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Magnetic Reconnection as a Mechanism for Energy Extraction from Rotating Black Holes

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Spinning black holes store rotational energy that can be extracted. When a black hole is immersed in an externally supplied magnetic field, reconnection of magnetic field lines within the ergosphere can generate negative energy (relative to infinity) particles that fall into the black hole event horizon while the other accelerated particles escape stealing energy from the black hole. We show analytically that energy extraction via magnetic reconnection is possible when the black hole spin is high (dimensionless spin a ∼ 1) and the plasma is strongly magnetized (plasma magnetization σ0 > 1/3). The parameter space region where energy extraction is allowed depends on the plasma magnetization and the orientation of the reconnecting magnetic field lines. For σ0 1, the asymptotic negative energy at infinity per enthalpy of the decelerated plasma that is swallowed by a maximally rotating black hole is found to be ∞ − − σ0/3. The accelerated plasma that escapes to infinity and takes away black hole energy asymptotes the energy at infinity per enthalpy ∞ + √ 3σ0. We show that the maximum power extracted from the black hole by the escaping plasma is P max extr ∼ 0.1M 2 √ σ0 w0 (here, M is the black hole mass and w0 is the plasma enthalpy density) for the collisionless plasma regime and one order of magnitude lower for the collisional regime. Energy extraction causes a significant spindown of the black hole when a ∼ 1. The maximum efficiency of the plasma ener-gization process via magnetic reconnection in the ergosphere is found to be ηmax 3/2. Since fast magnetic reconnection in the ergosphere should occur intermittently in the scenario proposed here, the associated emission within a few gravitational radii from the black hole is expected to display a bursty nature.
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Magnetic Reconnection as a Mechanism for Energy Extraction
from Rotating Black Holes
Luca Comisso1, and Felipe A. Asenjo2,
1Department of Astronomy and Columbia Astrophysics Laboratory,
Columbia University, New York, New York 10027, USA.
2Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´nez, Santiago 7941169, Chile.
Spinning black holes store rotational energy that can be extracted. When a black hole is immersed
in an externally supplied magnetic field, reconnection of magnetic field lines within the ergosphere
can generate negative energy (relative to infinity) particles that fall into the black hole event horizon
while the other accelerated particles escape stealing energy from the black hole. We show analyti-
cally that energy extraction via magnetic reconnection is possible when the black hole spin is high
(dimensionless spin a1) and the plasma is strongly magnetized (plasma magnetization σ0>1/3).
The parameter space region where energy extraction is allowed depends on the plasma magnetization
and the orientation of the reconnecting magnetic field lines. For σ01, the asymptotic negative
energy at infinity per enthalpy of the decelerated plasma that is swallowed by a maximally rotating
black hole is found to be
' −pσ0/3. The accelerated plasma that escapes to infinity and takes
away black hole energy asymptotes the energy at infinity per enthalpy
+'3σ0. We show that
the maximum power extracted from the black hole by the escaping plasma is Pmax
extr 0.1M2σ0w0
(here, Mis the black hole mass and w0is the plasma enthalpy density) for the collisionless plasma
regime and one order of magnitude lower for the collisional regime. Energy extraction causes a
significant spindown of the black hole when a1. The maximum efficiency of the plasma ener-
gization process via magnetic reconnection in the ergosphere is found to be ηmax '3/2. Since fast
magnetic reconnection in the ergosphere should occur intermittently in the scenario proposed here,
the associated emission within a few gravitational radii from the black hole is expected to display a
bursty nature.
PACS numbers: 52.27.Ny; 52.30.Cv; 95.30.Qd, 04.20.-q
Keywords: Black holes; General relativity; Relativistic plasmas; Magnetic reconnection
I. INTRODUCTION
Black holes are believed to play a key role in a number
of highly energetic astrophysical phenomena, from active
galactic nuclei to gamma-ray bursts to ultraluminous X-
ray binaries. The extraordinary amounts of energy re-
leased during such events may have two different origins.
It can be the gravitational potential energy of the matter
falling toward an existing or forming black hole during
accretion or a gravitational collapse. Or it can also be
the energy of the black hole itself. Indeed, a remarkable
prediction of general relativity is that a spinning black
hole has free energy available to be tapped. How this
occurs has fundamental implications for our understand-
ing of high energy astrophysical phenomena powered by
black holes.
It was shown by Christodoulou [1] that for a spinning
(Kerr) black hole having mass Mand dimensionless spin
parameter a, a portion of the black hole mass is “irre-
ducible”,
Mirr =Mr1
21 + p1a2.(1)
The irreducible mass has a one-to-one connection with
Electronic address: luca.comisso@columbia.edu
Electronic address: felipe.asenjo@uai.cl
the surface area of the event horizon, AH= 4π(r2
H+a2) =
16πM 2
irr, which is proportional to the black hole enthropy
SBH = (kBc3/4G~)AH[2–5], where kB,G,~, and cde-
note, respectively, the Boltzmann constant, the gravita-
tional constant, the reduced Planck constant, and the
speed of light in vacuum. Thus, the maximum amount
of energy that can be extracted from a black hole with-
out violating the second law of thermodynamics is the
rotational energy
Erot ="1r1
21 + p1a2#Mc2.(2)
For a maximally rotating black hole (a= 1), this gives
Erot = (1 1/2)M c2'0.29M c2. Therefore, a sub-
stantial fraction of black hole energy can, in principle, be
extracted [6].
The possibility of extracting black hole rotational en-
ergy was first realized by Penrose [7], who envisioned a
thought experiment in which particle fission (0 1 + 2)
occurs in the ergosphere surrounding a rotating black
hole. If the angular momentum of particle 1 is opposite
to that of the black hole and is sufficiently high, then
the energy of particle 1, as viewed from infinity, may be
negative. Hence, since the total energy at infinity is con-
served, the energy of particle 2 as measured from infinity
will be larger than that of the initial particle 0. When
the particle with negative energy at infinity (1) falls into
the black hole’s event horizon, the total energy of the
2
black hole decreases. Therefore, the energy of the escap-
ing particle 2, which is higher than that of the original
particle 0, is increased at the expense of the rotational
energy of the black hole.
Although the Penrose process indicates that it is pos-
sible to extract energy from a black hole, it is believed
to be impractical in astrophysical scenarios. Indeed, en-
ergy extraction by means of the Penrose process requires
that the two newborn particles separate with a relative
velocity that is greater than half of the speed of light
[8, 9], and the expected rate of such events is too rare
to extract a sizable amount of black hole’s rotational en-
ergy. On the other hand, Penrose’s suggestion sparked
the interest to find alternative mechanisms for extract-
ing black hole rotational energy, such as superradiant
scattering [10], the collisional Penrose process [11], the
Blandford-Znajek process [12] and the magnetohydrody-
namic (MHD) Penrose process [13]. Among them, the
Blandford-Znajek process, in which energy is extracted
electromagnetically through the magnetic field supported
by an accretion disk around the black hole, is thought
to be the leading mechanism for powering the relativis-
tic jets of active galactic nuclei (AGNs) [e.g. 14–17] and
gamma-ray bursts (GRBs) [e.g. 18–20].
While different mechanisms of energy extraction have
been carefully analyzed over the years, the possibility of
extracting black hole rotational energy as a result of rapid
reconnection of magnetic field lines has been generally
overlooked. An exploratory study conducted by Koide
and Arai [21] analyzed the feasibility conditions for en-
ergy extraction by means of the outflow jets produced in a
laminar reconnection configuration with a purely toroidal
magnetic field. In this simplified scenario, they suggested
that relativistic reconnection was required for energy ex-
traction, but the extracted power and the efficiency of
the reconnection process were not evaluated. This is
necessary for determining whether magnetic reconnec-
tion can play a significant role in the extraction of black
hole energy. The recent advent of general-relativistic ki-
netic simulations of black hole magnetospheres [22] do in-
deed suggest that particles accelerated during magnetic
reconnection may spread onto negative energy-at-infinity
trajectories, and that the energy extraction via negative-
energy particles could be comparable to the energy ex-
tracted through the Blandford-Znajek process.
In this paper we provide an analytical analysis of black
hole energy extraction via fast magnetic reconnection as
a function of the key parameters that regulate the pro-
cess: black hole spin, reconnection location, orientation
of the reconnecting magnetic field, and plasma magneti-
zation. Our main objective is to evaluate the viability,
feasibility conditions, and efficiency of magnetic recon-
nection as a black hole energy extraction mechanism. In
Section II we delineate how we envision the extraction
of black hole rotational energy by means of fast mag-
netic reconnection, and we derive the conditions under
which such energy extraction occurs. In Section III we
show that magnetic reconnection is a viable mechanism
of energy extraction for a substantial region of the pa-
rameter space. In Section IV we quantify the rate of en-
ergy extraction and the reconnection efficiency in order
to evaluate whether magnetic reconnection is an effective
energy extraction mechanism for astrophysical purposes.
We further compare the power extracted by fast mag-
netic reconnection with the power that can be extracted
through the Blandford-Znajek mechanism. Finally, we
summarize our results in Section V.
II. ENERGY EXTRACTION BY MAGNETIC
RECONNECTION
The possibility of extracting black hole rotational en-
ergy via negative-energy particles requires magnetic re-
connection to take place in the ergosphere of the spin-
ning black hole since the static limit is the boundary of
the region containing negative-energy orbits. Magnetic
reconnection inside the ergosphere is expected to occur
routinely for fast rotating black holes. Indeed, a con-
figuration with antiparallel magnetic field lines that is
prone to magnetic reconnection is caused naturally by
the frame-dragging effect of a rapidly spinning black hole.
In this paper, we envision the situation illustrated in Fig.
1, where the fast rotation of the black hole leads to an-
tiparallel magnetic field lines adjacent to the equatorial
plane. This scenario is also consistent with numerical
simulations of rapidly spinning black holes [e.g. 22–25].
The change in magnetic field direction at the equa-
torial plane produces an equatorial current sheet. This
current sheet forms dynamically and is destroyed by the
plasmoid instability (permitted by non-ideal magnetohy-
drodynamic effects such as thermal-inertial effects, pres-
sure agyrotropy, or electric resistivity) when the current
sheet exceeds a critical aspect ratio [26–28]. The forma-
tion of plasmoids/flux ropes (see circular sections in the
zoomed-in region of Fig. 1) drives fast magnetic recon-
nection [e.g. 31, 32], which rapidly converts the available
magnetic energy into plasma particle energy. Eventually,
the plasma is expelled out of the reconnection layer and
the magnetic tension that drives the plasma outflow re-
laxes. The field lines are then stretched again by the
frame-dragging effect and a current layer prone to fast
plasmoid-mediated reconnection forms again. This leads
to reconnecting current sheets that form rapidly and in-
termittently.
Magnetic reconnection in the plasma that rotates
around the black hole has the effect of accelerating part of
the plasma and decelerating another part. If the deceler-
ated plasma has negative energy at infinity and the accel-
erated one has energy at infinity larger than its rest mass
and thermal energies (see the example regions in orange
in Fig. 1(b)), then the plasma that escapes to infinity ac-
quires energy at the expense of the black hole rotational
energy when the negative-energy particles are swallowed
by the black hole as in the standard Penrose process [7].
Therefore, we want to examine when magnetic reconnec-
3
ϵ0
<
ϵ+
0
>
Δ
(b)
(a)
BH
BH
B
B
vr
out vr
out
βϕ>1
βϕ
<1
FIG. 1: Schematic illustration of the mechanism of energy
extraction from a rotating black hole by magnetic reconnec-
tion in the black hole ergosphere. A configuration with an-
tiparallel magnetic field lines adjacent to the equatorial plane
is favored by the frame-dragging effect of the rapidly spin-
ning black hole (panels (a) and (b) portray meridional and
equatorial views, respectively), and the resulting equatorial
current sheet is prone to fast plasmoid-mediated magnetic re-
connection (see circular structures in the zoomed-in region
[29]). Magnetic reconnection in the plasma that rotates in
the equatorial plane extracts black hole energy if the deceler-
ated plasma that is swallowed by the black hole has negative
energy as viewed from infinity, while the accelerated plasma
with a component in the same direction of the black hole ro-
tation escapes to infinity. The outer boundary (static limit)
of the ergosphere is indicated by the short-dashed lines in
both panels. In panel (b), long-dashed and solid lines indi-
cate magnetic field lines below and above of the equatorial
plane, respectively. Finally, the dashed lines in the zoomed
region indicate the two magnetic reconnection separatrices in-
tersecting at the dominant magnetic reconnection X-point.
tion in the ergosphere of the black hole redistributes the
angular momentum of the plasma in such a way to sat-
isfy these conditions. Furthermore, we want to evaluate
if the extraction of black hole rotational energy via fast
plasmoid-mediated reconnection can constitute a major
energy release channel.
We describe the spacetime around the rotating black
hole by using the Kerr metric in Boyer-Lindquist coordi-
nates xµ= (t, r, θ, φ), where ris the radial distance, θis
the polar angle, and φis the azimuthal angle. The Kerr
metric can be expressed in terms of the square of the line
element ds2=gµνdxµdxνas [e.g. 33]
ds2=gttdt2+ 2g dtdφ +gφφ2+grr dr2+gθθ2,(3)
where the non-zero components of the metric are given
by
gtt =2Mr
Σ1, g=2M2ar sin2θ
Σ,(4)
gφφ =A
Σsin2θ , gr r =Σ
, gθθ = Σ ,(5)
with
Σ = r2+ (aM)2cos2θ , (6)
∆ = r22Mr + (aM)2,(7)
A=r2+ (aM)22(aM)2∆ sin2θ . (8)
The only two parameters that appear in the metric are
the black hole mass, M, and the black hole dimensionless
spin, 0 a1. Here, and in all subsequent expressions,
we use geometrized units with G=c= 1.
The inner boundary of the ergosphere of the Kerr black
hole, which coincides with the outer event horizon, is
given by the radial distance
rH=M+M(1 a2)1/2,(9)
while the outer boundary (static limit) is given by
rE=M+M(1 a2cos2θ)1/2,(10)
which yields rE= 2Mat the equatorial plane θ=π/2.
In this paper we make the simplifying assumption that
magnetic reconnection happens in the bulk plasma that
rotates circularly around the black hole at the equatorial
plane. This corresponds to a Keplerian angular velocity
K=±M1/2
r3/2±aM3/2,(11)
as seen by an observer at infinity. The upper sign refers to
corotating orbits, while the lower sign applies to counter-
rotating orbits. Circular orbits can exist from r→ ∞
4
down to the limiting circular photon orbit, whose radius
is given by
rph = 2M1 + cos 2
3arccos(a).(12)
For a maximally rotating black hole (a= 1), one has
rph =M(corotating orbit) or rph = 4M(counter-
rotating orbit). However, for r > rph not all circular
orbits are stable. Non-spinning test particles can stably
orbit the black hole if they are at distances larger than
or equal to the innermost stable circular orbit [8]
risco =M3 + Z2(3 Z1)(3 + Z1+ 2Z2)1/2,
(13)
where
Z1= 1 + (1 a2)1/3[(1 + a)1/3+ (1 a)1/3],(14)
Z2= (3a2+Z2
1)1/2.(15)
For a maximally rotating black hole risco =M(corotat-
ing orbit) or risco = 9M(counter-rotating orbit). Here
we focus on corotating orbits since we are interested in
magnetic reconnection occurring inside the ergosphere.
We also assume that the plasma acceleration through
magnetic reconnection is localized in a small region (close
to the dominant reconnection X-point) compared to the
size of the black hole ergosphere.
In what follows, it is convenient to analyze the
plasma energy density in a locally nonrotating frame, the
so called “zero-angular-momentum-observer” (ZAMO)
frame [8]. In the ZAMO frame, the square of the line ele-
ment is given by ds2=dˆ
t2+P3
i=1 (dˆxi)2=ηµνdˆxµdˆxν,
where
dˆ
t=α dt , dˆxi=gii dxiαβidt (16)
(no implicit summation is assumed over i), with αindi-
cating the lapse function
α= gtt +g2
φt
gφφ !1/2
=∆Σ
A1/2
(17)
and βiindicating the shift vector (0,0, βφ), with
βφ=gφφ ωφ
α=ωφ
αA
Σ1/2
sin θ(18)
and ωφ=gφt/gφφ = 2M2ar/A being the angular veloc-
ity of the frame dragging. An advantage of this reference
frame is that equations become intuitive since the space-
time is locally Minkowskian for observers in this frame.
Hereinafter, quantities observed in the ZAMO frame are
denoted with hats. Vectors in the ZAMO frame are re-
lated to the vectors in the Boyer-Lindquist coordinates
as ˆ
b0=αb0and ˆ
bi=gii biαβib0for the contravari-
ant components, while ˆ
b0=b0+P3
i=1 (βi/gii)biand
ˆ
bi=bi/gii for the covariant components.
We evaluate the capability of magnetic reconnection to
extract black hole energy by examining the conditions for
the formation of negative energy at infinity and escaping
to infinity of the plasma accelerated/decelerated by the
reconnection process in the ergosphere (in this work we
do not address the origin of the plasma properties but
rather assume a plasma with a given particle density and
pressure). From the energy-momentum tensor in the one-
fluid approximation,
Tµν =pgµν +wU µUν+FµδFνδ 1
4gµν FρδFρδ ,(19)
where, p,w,Uµ, and Fµν are the proper plasma pressure,
enthalpy density, four-velocity, and electromagnetic field
tensor, respectively, one has the “energy-at-infinity” den-
sity e=αgµ0Tµ0. Therefore, the energy-at-infinity
density is given by
e=αˆe+αβφˆ
Pφ,(20)
where
ˆe=wˆγ2p+ˆ
B2+ˆ
E2
2(21)
is the total energy density and
ˆ
Pφ=wˆγ2ˆvφ+ˆ
B×
ˆ
Eφ(22)
is the azimuthal component of the momentum density,
with ˆγ=ˆ
U0=1P3
i=1 (dˆvi)21/2,ˆ
Bi=ijk ˆ
Fjk /2,
and ˆ
Ei=ηij ˆ
Fj0=ˆ
Fi0.
The energy-at-infinity density can be conveniently sep-
arated into hydrodynamic and electromagnetic compo-
nents as e=e
hyd +e
em, where
e
hyd =αˆehyd +αβφwˆγ2ˆvφ(23)
is the hydrodynamic energy-at-infinity density and
e
em =αˆeem +αβφˆ
B×
ˆ
Eφ(24)
is the electromagnetic energy-at-infinity density, with
ˆehyd =wˆγ2pand ˆeem = ( ˆ
B2+ˆ
E2)/2 indicating the
hydrodynamic and electromagnetic energy densities ob-
served in the ZAMO frame. In this paper we assume
an efficient magnetic reconnection process that converts
most of the magnetic energy into kinetic energy, so that
the electromagnetic energy at infinity is negligible with
respect to the hydrodynamic energy at infinity. Then,
from Eq. (23), we can evaluate the energy-at-infinity
density of the expelled plasma using the approximation
that the plasma element is incompressible and adiabatic,
which leads to [21]
e
hyd =αhγ+βφˆγˆvφ)wp
ˆγi.(25)
5
To analyze the localized reconnection process, we in-
troduce the local rest frame xµ0= (x00, x10, x20, x30) of
the bulk plasma that rotates with Keplerian angular ve-
locity ΩKin the equatorial plane. We choose the frame
xµ0in such a way that the direction of x10is parallel
to the radial direction x1=rand the direction of x30
is parallel to the azimuthal direction x3=φ. The ori-
entation of the reconnecting magnetic field lines is kept
arbitrary as it ultimately depends on the large scale mag-
netic field configuration, the black hole spin, and is also
time dependent. Indeed, the complex nonlinear dynamics
around the spinning black hole induces magnetic field line
stretching, with magnetic reconnection causing a topo-
logical change of the macroscopic magnetic field configu-
ration on short time scales. Therefore, here we introduce
the orientation angle
ξ= arctan v10
out/v30
out,(26)
where v10
out and v30
out are the radial and azimuthal compo-
nents of the outward-directed plasma in the frame xµ0.
Accordingly, the plasma escaping from the reconnection
layer has velocities v0
±=vout(±cos ξe0
3sin ξe0
1), with
vout indicating the magnitude of the outflow velocity ob-
served in the frame xµ0and the subscripts + and in-
dicating the corotating and counterrotating outflow di-
rection, respectively. In the plasmoid-mediated recon-
nection regime, a large fraction of the plasma is evac-
uated through plasmoid-like structures [34], which can
also contain a significant component of nonthermal par-
ticles. Such particles gain most of their energy from the
motional electric field [e.g. 35] and are carried out by
the plasmoids (where most of them are trapped) in the
outflow direction [e.g. 36].
The outflow Lorentz factor ˆγand the outflow veloc-
ity component ˆvφobserved by the ZAMO can be con-
veniently expressed in terms of the Keplerian velocity in
the ZAMO frame and the outflow velocities in the local
frame xµ0. From Eq. (11), we can express the corotating
Keplerian velocity observed in the ZAMO frame as
ˆvK=A
1/2(M/r)1/2a(M/r)2
r3a2M3βφ.(27)
Then, using ˆvφ
±= (ˆvK±vout cos ξ)/(1 ±ˆvKvout cos ξ) for
the azimuthal components of the two outflow velocities
and introducing the Lorentz factors ˆγK= (1 ˆv2
K)1/2
and γout = (1 v2
out)1/2, we can write the energy-at-
infinity density of the reconnection outflows as
e
hyd,±=αˆγK"1+ˆvKβφγoutw
±cos ξˆvK+βφγoutvout w
p
(1±cos ξˆvKvout )γoutˆγ2
K#,(28)
where the subscripts + and indicate the energy-
at-infinity density associated with corotating (v0
+) and
counterrotating (v0
) outflow directions as observed in
the local frame xµ0.
The outflow velocity vout can be evaluated by assum-
ing that the local current sheet at the dominant X-point
has a small inverse aspect ratio δX/LX1, where δX
and LXare the half-thickness and half-length of this local
current sheet. If we consider that the rest frame rotat-
ing with Keplerian velocity is in a gravity-free state and
neglect general relativistic corrections [37–39], then, the
conservation of momentum along the reconnection neu-
tral line gives
2
outv2
out/LX+B2
upδ2
X/L3
X'(BupX)(Bup δX/LX),
(29)
where Bup is the local magnetic field strength immedi-
ately upstream of the local current sheet. Here we have
used Maxwell’s equations to estimate the current density
at the neutral line in addition to the outflow magnetic
field strength [40, 41]. We also assumed that the ther-
mal pressure gradient force in the outflow direction is
small compared to the magnetic tension force, as verified
by numerical simulations of relativistic reconnection with
antiparallel magnetic fields [43]. Then, from Eq. (29) one
gets
vout '"1δ2
X/L2
Xσup
1 + (1 δ2
X/L2
X)σup #1/2
,(30)
where σup =B2
up/w0is the plasma magnetization imme-
diately upstream of the local current sheet at the dom-
inant X-point. Consequently, for δX/LX1, the out-
flow velocity reduces to vout '[σup/(1 + σup)]1/2.
The local magnetic field Bup can be connected to the
asymptotic macro-scale magnetic field B0by considering
force balance along the inflow direction. In the magnet-
ically dominated regime, thermal pressure is negligible,
and the inward-directed magnetic pressure gradient force
must be balanced by the outward-directed magnetic ten-
sion (the inertia of the inflowing plasma is negligible if
δX/LX1). Then, from geometrical considerations one
gets [43]
Bup =1(tan ϕ)2
1 + (tan ϕ)2B0,(31)
where ϕis the opening angle of the magnetic reconnec-
tion separatrix. Estimating tan ϕ'δX/LX, we have
simply
vout 'σ0
1 + σ01/2
, γout '(1 + σ0)1/2,(32)
where we have defined σ0=B2
0/w0as the plasma magne-
tization upstream of the reconnection layer. Accordingly,
in the magnetically dominated regime σ01, the recon-
nection outflow velocity approaches the speed of light.
We finally note that in the presence of significant embed-
ding of the local current sheet, the scaling of the outflow
6
velocity could be weakened with respect to B0, while Eq.
(30) remains accurate [36, 43].
We must point out that in the plasmoid-mediated re-
connection regime considered here, the continuous forma-
tion of plasmoids/flux ropes prevents the formation of ex-
tremely elongated “laminar” reconnection layers, thereby
permitting a high reconnection rate [e.g. 31, 32]. De-
pending on the plasma collisionality regime, plasmoid-
mediated reconnection yields an inflow velocity (as ob-
served in the frame xµ0)
vin =(O(102) for δX> `k[4447]
O(101) for δX.`k[42,43] ,(33)
where `kis the relevant kinetic scale that determines
the transition between the collisional and collisionless
regimes. The collisional regime is characterized by δX>
`k, while the collisionless regime occurs if δX.`k. For
a pair (ee+) dominated plasma, we have [41] `k=
γth,e λe, where λeis the nonrelativistic plasma skin
depth and γth,e is the electron/positron thermal Lorentz
factor. If there is also a significant ion component, then
[31] `k=γth,i λi, where λiis the nonrelativistic ion in-
ertial length and γth,i is the ion thermal Lorentz factor.
We emphasize that the reconnection rate is independent
of the microscopic plasma parameters when magnetic re-
connection proceeds in the plasmoid-mediated regime. In
particular, plasmoid-mediated reconnection in the colli-
sionless regime has an inflow velocity vin that is a sig-
nificant fraction of the speed of light, which potentially
allows for a high energy extraction rate from the black
hole (see Sec. IV).
The expression for the energy at infinity associated
with the accelerated/decelerated plasma as a function
of the critical parameters (a,r/M,σ0,ξ) can be finally
obtained by substituting the magnetization dependence
of the outflow velocity into Eq. (28). Then, the hydro-
dynamic energy at infinity per enthalpy
±=e
hyd,±/w
becomes
±=αˆγK"1+βφˆvK(1+σ0)1/2±cos ξβφ+ ˆvKσ1/2
0
1
4
(1+σ0)1/2cos ξˆvKσ1/2
0
ˆγ2
K(1 + σ0cos2ξˆv2
Kσ0)#,(34)
where we have assumed a relativistically hot plasma with
polytropic index Γ = 4/3. Similarly to the original Pen-
rose process [7], energy extraction from the black hole
through magnetic reconnection occurs when
<0 and ∆
+>0,(35)
where
+=
+1Γ
Γ1
p
w=
+(36)
for a relativistically hot plasma. Therefore, black hole
rotational energy is extracted if the decelerated plasma
ϵ+
σ
ϵ-
-σ/

-
σ
ϵ+
ϵ-
FIG. 2: Energy at infinity per enthalpy
+(gray line) and
(orange line) for maximal energy extraction conditions
(a, r/M 1 and ξ0). Energy extraction requires σ0>
1/3. For σ01,
+'3σ0(dash-dotted black line) and
' −pσ0/3 (dashed black line).
acquires negative energy as measured at infinity, while
the plasma that is accelerated acquires energy at infinity
larger than its rest mass and thermal energies.
The energy at infinity per enthalpy
±given by Eq.
(34) depends on the black hole spin aand the X-point
distance r/M, as well as the plasma magnetization σ0and
the orientation angle ξ, which encodes the information
of the magnetic field configuration surrounding the black
hole. Equations (34)-(36) indicate that energy extraction
is favored by lower values of the orientation angle ξand
higher values of the magnetization σ0. It is instructive to
consider the limit a1, ξ0, and rM(the metric
(3) has a coordinate singularity at the event horizon that
can be removed by a coordinate transformation). In this
case, from Eq. (34) we obtain
+>0 and
<0 when
σ0>1/3.(37)
Therefore, in principle, it is possible to extract rotational
energy via magnetic reconnection for values of σ0below
unity. However, higher σ0values are required to extract
sizable amounts of energy. If, in addition to a, r/M 1
and ξ0, we also consider σ01, from Eq. (34) we
obtain
+'p3gφφ ωφγoutvout '3σ0,(38)
' −rgφφ
3ωφγoutvout ' −rσ0
3.(39)
These relations give us the energy at infinity per enthalpy
of the accelerated (+) and decelerated () plasma in the
maximal energy extraction regime (as can be seen from
Fig. 2, they provide a fairly accurate estimate already at
values of σ0moderately larger then unity).
In the next sections, we will show that magnetic re-
connection is a viable mechanism for extracting energy
7
ξ=π12/
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r/M
a
ϵ00 0
1σ)( <
=
0
ϵ0 0
3σ)( <
=
0
ϵ0 0
1σ)( <
=
0
ϵ0
3)( <
=
0
σ
ϵ0
1
0
σ)( <
=
FIG. 3: Regions of the phase-space {a, r /M}where the en-
ergies at infinity per enthalpy from Eq. (34) are such that
+>0 (gray area) and
<0 (orange to red areas),
for a reconnecting magnetic field having orientation angle
ξ=π/12 and different values of the magnetization param-
eter σ0∈ {1,3,10,30,100}. The area with
<0 increases
monotonically as σ0increases. The solid black line indicates
the limit of the outer event horizon, Eq. (9), the dashed
black line represents the limiting corotating circular photon
orbit, Eq. (12), while the dash-dotted black line corresponds
to the innermost stable circular orbit, Eq. (13). The limit
r/M = 2 corresponds to the outer boundary of the ergosphere
at θ=π/2.
from rotating black holes for a significant region of the
parameter space, we will evaluate the rate of black hole
energy extraction, and we will determine the efficiency of
the reconnection process.
III. ENERGY EXTRACTION ASSESSMENT IN
PHASE SPACE
We analyze the viability of energy extraction via mag-
netic reconnection by considering solutions of Eq. (34).
In particular, in Figs. 3 and 4 we display the regions of
the phase-space {a, r/M}where
<0 and ∆
+>0,
which correspond to the conditions for energy extraction.
This is done for a reconnecting magnetic field with orien-
tation angle ξ=π/12 and different values of the magne-
tization parameter σ0∈ {1,3,10,30,100}(Fig. 3), and
for a plasma magnetization σ0= 100 and different values
of the orientation angle ξ∈ {π/20, π/12, π/6, π/4}(Fig.
4).
As the magnetization of the plasma increases, the re-
gion of the phase-space {a, r/M}where magnetic recon-
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r/M
a
=100
0
σ
ϵ6 0
ξ)( <
=/
ϵ2 0
1ξ)( <
=/
ϵ4 0
ξ)( <
=/
ϵ2 0
0ξ)( <
=/
FIG. 4: Regions of the phase-space {a, r /M}where the en-
ergies at infinity per enthalpy from Eq. (34) are such that
+>0 (gray area) and
<0 (green areas), for plasma
magnetization σ0= 100 and different values of the orientation
angle ξ∈ {π/20, π/12, π /6, π/4}. Other lines are the same as
in Figure 3. The area with
<0 increases monotonically
as ξdecreases.
nection extracts black hole rotational energy extends to
larger r/M values and lower values of the dimensionless
spin a(Fig. 3). From Eq. (34) we can see that
is a
monotonically decreasing function of σ0, while
+mono-
tonically increases with σ0.
+>0 is easily satisfied
for rph < r < rE,a > 0, and ξ < π/2. On the other
hand,
<0 requires σ01 in order for reconnec-
tion to extract black hole energy in an extended region
of the phase-space {a, r/M}. High values of the plasma
magnetization can extend the energy extraction region
up to the outer boundary of the ergosphere, while energy
extraction for moderate values of the spin parameter a
is subject to the occurrence of particle orbits inside the
ergosphere.
Energy extraction via magnetic reconnection is also fa-
vored by reconnection outflows whose orientation is close
to the azimuthal direction. The region of the phase-
space {a, r/M}where energy extraction occurs increases
to larger r/M values and lower avalues as the orien-
tation angle ξdecreases. Notwithstanding, even an an-
gle as large as ξ=π/4 admits a feasible region of the
phase-space where magnetic reconnection extracts rota-
tional energy. The increase of the energy extraction re-
gion for decreasing angle ξis due to the fact that only
the azimuthal component of the outflow velocity con-
tributes to the extraction of rotational energy. For an
angle ξ=π/20, the extraction of black hole energy hap-
pens for X-points up to r/M 1.96 (for σ0= 100), while
8
ξ0 can extend this margin up to the outer boundary
of the ergosphere.
The ergosphere of spinning black holes (rH< r < rE)
can reach very high plasma magnetizations (e.g, σ0
100 close to the event horizon of the black hole M87*
[48]). Furthermore, for rapidly spinning (aclose to unity)
black holes, we expect a reconnecting magnetic field with
small orientation angle, ξ.π/6, as the strong frame-
dragging effect inside the ergosphere stretches the mag-
netic field lines along the azimuthal direction [e.g. 49, 50].
Therefore, the plots shown in Figs. 3 and 4 indicate that
magnetic reconnection is a viable mechanism for extract-
ing energy from rotating black holes with dimensionless
spin aclose to unity. On the other hand, energy ex-
traction via magnetic reconnection becomes negligible
for spin values a.0.8. The availability of reconnec-
tion regions inside the ergosphere decreases as the spin
parameter decreases, with no circular orbits inside the
ergosphere for spin a1/2. Magnetic reconnection
could still be capable of extracting energy in such cases
if a circular orbit is sustained thanks to the help of the
magnetic field or if one considers non-circular orbits.
IV. ENERGY EXTRACTION RATE AND
RECONNECTION EFFICIENCY
We now evaluate the rate of black hole energy extrac-
tion. This depends on the amount of plasma with nega-
tive energy at infinity that is swallowed by the black hole
in the unit time. Therefore, a high reconnection rate can
potentially induce a high energy extraction rate. The
power Pextr extracted from the black hole by the escap-
ing plasma can be estimated as
Pextr =
w0AinUin ,(40)
where Uin =O(101) for the collisionless regime, while
Uin =O(102) for the collisional one. Ain is the cross-
sectional area of the inflowing plasma, which can be es-
timated as Ain (r2
Er2
ph) for rapidly spinning black
holes. In particular, for a1 one has (r2
Er2
ph) =
(r2
Er2
H)=3M2.
We show in Fig. 5 the ratio Pextr/w0as a function of
the dominant X-point location r/M for a rapidly spin-
ning black hole with a= 0.99 and magnetic reconnec-
tion in the collisionless regime. This is done for a typ-
ical reconnecting magnetic field with orientation angle
ξ=π/12 and different values of the magnetization pa-
rameter σ010,102,103,104,105(top panel), and for
a typical magnetization σ0= 104and different values of
the orientation angle ξ∈ {0, π/20, π/12, π/8, π/6}(bot-
tom panel). The power extracted from the black hole in-
creases monotonically for increasing values of the plasma
magnetization and for lower values of the orientation an-
gle. It reaches a peak for X-point locations that are close
to the limiting circular orbit until it drops off. The peak
of the extracted power can continue to raise up to a max-
imum value that is achieved for r/M 1 if a1. The
     




/
/
= ξ=π/
σ=
σ=
σ=
σ=
σ=
ξ=π/ξ=π/ξ=π/
ξ=π/ ξ=
     




/
/
= σ=
FIG. 5: Pextr/w0=
AinUin as a function of the dominant
X-point location r/M for a rapidly spinning black hole with
a= 0.99 and reconnection inflow four-velocity Uin = 0.1 (i.e.,
collisionless reconnection regime).
is evaluated using Eq.
(34), while Ain = (r2
Er2
ph). We have also set M= 1.
Different colors (from indigo to red) refer to different plasma
magnetizations (from σ0= 10 to σ0= 105) and ξ=π/12
(top panel) or different orientation angles (from ξ=π/6 to
ξ= 0) and σ0= 104(bottom panel). The vertical dashed line
indicates the limiting circular orbit rph(a= 0.99).
theoretical limit of the maximum power is given by
Pmax
extr 'pσ0/3w0AinUin 0.1M2σ0w0,(41)
which follows directly from Eqs. (39) and (40). We can
see from Fig. 5 that the peak of the extracted power
is already close to the maximum theoretical limit when
ξ.π/12.
The proposed mechanism of energy extraction via mag-
netic reconnection generates energetic plasma outflows
that steal energy from the black hole, but it also necessi-
tates magnetic field energy to operate. Magnetic energy
is indeed needed in order to redistribute the angular mo-
mentum of the particles in such a way to generate parti-
cles with negative energy at infinity and particles escap-
ing to infinity. Therefore, it is convenient to define the
9
=
=
=
=
=
     






/
η
σ= ξ=π/
FIG. 6: Efficiency ηof the reconnection process as a func-
tion of the dominant X-point location r/M for a reconnection
layer with upstream plasma magnetization σ0= 100 and re-
connecting magnetic field having orientation angle ξ=π/20.
Different colors (from indigo to red) refer to different black
hole spin values (from a= 0.9 to a= 1).
efficiency of the plasma energization process via magnetic
reconnection as
η=
+
++
.(42)
Extraction of energy from the black hole takes place when
η > 1. Figure 6 shows the efficiency ηas a function of
the dominant X-point location r/M for a reconnection
layer with magnetization parameter σ0= 100, orienta-
tion angle ξ=π/20, and different black hole spin val-
ues a∈ {0.90,0.96,0.99,0.999,1}. The efficiency ηsig-
nificantly increases for reconnection X-points that are
closer to the black hole event horizon and falls off below
unity when the inner radius reaches rph. The maximum
efficiency can be evaluated by considering the optimal
energy extraction conditions (a, r/M 1, ξ0) and
σ01. In this case, Eq. (42) gives
ηmax '3σ0
3σ0pσ0/3= 3/2.(43)
Therefore, the additional energy extracted from the black
hole, while non-negligible, does not extensively modify
the energetics of the escaping plasma.
We can also compare the power extracted from the
black hole by fast magnetic reconnection with the one
that can be extracted via the Blandford-Znajek mecha-
nism, in which the rotational energy is extracted electro-
magnetically through a magnetic field that threads the
black hole event horizon. For maximum efficiency condi-
tions [51–53], the rate of black hole energy extraction via
the Blandford-Znajek mechanism is given by [12, 54]
PBZ 'κΦ2
BH 2
H+χ4
H+ζ6
H,(44)
=
=
=
=
=
   





σ
/
= ξ=π/
FIG. 7: Power ratio Pextr/PBZ as a function of the plasma
magnetization σ0for a black hole with dimensionless spin
a= 0.99 and a reconnecting magnetic field having orien-
tation angle ξ=π/12. Different colors (from indigo to
red) refer to different dominant X-point locations r/M
{1.7,1.6,1.5,1.4,1.3}. We considered Uin = 0.1 (i.e., colli-
sionless reconnection regime), Ain = (r2
Er2
ph), and κ= 0.05.
where ΦBH =1
2RθRφ|Br|dAθφ is the magnetic flux
threading one hemisphere of the black hole horizon (with
dAθφ =g dθdφ indicating the area element in the θ-
φplane), ΩH=a/2rHis the angular frequency of the
black hole horizon, while κ,χ, and ζare numerical con-
stants. The numerical prefactor κdepends on the mag-
netic field geometry near the black hole (κ0.053 for a
split monopole geometry and κ0.044 for a parabolic
geometry), while χ1.38 and ζ≈ −9.2 [54]. Equation
(44) is a generalization of the original Blandford-Znajek
scaling [12] PBZ 'κΦ2
BH(a/4M)2, which is recovered in
the small spin limit a1.
In order to provide a rough order of magnitude es-
timate of the power extracted during the occurrence of
fast magnetic reconnection with respect to the Blandford-
Znajek process, we assume ΦBH ∼ |Br|r2
HB0sin ξ r2
H
(we point out that a precise evaluation of ΦBH requires
direct numerical simulations that reproduce the detailed
magnetic field configuration at all latitudes, while the
angle ξis a good estimate for the magnetic field configu-
ration only at low latitudes [e.g. 49, 50]). Then, we can
evaluate the ratio Pextr/PBZ as
Pextr
PBZ
AinUin
κ2
Hr4
Hσ0sin2ξ(1 + χ2
H+ζ4
H).(45)
Figure 7 shows the ratio Pextr/PBZ given by the right-
hand side of Eq. (45) as a function of the plasma magne-
tization σ0for the fast collisionless reconnection regime.
Pextr/PBZ 1 for an extended range of plasma mag-
netizations. For σ01, the force-free electrodynamics
approximation (the inertia of the plasma is ignored, i.e.
w00) that is used to derive the extracted power in
the Blandford-Znajek process becomes invalid. In this
10
case, magnetic reconnection is an effective mechanism
of energy extraction provided that the plasma magneti-
zation is sufficient to satisfy the condition
<0 (as
well as ∆
+>0). On the other hand, for σ0→ ∞,
energy extraction via fast magnetic reconnection is al-
ways subdominant to the Blandford-Znajek process since
Pextr/PBZ 0 in this limit. If we neglect higher order
corrections with respect to Ω2
H(which leads to an over-
prediction of PBZ by about 25% as a1 [54]), and
recalling that ΩH= 1/2Mfor a1, we can estimate
the ratio Pextr/PBZ for a rapidly spinning black hole as
Pextr
PBZ
κ σ0sin2ξ,(46)
where we considered plasmoid-mediated reconnection in
the collisionless regime. Therefore, the power extracted
via fast collisionless magnetic reconnection can exceed
the one extracted through the Blandford-Znajek process
for an extended range of plasma magnetizations if there is
a significant toroidal component of the magnetic field in
the black hole ergosphere. Note that this energy extrac-
tion mechanism is expected to be bursty in nature, with
a continuous build-up of the magnetic field configuration
storing the magnetic energy that is eventually dissipated
via fast magnetic reconnection.
V. CONCLUSIONS
In this paper, we envisioned the possibility of extract-
ing black hole rotational energy via fast magnetic recon-
nection in the black hole ergosphere. We considered a
configuration with antiparallel magnetic field lines near
the equatorial plane, which is induced by the frame drag-
ging of the spinning black hole. The change in magnetic
field direction at the equatorial plane produces an equa-
torial current sheet that is disrupted by the plasmoid in-
stability when its aspect ratio reaches a critical value (for
a collisionless relativistic pair plasma, the critical aspect
ratio condition is derived in Ref. [55]). The formation of
plasmoids/flux ropes drives fast magnetic reconnection,
which rapidly converts the available magnetic energy into
plasma particle energy. When the plasma is expelled
out of the reconnection layer, the magnetic tension that
drives the plasma outflow relaxes. The field lines are then
stretched again as a consequence of the frame dragging
and a current layer prone to fast plasmoid-mediated re-
connection forms again. This process leads to reconnect-
ing current sheets that form rapidly and intermittently.
Magnetic reconnection accelerates part of the plasma
in the direction of the black hole rotation, while another
part of the plasma is accelerated in the opposite direc-
tion and falls into the black hole. Black hole energy ex-
traction occurs if the plasma that is swallowed by the
black hole has negative energy as viewed from infinity,
while the accelerated plasma that gains energy from the
black hole escapes to infinity. Therefore, differently from
the Blandford-Znajek process, in which the extraction of
rotational energy is obtained through a purely electro-
magnetic mechanism, the energy extraction mechanism
described here requires non-zero particle inertia. This
mechanism is also different from the original Penrose pro-
cess, since dissipation of magnetic energy is required to
produce the negative-energy particles. Clearly, all mech-
anisms extract black hole rotational energy by feeding the
black hole with negative energy and angular momentum.
We showed analytically that energy extraction via
magnetic reconnection is possible when the black hole
spin is high (dimensionless spin a1) and the plasma is
strongly magnetized (plasma magnetization σ0>1/3).
Magnetic reconnection is assumed to occur in a circu-
larly rotating plasma with a reconnecting field having
both azimuthal and radial components. The region of
the phase-space {a, r/M}where magnetic reconnection
is capable of extracting black hole energy depends on
the plasma magnetization σ0and the orientation ξof the
reconnecting magnetic field. We showed that high val-
ues of the plasma magnetization and mostly azimuthal
reconnecting fields can expand the energy extraction re-
gion up to the outer boundary of the ergosphere. For
a dimensionless spin parameter that approaches unity,
the extraction of black hole energy is maximal when the
dominant reconnection X-point (where the two magnetic
reconnection separatrices intersect) is close to the event
horizon. For σ01, we showed that the asymptotic neg-
ative energy at infinity per enthalpy of the plasma that
is swallowed by the black hole is
' −γoutvout /3'
pσ0/3. On the other hand, the plasma that escapes
to infinity and takes away black hole energy asymptotes
the energy at infinity per enthalpy
+'3γoutvout '
3σ0.
We calculated the power extracted from the black hole
by the escaping plasma and evaluated its maximum when
the dominant reconnection X-point is close to the event
horizon. This corresponds to Pmax
extr 0.1M2σ0w0for
the collisionless plasma regime and one order of mag-
nitude lower for the collisional regime. The overall effi-
ciency of the plasma energization process via magnetic re-
connection can reach a maximum of ηmax '3/2. There-
fore, the additional energy extracted from the black hole,
while important, does not extensively modify the energet-
ics of the escaping plasma. On the other hand, the power
extracted via fast magnetic reconnection can induce a
significant reduction of the rotational energy of the black
hole, dErot/dt =
w0AinUin . This is effective when ais
close to unity. Therefore, if we consider a black hole with
dimensionless spin parameter close to unity and define
$= 1 a1, we have dErot/dt ' −(M/4$)d$/dt
and the spindown time can be obtained as
tsd =O(10)
2σ0w0M($f$i),(47)
where the subscripts f and i are used to label final and
initial values, respectively. This indicates that mag-
netic reconnection can cause a significant spindown of
11
the black hole when a1. For example, fast mag-
netic reconnection in the ergosphere can reduce the black
hole dimensionless spin from a= 0.999 to a= 0.99 in
tsd 1/(σ0w0M). On the other hand, at lower spin
values, especially for a < 0.9, magnetic reconnection loses
its efficacy as the plasma available in the ergosphere di-
minishes.
Various systems hosting a black hole are expected to
have magnetization σ0&1 in the ergosphere. For the
typical conditions around supermassive black holes in ac-
tive galactic nuclei (AGNs), the energy density of the
electromagnetic field far exceeds the enthalpy density
of the plasma and σ0104or larger [48, 56, 57] is
foreseeable. Likewise, long and short gamma-ray bursts
(GRBs) may have σ01 or larger [58–61] in the ergo-
sphere (a central black hole is assumed). Under these
magnetization conditions (in addition to a1), mag-
netic reconnection is capable of extracting energy from
the black hole. For σ01104, we have shown that
the bursty energy extraction rate occurring during fast
magnetic reconnection can exceed the more steady en-
ergy extraction rate expected from the Blandford-Znajek
mechanism. On the other hand, as the plasma magneti-
zation increases, energy extraction via fast magnetic re-
connection becomes always subdominant since it requires
non-vanishing plasma inertia.
In the scenario proposed here, fast magnetic reconnec-
tion occurs rapidly and intermittently, so that the as-
sociated emission within a few gravitational radii from
the black hole is expected to be bursty in nature. This
bursty behavior of fast magnetic reconnection might be
responsible for triggering flares in the vicinity of rotat-
ing black holes. Indeed, frequent X-ray and near-infrared
flares are detected on a regular basis from the Galactic
Center black hole Sgr A* [e.g. 62–65], and magnetic re-
connection close to the black hole is often conjectured to
induce these flares [e.g. 25, 56, 66]. Recent observations
by the GRAVITY collaboration [67] have been able to pin
down the motion of near-infrared flares originating near
the last stable circular orbit of Sgr A*. Reconnection
layers originate naturally in the ergosphere of rotating
black holes and produce plasmoids/flux ropes that are
filled with energized plasma with an energy budget that
can exceed the energy originally stored in the magnetic
field.
In this paper we have assumed that the plasma rotates
circularly around the black hole. This assumption may be
relaxed in order to treat more complex scenarios in which
reconnection occurs in non-circular orbits. In this case,
the plasma could approach the event horizon even when
the black hole spin is not particularly high, expanding
the parameter space region where magnetic reconnection
can extract black hole energy. Another situation that
could increase the efficacy of magnetic reconnection is the
simultaneous presence of equatorial and non-equatorial
current sheets [25], which may result in an increase of the
extracted power to some degree. Finally, for reconnecting
magnetic fields that have a significant radial component,
particle acceleration owing to the reconnection electric
field can increase the rate of energy extraction and the
overall efficiency of the reconnection process.
Acknowledgments
We gratefully acknowledge discussions with Lorenzo
Sironi, Daniel Groˇselj, Russell Kulsrud, Manasvi Lingam,
Yi-Hsin Liu, Joonas N¨attil¨a, Kyle Parfrey, Bart Rip-
perda, Daniel Siegel, and Yajie Yuan. L.C. acknowl-
edges support by the NASA ATP NNX17AG21G and
NSF PHY-1903412 grants. F.A.A. acknowledges support
by the Fondecyt-Chile Grant No. 1180139.
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... [13][14][15] Beyond the more exotic Hawking radiation, another extraction mechanism has recently been proposed. In Ref. [16], the authors showed that, when a Kerr BH is immersed in an externally supplied magnetic field, reconnection of magnetic field lines within the ergosphere can generate negative energy particles (relative to infinity) that fall into the event horizon and positive energy particles which steal energy from the black hole. In other words, Magnetic reconnection accelerates part of the plasma in the direction of the black hole rotation and another part in the opposite direction which falls into the black hole. ...
... This rapidly converts the available magnetic energy into plasma particle energy. Comisso and Asenjo [16] analytically found that this channel is several times more efficient than the BZ one, but energy extraction is possible only for an extreme rotating black hole, a ∼ 1, and in presence of strongly magnetized plasma, σ > 1/3, where σ is the plasma magnetization. In this scenario, the maximum power extracted is when the dominant reconnection point is close to the event horizon and corresponds to P max extr ∼ 0.1M 2 √ σ w 0 , where M is the black hole mass, w 0 the plasma enthalpy density and we are considering a collisionless plasma regime. 1 The minimum σ value for extracting energy is ∼ 1/3 but the efficiency of such a process is greater than 1 only for σ 1. ...
... Here, the authors computed the energy and angular momentum extraction rate of a split-monopole magnetosphere. In the wake of Ref. [16], instead, the role of a Lorentz parameter l in energy extraction has recently been investigated Ref. [20]. The author found that energy extraction power from a rotating BH solution with broken fundamental Lorentz symmetry is, in some cases, more efficient than in the classical Kerr solution. ...
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Black holes can accumulate a large amount of energy, responsible for highly energetic astrophysical phenomena. Although the Blandford–Znajek is considered to date the leading mechanism for powering jets and GRBs, recently fast magnetic reconnection (MR) of the magnetic field was proposed as a new way to extract energy. In this paper, we investigate this phenomena in a bumblebee Kerr–Sen BH, which differentiates from standard Kerr solution via a Lorentz symmetry breaking parameter $$\ell $$ ℓ and an electric charge one b . We find that the presence of the charge parameter strongly changes the simple Kerr case, making this extraction mechanism possible even for not extremely rotating black holes ( $$a\sim 0.7$$ a ∼ 0.7 ). We also show that, under appropriate circumstances, MR is more efficient compared to the Blandford–Znajek mechanism. We finally compare these results with dark energy (quintessence) black-hole solutions. In this case, we find that a Kerr black hole is indistinguishable from a rotational Kiselev one, whatever the energy-matter that surrounds it (including ordinary matter, such as dust and radiation).
... This rapidly converts the available magnetic energy into plasma particle energy. Comisso and Asenjo [16] analytically found that this channel is several times more efficient than the BZ one, but energy extraction is possible only for an extreme rotating black hole, a ∼ 1, and in presence of strongly magnetized plasma, σ > 1/3, where σ is the plasma magnetization. In this scenario, the maximum power extracted is when the dominant reconnection point is close to the event horizon and corresponds to P max extr ∼ 0.1M 2 √ σw 0 , where M is the black hole mass, w 0 the plasma enthalpy density and we are considering a collisionless plasma regime 1 . ...
... We also found that, given a value of the charge b (with b < 0.4), the smallest value of in order to have r isco < r out is = −b. Following Ref. [16], we assume that magnetic reconnection happens in the the bulk plasma which stably rotates around the black hole. Since the orbit is supposed circular, the angular velocity is keplerian, w K =φ/ṫ. ...
... where p, w, u µ and F µν are plasma pressure, enthalpy density, velocity and electromagnetic tensor, respectively. Neglecting electromagnetic component (assuming a highly efficient transformation of magnetic energy into kinetic energy), the energy density at infinity [16] is ...
Preprint
Black holes can accumulate a large amount of energy, responsible for highly energetic astrophysical phenomena Recently, fast magnetic reconnection (MR) of the magnetic field was proposed as a new way to extract energy and in this paper, we investigate this phenomena in a bumblebee Kerr-Sen BH. We find that the presence of the charge parameter strongly changes the simple Kerr case, making this extraction mechanism possible even for not extremely rotating black holes ($a \sim 0.7$). We also show that, under appropriate circumstances, MR is more efficient compared to the Blandford-Znajek mechanism. We finally compare these results with quintessence black-hole solutions not finding and enhancement respect to Kerr solution.
... Energy extraction from black holes, which is connected to various significant astrophysical events, including black hole jets and therefore Gamma-ray bursts (GRBs), is one area where NLED effects have not yet been properly studied [60]. The Blandford-Znajeck (BZ) process [61-66] and the (very recent) magnetic reconnection mechanism [67,68] are the two different energy extraction techniques used today, omitting the Penrose process [69]. The BZ mechanism is still the most widely accepted theory to explain high energy phenomena [70,71], even if there are still open questions (in certain models or combinations) [72][73][74]. ...
Preprint
Non-linear electrodynamics (NLED) is a generalization of Maxwell's electrodynamics for strong fields, where vacuum polarization in quantum electrodynamics (QED) results in nonlinear interaction between the electromagnetic fields (EMF). This interaction might lead to a new field of nonlinear electrodynamics, which could have significant implications for the study of black holes and cosmology and have been extensively studied in the literature, extending from quantum to cosmological contexts. Recently, its application to black holes, inflation and dark energy has caught on, being able to provide an accelerated Universe and address some current theoretical inconsistencies, such as the Big Bang singularity. In this work, we have analyzed the Blandford-Znajeck mechanism in light of this promising theoretical context, providing the general form of the extracted power up to second order in the black hole spin parameter a. We have found that, depending on the NLED model, the emitted power can be extremely increased or decreased, and that the magnetic field lines around the black hole seems to become vertical quickly. Considering only separated solutions, we have found that no monopole solutions exist and this could have interesting astrophysical consequences. Finally, we have tried to constrain the NLED parameters by forcing the amplification of primordial magnetic fields, finding that this could be a good way to study NLED only in some models. Last but not least, we attempted to confine the NLED parameters by inducing the amplification of primordial magnetic fields, however this approach proved to be effective for NLED research only in certain models.
... By referring to seminal paper [31], one will face the postulation stating that if the superradiance arising from a perturbed black hole becomes reflected toward the event horizon of the black hole repeatedly; then, due to an initial small perturbation, an exponential growth without any bound arises which results in a new phenomenon known as black hole bomb [32]. The mentioned effect is one of the most considered processes around black holes, whose key characteristic is trapping the radiation between the event horizon and a reflect- 1 It is worthy to mention it is not the only way for energy extraction from a black hole, rather there are other theoretical mechanisms such as the Penrose process [9] and magnetic reconnection [10,11]. The fact that the superradiance effect, the contrary Hawking radiation [12] is justifiable in the context of classic physics without any origination in quantum mechanics has caused a lot of renewed interest in studying it via employing various classes of black hole solutions admitted by extended theories of gravity (e.g., see [13]- [30]). ...
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... Our Galaxy and other spiral galaxies are endowed with coherent large scale magnetic fields with typical length ≥ 10 kpc and strength of ∼ 3 × 10 −6 G. They play important roles in a multitude of astrophysical phenomena, such as the confinement of cosmic rays, the transfer of angular momentum away from protostellar clouds (allowing their collapse in stars), the genesis of gamma ray-bursts (GRBs) and, recently, the extraction of energy from BHs [1][2][3][4]. While local strong magnetic fields (up to 10 14 G) come out from stars or compact objects (like neutron stars), galactic and intergalactic magnetic fields still have no explanation, being one of the long standing problem of astrophysics and cosmology. ...
Preprint
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... refs. [3] - [9]. ...
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Magnetic reconnection in strongly magnetized astrophysical plasma environments is believed to be the primary process for fast energy release and particle energization. Currently, there is strong interest in relativistic magnetic reconnection in that it may provide a new explanation for high-energy particle acceleration and radiation in strongly magnetized astrophysical systems. We review recent advances in particle acceleration and reconnection physics in the magnetically dominated regime. Much discussion is focused on the physics of particle acceleration and power-law formation as well as the reconnection rate problem. In addition, we provide an outlook for studying reconnection acceleration mechanisms and kinetic physics in the next step.
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Events GW170817 and GRB 170817A provide the best confirmation so far that compact binary mergers where at least one of the companions is a neutron star can be the progenitors of short gamma-ray bursts (sGRBs). An open question for GW170817 remains the values and impact of the initial neutron star spins. The initial spins could possibly affect the remnant black hole mass and spin, the remnant disk, and the formation and lifetime of a jet and its outgoing electromagnetic Poynting luminosity. Here we summarize our general relativistic magnetohydrodynamic simulations of spinning, neutron star binaries undergoing merger, and delayed collapse to a black hole. The binaries consist of two identical stars, modeled as Γ=2 polytropes, in quasicircular orbit, each with spins χNS=−0.053, 0, 0.24, or 0.36. The stars are endowed initially with a dipolar magnetic field extending from the interior into the exterior, as in a radio pulsar. Following the merger, the redistribution of angular momentum by magnetic braking and magnetic turbulent viscosity in the hypermassive neutron star (HMNS) remnant, along with the loss of angular momentum due to gravitational radiation, induces the formation of a massive, nearly uniformly rotating inner core surrounded by a magnetized Keplerian disklike envelope. The HMNS eventually collapses to a black hole, with spin a/MBH≃0.78 independent of the initial spin of the neutron stars, surrounded by a magnetized accretion disk. The larger the initial neutron star spin the heavier the disk. After Δt∼3000M−4000M∼45(MNS/1.625 M⊙) ms−60(MNS/1.625 M⊙) ms following merger, a mildly relativistic jet is launched. The lifetime of the jet [Δt∼100(MNS/1.625 M⊙) ms−140(MNS/1.625 M⊙) ms] and its outgoing Poynting luminosity [LEM∼1051.5±1 erg/s] are consistent with typical sGRBs, as well as with the Blandford-Znajek mechanism for launching jets and their associated Poynting luminosities.
Article
We analytically explore the effects of the gravitational electromotive force on magnetic reconnection around Schwarzschild black holes through a generalized general-relativistic magnetohydrodynamic model that retains two-fluid effects. It is shown that the gravitational electromotive force can couple to collisionless two-fluid effects and drive magnetic reconnection. This is allowed by the departure from quasineutrality in curved spacetime, which is explicitly manifested as the emergence of an effective resistivity in Ohm’s law. The departure from quasineutrality is owed to different gravitational pulls experienced by separate parts of the current layer. This produces an enhancement of the reconnecion rate due to purely gravitational effects.
Article
Black holes drive powerful plasma jets to relativistic velocities. This plasma should be collisionless, and self-consistently supplied by pair creation near the horizon. We present general-relativistic collisionless plasma simulations of Kerr-black-hole magnetospheres which begin from vacuum, inject e± pairs based on local unscreened electric fields, and reach steady states with electromagnetically powered Blandford-Znajek jets and persistent current sheets. Particles with negative energy at infinity are a general feature, and can contribute significantly to black-hole rotational-energy extraction in a variant of the Penrose process. The generated plasma distribution depends on the pair-creation environment, and we describe two distinct realizations of the force-free electrodynamic solution. This sensitivity suggests that plasma kinetics will be useful in interpreting future horizon-resolving submillimeter and infrared observations.
Article
We report the detection of continuous positional and polarization changes of the compact source SgrA* in high states ('flares') of its variable near- infrared emission with the near-infrared GRAVITY-Very Large Telescope Interferometer (VLTI) beam-combining instrument. In three prominent bright flares, the position centroids exhibit clockwise looped motion on the sky, on scales of typically 150 micro-arcseconds over a few tens of minutes, corresponding to about 30% the speed of light. At the same time, the flares exhibit continuous rotation of the polarization angle, with about the same 45(+/-15)-minute period as that of the centroid motions. Modelling with relativistic ray tracing shows that these findings are all consistent with a near face-on, circular orbit of a compact polarized 'hot spot' of infrared synchrotron emission at approximately six to ten times the gravitational radius of a black hole of 4 million solar masses. This corresponds to the region just outside the innermost, stable, prograde circular orbit (ISCO) of a Schwarzschild-Kerr black hole, or near the retrograde ISCO of a highly spun-up Kerr hole. The polarization signature is consistent with orbital motion in a strong poloidal magnetic field.