda^{2}}.$

The expansion in series around of $\Lambda\ll 1$ in the mass condition of Eq.
(49), leads to

$\frac{M}{Q}=1-\frac{Q^{2}}{3}\left(\frac{\Lambda}{2}\right)-\frac{Q^{4}}{2}\left(\frac{\Lambda}{2}\right)^{2}-\frac{3Q^{6}}{2}\left(\frac{\Lambda}{2}\right)^{3}+\mathcal{O}\left(\left(\frac{\Lambda}{2}\right)^{4}\right),$
(50)

where is evident that for $\Lambda=0$, the condition of the extreme RN BH
$M=Q$ is recovered; however, for $\Lambda\neq 0$, Eq. (50) suggests that the
charge $Q$ can exceed the mass $M$, $Q>M$, for and extreme RNdS BH.

####  IV.1.4 Case III: Three degenerate horizons $r_{c}=r_{h}=r_{-}$

The third case occurs when the three horizons coincide $r_{c}=r_{h}=r_{-}$; in
this scenario the horizons are given by,

$r_{c}=r_{h}=r_{-}=\sqrt{\frac{1}{2\Lambda}},$ (51)

when the three horizons degenerate into one, the value of the single horizon
can be written down in terms of a single parameter $M$, $Q$ or $\Lambda$,
using the relations

$Q=\frac{1}{2\sqrt{\Lambda}},\quad M=\sqrt{\frac{1}{2\Lambda}}.$ (52)

###  IV.2 Generalized ergosphere of the RNdS BH for zero angular momentum

To obtain the generalized ergosphere is necessary to determine the regions
where the effective potential in Eq. (8) becomes negative; the condition to
solve is Eq. (10). As we discussed in Sec. II.2 the maximum radius of the
generalized ergosphere is reached when the angular momentum is zero, then
imposing $l=0$ and using Eqs. (10) and (43) it is obtained

$r^{2}-2Mr+\mathcal{Q}-\frac{\Lambda}{3}r^{4}=0,\quad\mathcal{Q}=Q^{2}\left(1-e^{2}\right),$
(53)

Figure 8:  The effective potential for RNdS BH; the BH parameters fixed as
$Q=0.5M$, $\Lambda=0.01/M^{2}$, and for the test particle $m=1$, $q=-10$. The
grey area is the ergosphere. (a) The vertical dot-dashed line represents the
event horizon $r_{h}$ of RNdS BH; the dots are the radii of the generalized
ergosphere for RN BH and RNdS BH labelled as $r_{e}$ and $r_{e\Lambda}$,
respectively. The introduction of $\Lambda$ increases the outer radius of the
generalized ergosphere; the largest ergosphere is for $L=0$ (b) $\Lambda$
produces two regions where negative energy states are allowed; one first
generalized ergosphere in the spherical shell $r_{h}<r<r_{eg_{1}}$ and a
second one, a cosmological ergosphere, delimited by $r_{eg_{2}}<r<r_{c}$.

this condition has the same form as $\Delta=0$ that give us the horizon of the
RNdS BH with the change $Q^{2}\to\mathcal{Q}$. Then the horizons given by Eqs.
(46) are valid for the generalized ergosphere changing $Q^{2}\to\mathcal{Q}$.
Since the radius of the generalized ergosphere must be limited by the event
horizon $r_{h}$ and the cosmological horizon $r_{c}$, the only physical
solutions is,

$\displaystyle r_{eg_{1}}=\alpha-\sqrt{\alpha^{2}-\beta_{+}},\qquad\quad
r_{eg_{2}}=\alpha+\sqrt{\alpha^{2}-\beta_{+}},$ (54)
$\displaystyle\beta_{\pm}=2\alpha^{2}-3A_{1}\pm\frac{3M}{2\alpha\Lambda},\qquad\mathcal{Q}=Q^{2}\left(1-e^{2}\right),$
$\displaystyle A_{1}=\frac{1}{2\Lambda},\qquad
A_{2}=\frac{1-4\mathcal{Q}\Lambda}{16\Lambda^{2}},\qquad
A_{3}=-\left(\frac{3M}{4\Lambda}\right)^{2},$
$\displaystyle\alpha=\sqrt{A_{1}+2\sqrt{A_{2}}\cos\Bigr{[}\frac{1}{3}\arccos\left(\frac{A_{1}\left(3A_{2}-A_{1}^{2}\right)-A_{3}}{2A_{2}^{3/2}}\right)\Bigr{]}}.$

Fig. 8 displays the effective potential for RNdS BH; Fig. 8(a) shows that the
radius of the ergosphere $r_{eg_{1}}$ of the RNdS BH increases in presence of
the cosmological constant in relation with the generalized ergosphere for RN
BH, $r_{e+}$. Moreover it is shown that the largest radius of the ergosphere
is reached when the angular momentum is $l=0$. In contrast with the Kerr BH or
the RN BH where there is only one ergoregion, the existence of two radius in
Eq. (54) implies two ergoregions with NES as shown in Fig. 8(b); the first of
them is the generalized ergosphere of BH in $r_{h}<r<r_{eg_{1}}$. The second
one is due to the presence of the cosmological constant $\Lambda>0$ and is
$r_{eg_{2}}<r<r_{c}$, we will refer to this region as the cosmological
esgosphere. The existence of two ergospheres is discussed in Sourav2018  for
the Kerr-de Sitter spacetime with chargeless particles.

Similar to the case of the RN BH generalized ergosphere, the ergosphere in the
RNdS BH depends on the charge mass ratio $e$ of the splitting particle, but an
important difference lies in the fact that the cosmological constant generates
a second region where NES are allowed, and both regions increase if $e$
increases, and decrease in the opposite case. An interesting scenario occurs
when $e$ is large enough and both ergospheres increase until
$r_{eg_{1}}=r_{eg_{2}}$ and the effective potential is negative for any $r$ in
the domain $r_{h}<r<r_{c}$. The critic value $e_{crit}$ that produces
$r_{eg_{1}}=r_{eg_{2}}$ is defined by the solution of

$\displaystyle\qquad\quad\mathcal{P}=0,\quad\frac{d\mathcal{P}}{dr}=0,$ (55)
$\displaystyle\mathcal{P}$
$\displaystyle=g_{\phi\phi}\left(\mathcal{E}+eA_{t}\right)^{2}+g_{tt}\left(l^{2}+g_{\phi\phi}\delta\right).$

Considering the condition that maximizes the radii of both ergospheres, namely
$l=0$, and imposing $E=0$ that implies that $r$ is the radius of the
ergosphere, the solution of Eq. (55) for the critic charge $e_{crit}$ is,

$e_{crit}=-\sqrt{1+\frac{r_{crit}(r_{crit}-3M)}{2Q^{2}}},$ (56)

where $r_{crit}$ is the value of $r$ that makes
$r_{crit}=r_{eg_{1}}=r_{eg_{2}}$ and is given by,

$r_{crit}=\sqrt{\frac{2}{\Lambda}}\cos\left(\frac{1}{3}\arccos\left(-3M\sqrt{\frac{\Lambda}{2}}\right)\right).$
(57)

We can distinguish three different scenarios according to the value of $e$
respect to $e_{crit}$, that are shown in Fig. 9. The first case occurs when
$e<e_{crit}$ and is shown in Fig. 9 (a) where the effective potential exhibits
two regions (gray areas) with NES, one of them delimited by $r_{h}$ and
$r_{eg_{1}}$, and the other one by $r_{eg_{2}}$ and $r_{c}$. In Fig. 9(b) is
illustrated the generalized ergosphere for the case $e<e_{crit}$ in the
equatorial plane. The second case occurs when the charge mass ratio $e_{crit}$
is reached by the test particle, in this scenario the radii of both
ergospheres coalesce at $r_{crit}=r_{eg_{1}}=r_{eg_{2}}$.

Figure 9: . The effective potential for RNdS BH in the cases: $e<e_{crit}$,
$e=e_{crit}$ and $e>e_{crit}$ are shown. The parameters are fixed as $Q=0.5M$,
$\Lambda=0.05/M^{2}$ and $l=0$; $e_{crit}=-4.4056$, the dots represent the
event horizon and cosmological horizon, $r_{h}=2.0113M$ and $r_{c}=6.4651M$,
respectively. (a) The effective potential for $e=-3.5$, the black dots
represent the radii of the generalized ergosphere where $r_{eg_{1}}=3.5297M$
and $r_{eg_{2}}=5.9005M$, (b) the generalized ergospheres $e=-3.5$ in the
equatorial plane. (c) The effective potential for $e=e_{crit}=-4.4056$, the
black dot represents the outer radius of the ergosphere that is degenerate in
$r_{crit}=4.88445$; (d) the generalized ergospheres for $e=e_{crit}=-4.4056$
in the equatorial plane. (e) The effective potential with $e=-6$, NES are
allowed in the range $r_{h}<r<r_{c}$ (f) the generalized ergospheres for
$e=-6$ in the equatorial plane. The gray area represents the domain of $r$
where NES can exist.

Fig. 9(c) displays the effective potential with allowed NES in the range
$r_{h}<r<r_{c}$, except at $r_{crit}$ where the potential vanishes. In Fig.
9(d) is shown the generalized ergosphere for the case $e=e_{crit}$ in the
equatorial plane. The third case occurs when the charge mass ratio $e$ exceeds
the value of $e_{crit}$, Figs. 9(e) and 9(f) exhibit the effective potential
and generalized ergosphere in the equatorial plane when $e>e_{crit}$, in this
case the potential is negative in the whole range $r_{h}<r<r_{c}$, in this
region the NES are allowed.
In Fig. 10 is shown how $e_{crit}$ depends on the cosmological constant
$\Lambda$, given by Eq. (48). When $\Lambda=0$ we recover the case of RN BH
(without cosmological horizon). The divergence of $|e_{crit}|$ occurs when the
generalized ergosphere covers the whole vicinity of the RNdS BH case.
Moreover, for $\Lambda\neq 0$ the value of $|e_{crit}|$ decreases
monotonically until the inequality in Eq. (48) is saturated, namely,

$\Lambda=\frac{6\left(M+\sqrt{9M^{2}-8Q^{2}}\right)}{\left(3M+\sqrt{9M^{2}-8Q^{2}}\right)^{3}},$
(58)

where $|e_{crit}|=0$, and the region delimited by the event horizon $r_{h}$
and the cosmological horizon $r_{c}$ diminishes.

Figure 10:  $e_{crit}$ as a function of $\Lambda$. The left vertical line
corresponds to $|e_{crit}|$ for RN BH ($\Lambda=0$). The right line correspond
to $|e_{crit}|$ when the event horizon and cosmological horizon are the same.
We fixed $Q=0.5M$.

Finally, in the case $l\neq 0$ there are as well two regions with NES, while
the condition to determine the radius of the generalized ergosphere and the
cosmological ergosphere is,

$r^{2}\left(\Delta-Q^{2}e^{2}\right)+l^{2}\Delta=0,$ (59)

where $\Delta$ is given by Eq. (45), and Eq. (59) is a sixth degree polynomial
in $r$ that we solve numerically.

###  IV.3 Efficiency of the energy extraction in the RNdS BH

Considering Eq. (25) and $g_{tt}$, Eq. (43), we analyze the efficiency of the
energy extraction from RNdS BH. The first scenario we consider is the case
when the ergosphere reaches it maximum for a particle with charge $e_{1}$ and
$L_{1}=0$. The efficiency in terms of the masses and charges of the particles
involved in the process for an arbitrary break-up point $r_{*}$ can be
determined with Eq. (28); it is only necessary to satisfy the conditions
$m_{0}\geq m_{1}+m_{2}$ and $e_{1}Q<0$.

Figure 11:  The efficiency of the extraction process as a function of the
break up point $r_{*}$ for RNdS BH for different values of $\Lambda$. The dot-
dashed lines represent the event horizons $r_{h}$ and cosmological horizons
$r_{c}$. The fixed parameters are $Q=0.5M$, $m_{0}=2.3$, $m_{1}=1$,
$m_{2}=1.1$, $q_{0}=4.5$, $q_{1}=-3.5$ and $q_{2}=8$. For $\Lambda=0$ the BH
presents one event horizon at $r_{h}=1.86603M$ and the ergosphere radius is
$r_{e+}=2.95256M$. For $\Lambda=0.04/M^{2}$ the BH has one event horizon at
$r_{h}=1.97645M$ and a cosmological horizon at $r_{c_{1}}=7.425615M$; moreover
the radii of the generalized ergosphere are $r_{eg_{1}}=3.33859M$ and
$r_{eg_{2}}=7.02733M$. For $\Lambda=0.05/M^{2}$ the BH has one event horizon
at $r_{h}=2.01131M$ and a cosmological horizon at $r_{c_{2}}=6.46512M$,
moreover the radii of the generalized ergosphere are $r_{eg_{3}}=3.52977M$ and
$r_{eg_{4}}=5.90049M$.

The effect produced by the cosmological constant in the efficiency as a
function of $r_{*}$ is shown in Fig. 11 for different values of $\Lambda$: (i)
$\Lambda=0$ (blue curve) corresponds to the efficiency of RN BH, the region
where the efficiency $\eta$ is positive lies in the region
$r_{h}<r_{*}<r_{e+}$, where $r_{h}=1.86603M$ is the event horizon and
$r_{e+}=2.95256M$ is the radius of the generalized ergosphere. (ii) For
$\Lambda=0.04/M^{2}$, we can identify two regions where $\eta>0$,
$r_{h}<r_{*}<r_{eg_{1}}$ and $r_{eg_{2}}<r_{*}<r_{c_{1}}$, where
$r_{h}=1.97645M$, $r_{c_{1}}=7.425615M$, $r_{eg_{1}}=3.33859M$ and
$r_{eg_{2}}=7.02733M$, that correspond to the regions where NES are allowed.
(iii) For $\Lambda=0.05/M^{2}$, there are two regions where $\eta>0$,
$r_{h}<r_{*}<r_{eg_{3}}$ and $r_{eg_{4}}<r_{*}<r_{c_{2}}$, where
$r_{h}=2.01131M$, $r_{c_{2}}=6.46512M$, $r_{eg_{3}}=3.52977M$ and
$r_{eg_{4}}=5.90049M$. Note that $\eta>0$ grows as $\Lambda$ increases.

Analysing how $\Lambda$ affects the efficiency of the energy extraction, we
highlight the existence of two regions where the efficiency is positive for
$\Lambda>0$ in constrast with the case $\Lambda=0$, with only one region with
$\eta>0$; consequently, for $\Lambda>0$ there are two break up points $r_{*}$
with the same efficiency. On the other hand, if $r_{*}$ is outside of the
generalized ergosphere then the outgoing particle has less energy than the
ingoing particle and the efficiency becomes negative. Additionally,
$\Lambda>0$ makes the extraction process more efficient at any break-up point
where the NES are allowed in contrast to the case $\Lambda=0$; as $\Lambda$
increases the regions where the efficiency is positive also increase. Finally,
it is worth to mention that the maximum efficiency is reached when the break-
up point $r_{*}$ is located at the event horizon or the cosmological horizon;
the efficiency at those points is independent of $\Lambda$ and the rest of the
BH parameters $M$ and $Q$, and is given by $\eta_{r_{h}}=-q_{1}/q_{0}$,
Eq.(26).
Analogously to Sec. IV.2, we can distinguish three scenarios for the RNdS BH
according to the value of charge mass ratio $e_{1}$ and $L_{1}=0$. In Fig. 12,
are illustrated the three possible cases: (i) $e_{1}<e_{crit}$ with
$\Lambda>0$ there are two region where the energy extraction is possible. In
this scenario $r_{crit}$ is located outside of the ergospheres and represents
the break up point where the outgoing particle has less energy than the
incident particle.

Figure 12:  It is shown the efficiency $\eta$ for the case $e_{1}<e_{crit}$,
as a function of the break up point $r_{*}$. the rest of the parameters are
fixed: $Q=0.5M$, $\Lambda=0.05/M^{2}$, $m_{0}=2.3$, $m_{1}=1$, $m_{2}=1.1$,
$q_{0}=4.5$, $q_{2}=8$, and $q_{1}=e_{1}=-3.5$. The critic charge and critic
radii are $e_{crit}=-4.4056$ and $r_{crit}=4.8844$, respectively; while the
event horizon and cosmological horizon are $r_{h}=2.0113$ and $r_{c}=6.4651$.

(ii) $e_{1}=e_{crit}$, see Fig. 13, in this scenario both ergospheres join at
$r_{crit}$ and the entire region delimited by the event horizon $r_{h}$ and
the cosmological horizon $r_{c}$ is viable for the extraction process, except
the point $r_{crit}$ where the efficiency vanishes.

Figure 13:  The efficiency $\eta$ versus the break up point $r_{*}$ for RNdS
in the case $e_{1}=e_{crit}$. The rest of the parameters fixed as $Q=0.5M$,
$\Lambda=0.05/M^{2}$, $m_{0}=2.3$, $m_{1}=1$, $m_{2}=1.1$, $q_{0}=3.5944$,
$q_{2}=8$, and $q_{1}=e_{1}=-4.4056$. The critic charge and critic radius are
$e_{crit}=-4.4056$ and $r_{crit}=4.8844$, respectively. Both ergospheres join
at $r_{crit}$. The event horizon and cosmological horizon are $r_{h}=2.0113$
and $r_{c}=6.4651$. Figure 14:  The efficiency $\eta$ as a function of the
break up point $r_{*}$ for RNdS in the case $e_{1}>e_{crit}$. We fixed
$Q=0.5M$, $\Lambda=0.05/M^{2}$, $m_{0}=2.3$, $m_{1}=1$, $m_{2}=1.1$,
$q_{0}=2$, $q_{2}=8$, and $q_{1}=e_{1}=-6$. The critic charge and critic
radius are $e_{crit}=-4.4056$ and $r_{crit}=4.8844$, respectively. The
efficiency is positive in the whole region and $r_{crit}$ corresponds to the
minimum efficiency. The event horizon and cosmological horizon are
$r_{h}=2.0113$ and $r_{c}=6.4651$.

(iii) $e_{1}>e_{crit}$, in this scenario the efficiency $\eta$ is always
positive for any $r$ between the horizons, see Fig. 14. In the three cases
$r_{crit}$ corresponds to a minimum efficiency that surprisingly is
independent of $Q$, according to Eq. (57).
So far we have discussed the case $L_{1}=0$; nevertheless, Eq. (25) allows
$L_{1}\neq 0$; in this case there are two possible efficiencies, as we discuss
in Sec. II.4. Fig. 15 displays the efficiency of the energy extraction from
RNdS BH showing a duplicity for $L_{1}\neq 0$; one can see that the efficiency
is larger for $\epsilon=+1$ than for $\epsilon=-1$, as in the RN BH; while
$L_{1}\neq 0$ diminishes the generalized ergosphere in contrast to $L_{1}=0$.

Figure 15:  The efficiency of the extraction process as a function of the
break up point $r_{*}$ for RNdS BH for different values of $L_{1}$. The
parameters fixed as $Q=0.5M$, $m_{0}=2.3$, $m_{1}=1$, $m_{2}=1.1$,
$q_{0}=4.5$, $q_{1}=-3.5$, $q_{2}=8$ and $\Lambda=0.04/M^{2}$. For $L_{1}=0$
the generalized ergospheres are $r_{h}<r<r_{eg_{1}}$ and $r_{eg_{2}}<r<r_{c}$
and they reach their maxima in contrast with the case $L_{1}\neq 0$. For
$L_{1}/M=2$ there are two possible results for the efficiency $\eta$: for
$\epsilon=+1$ $\eta$ is larger than for $\epsilon=-1$. The ergospheres are
defined by $r_{h}<r<r_{eg_{3}}$ and $r_{eg_{4}}<r<r_{c}$. The event horizon
and cosmological horizon are given by $r_{h}=1.97645M$ and $r_{c}=7.425615M$,
respectively. The radius of the ergospheres are $r_{eg_{1}}=3.33859M$ and
$r_{eg_{2}}=7.02733M$ for $L_{1}=0$; $r_{eg_{3}}=2.97645M$ and
$r_{eg_{4}}=7.06201M$ for $L_{1}=2$.

###  IV.4 Examples of the energy extraction from RNdS

Once we have described the efficiency $\eta$, in this subsection some concrete
examples of the energy extraction in RNdS illustrate two remarkably
characteristics, the first one is the same resulting efficiency for different
break up points $r_{*}$ and the second one is the possibility of a different
efficiency for the same break up point $r_{*}$. In Fig. 16 is illustrated the
extraction process from a RNdS BH; the parameters of the BH, $Q$ and $\Lambda$
are fixed and $L_{1}=0$ then the efficiency is unique, and we consider that
the break up point is inside the generalized ergosphere,
$r_{h}<r_{*}<r_{eg_{1}}$ or in the cosmological ergosphere
$r_{eg_{2}}<r_{*}<r_{c}$; $T_{0}$ and $T_{2}$ describe the trajectory of the
ingoing and the outgoing particle, while $T_{1}$ describes the trajectory of
the particle with negative energy. In Fig. 16 are shown two cases at two
different break up points, where both cases have the same efficiency.
Recalling that the efficiency is a ratio between the energy gained by the
outgoing particle and the energy of the incident particle, then, although the
efficiency is the same, the energy extracted from the BH is different, as
Table 4 indicates, along the values of the rest of the parameters involved.

Figure 16:  Two examples of energy extraction from RNdS BH with the same efficiency $\eta=0.1$ for two different processes, corresponding to two different break up points $r_{1_{*}}=2.262794$ and $r_{2_{*}}=6.406015M$. The BH parameters are $Q=0.8M$ and $\Lambda=0.05/M^{2}$. The outer radius of the generalized ergosphere is $r_{eg_{1}}=3.1222M$ and the inner radius of the cosmological ergosphere is $r_{eg_{2}}=6.1027M$; the event horizon is $r_{h}=1.70853M$ and the cosmological horizon is $r_{c}=6.5188M$; the shaded areas are the ergoregions. Table 4: Example of parameters for energy extraction from an extreme RNdS BH, characterized by $Q=0.8M$ and $\Lambda=0.05/M^{2}$, with an efficiency $\eta=0.1$. The break-up point is $r_{*}=2.262794M$ and the radius of the ergosphere is $r_{eg_{1}}=3.1222M$, while the cosmological ergosphere is at $r_{eg_{2}}=6.1027M$. The event horizon $r_{h}=1.70853M$ and the cosmological horizon $r_{c}=6.5188M$. These parameters generate the trajectories $T_{i}$ in Fig. 16(a). $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.197 | 6 | 3.12383 | 2.43878 |
1 | 1 | -2 | -0.31238 | 0 |
2 | 1 | 8 | 3.43621 | 2.43878 |
Table 5: A set of parameters for the energy extraction from an extreme RNdS BH with efficiency $\eta=0.1$. The BH parameters are $Q=0.8M$ and $\Lambda=0.05/M^{2}$. The break-up point is $r_{*}=6.406015M$ and the radius of the ergosphere and cosmological horizons are $r_{eg_{1}}=3.1222M$ and $r_{eg_{2}}=6.1027M$, respectively. The event horizon is $r_{h}=1.70853M$ and cosmological horizon is $r_{c}=6.5188M$. This set of parameters generates the trajectories $T^{*}_{i}$ in Fig. 16(b). $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.197 | 6 | 1.10343 | 6.904246 |
1 | 1 | -2 | -0.11034 | 0 |
2 | 1 | 8 | 1.21377 | 6.904246 |
Table 6: Example of parameters for the energy extraction from an extreme RNdS BH with efficiency $\eta=0.0962097$. The BH parameters are $Q=1.00892M$ and $\Lambda=0.05/M^{2}$. The break up point is $r_{*}=2M$. These parameters generate the trajectories $T_{0}$, $T_{1}$ and $T_{2}$ in Fig. 17. $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
0 | 2.3 | 6 | 4.11637 | 2.03159 |
1 | 1 | -2 | -0.396035 | 2 |
2 | 1 | 8 | 4.51242 | 0.03159 |
Table 7: Illustration of the energy extraction from an extreme RNdS BH with efficiency $\eta=0.0784147$. The BH parameters are $Q=1.00892M$ and $\Lambda=0.05/M^{2}$. The break up point is $r_{*}=2M$. These parameters generate the trajectories $T^{*}_{0}$, $T_{1}$ and $T^{*}_{2}$ in Fig. 17. $i$ | $m_{i}$ | $q_{i}$ | $E_{i}$ | $L_{i}/M$ |
---|---|---|---|---|---
$0^{*}$ | 2.3 | 6 | 5.050522 | 8.12841 |
1 | 1 | -2 | -0.396035 | 2 |
$2^{*}$ | 1 | 8 | 5.44656 | 6.12841 |

In Fig. 17 are illustrated two different processes of extraction with the same
break-up point $r_{*}$ for a given set of parameters ($m_{0}$, $m_{1}$,
$m_{2}$, $q_{1}$, $q_{2}$, $L_{1}$); $E_{2}$ and $L_{2}$ are calculated with
Eq. (19), that generates two different scenarios depending on $\epsilon=\pm
1$, as discussed in Sec. II.4. The trajectories $T_{0}$, $T_{1}$, $T_{2}$
correspond to $\epsilon=+1$, and $T^{*}_{0}$, $T^{*}_{1}$, $T_{2}$ correspond
to $\epsilon=-1$. The explicit parameter values in each trajectory are
indicated in Table 6 and 7.

Figure 17:  Examples of energy extraction from RNdS BH for the same break up
point but with two different efficiencies. The first case is described with
the trajectories $T_{0}$, $T_{1}$ and $T_{2}$, while the second case with
$T^{*}_{0}$, $T_{1}$ and $T^{*}_{2}$; the whole set of parameters for each
process are in Table 6 and 7. The BH parameters are fixed as $Q=1.00892M$,
$\Lambda=0.05/M^{2}$. The radius of the generalized ergosphere is
$r_{eg_{1}}=3.09176M$ and the radius of the cosmological ergosphere is
$r_{eg_{2}}=5.94369M$; the event horizon is $r_{h}=1.03719M$ and the
cosmological horizon is $r_{c}=6.56863M$, the shaded areas are the
ergoregions.

##  V Conclusions

We have analyzed in detail the process of energy extraction from the static
spherically symmetric charged BH, where the energy extraction is possible with
a charged test particle (particle $0$) that splits into two particles, one of
them penetrates the horizon (particle $1$) while the second one escapes to
infinity (particle $2$) with a greater energy than the initial particle. We
determined the energy conditions for the particles involved in the extraction
process.

We extend the study in DenardoRuffini1972  to the case of nonvanishing angular
momentum $L_{1}\neq 0$ for the Reissner-Nordstrom BH; in this case the
particles should satisfy $m_{0}>m_{1}+m_{2}$ and it turns out that the angular
momentum $L_{2}$ and energy $E_{2}$ can take two possible values,
consequently, there are two efficiencies for the same set of parameters
($m_{0}$, $m_{1}$, $m_{2}$, $q_{1}$, $q_{2}$) at a given break-up point
$r_{*}$, see Eq. (25), that allows two scenarios with different efficiencies
depending on the choice of $\epsilon=\pm 1$. For $L_{1}=0$ the energy
conditions in DenardoRuffini1972  are recovered. Additionally, in Sec. II.6 we
determined the conditions that maximize the efficiency of the energy
extraction process. Eq. (25) defines the region where the extraction process
is viable, that is called the generalized ergosphere; this region depends on
the parameters of the BH and the charge mass ratio $e$; the condition $eQ<0$
must be fulfilled in order that the energy extraction be viable, where $Q$ is
the electric charge of the BH. From Eq.(25) the maximum of the efficiency
depends on the location of the break-up point $r_{*}$ and in case it
approaches the BH horizon the efficiency is $\eta=-q_{1}/q_{0}$.

The analysis of RN with negative cosmological constant, namely, the Reissner
Nordström-anti-de Sitter BH (RNAdS BH) was studied in detail by Lemos2024 . In
this paper we examined the energy extraction from the Reissner-Nordstrom de
Sitter black hole RNdS BH, with $\Lambda>0$. For $L_{1}=0$ we derive the
analytic expression of the maximum radius of the generalized ergosphere, that
depends on the RNdS BH parameters ($M$, $Q$, $\Lambda$) and the charge mass
ratio $e$ of the test particle. In contrast with RN BH, introducing a positive
cosmological constant $\Lambda>0$ generates two regions where the negative
energy states (NES) are allowed and therefore the energy extraction is viable;
these regions share the same characteristics as the ones for RN BH, namely,
for a given $e$ both regions reach their maximum size when $L_{1}=0$; also if
$e$ increases, the sizes of the ergospheres increase as well. Also, the
introduction of $\Lambda$ allows to define a critical charge $e_{crit}$ that
corresponds to a critical radius $r_{crit}$ where both ergospheres join such
that NES can exist throughout the whole domain of $r_{h}<r<r_{c}$; analytic
expressions for $e_{crit}$ and $r_{crit}$ are given and the cases
$e_{crit}<r_{crit}$, $e_{crit}=r_{crit}$ and $e_{crit}>r_{crit}$ are
illustrated. Finally, the existence of two ergospheres permits to extract
energy with the same efficiency for different break-up points.

The analysis carried out in this work applies not only to the Reissner-
Nordström BH, but to any electrostatic and spherically symmetric BH, for
instance, solutions of BH coupled to nonlinear electrodynamics Nora2023 .

##  VI Acknowledgments

NB acknowledges partial support by CONAHCyT Project CBF2023-2024-811. ICM
acknowledges financial support of SNI-CONAHCyT, Mexico, grant CVU No. 173252.
AB acknowledges financial support by CONAHCyT, Mexico, through the PhD
Scholarship 814092.

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