Non-linear Electrodynamics in Blandford-Znajeck Energy Extraction A. Carleo1 2 3G. Lambiase 1 2yand A. Övgün4z 1Dipartimento di Fisica Università di Salerno Via Giovanni Paolo II 132 I-84084 Fisciano SA Italy

2025-05-02 0 0 1.31MB 17 页 10玖币

侵权投诉

Non-linear Electrodynamics in Blandford-Znajeck Energy Extraction

A. Carleo,1, 2, 3, ∗G. Lambiase ,1, 2, †and A. Övgün 4, ‡

1Dipartimento di Fisica, Università di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy

2INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy

3INAF, Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy

4Physics Department, Eastern Mediterranean University,

Famagusta, 99628 North Cyprus via Mersin 10, Turkey

Non-linear electrodynamics (NLED) is a generalization of Maxwell’s electrodynamics for strong

ﬁelds. It 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 report two new ways to investigate these non-linear theories. First,

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 (not considered here). Last but not

least, we attempted to conﬁne the NLED parameters by inducing the ampliﬁcation of primordial

magnetic ﬁelds (‘seeds’), thus admitting non-linear theories already during the early stages of the

Universe. However, the latter approach proved to be useful for NLED research only in certain models.

Our (analytical) results emphasize that the existence and behavior of non-linear electromagnetic

phenomena strongly depend on the physical context and that only a power-low model seems to have

any chance to compete with Maxwell.

I. INTRODUCTION

Maxwell’s electromagnetic theory (MED) is a widely used fundamental theory in both quantum physics and the

context of cosmology. It is a well-known and recognized theory. In 1933 and 1934 Born and Infeld made the ﬁrst

attempts to change equations of MED [1,2] and tried to eliminate the divergence of the electron’s self-energy in classical

electrodynamics. The Born-Infeld electrodynamics model does not contain any singularities because its electric ﬁeld

starts at its highest value at the center (which is equal to the nonlinearity parameter b), and then gradually decreases

until it behaves like the electric ﬁeld of Maxwell at longer distances. This model also ensures that the energy of a single

point charge is limited. The parameter bhas a connection to the tension of strings in the theory [3,4], and there have

been studies done to determine potential constraints for the value of bin [5–11]. In contrast to the Euler-Heisenberg

electrodynamics [12], the Born-Infeld model does not show vacuum birefringence when subjected to an external electric

ﬁeld. The Born-Infeld theory maintains both causality and unitarity principles. The Born-Infeld electrodynamics has

served as inspiration for other models that are free of singularities and possess similar properties. For instance, various

models presented by Kruglov in [13–24]. Fang and Wang have presented a fruitful method for ﬁnding black hole

solutions that have either electric or magnetic charges, in a theory that combines General Relativity with a nonlinear

electrodynamics [25]. Since then, numerous models have been advocated, and the eﬀects of these theories—known

as Non Linear Electrodynamics (NLED)—have been investigated in a wide range of contexts, not just those related

to cosmology and astrophysics [26–44], but also in non-linear optics [45], high power laser technologies and plasma

physics [46,47], nuclear physics [48,49], and supeconductors [36]. Many gravitational non-linear electrodynamics

(G-NED), extensions of the Reissner-Nordstrom (RN) solutions of the Einstein- Maxwell ﬁeld equations have gained

a lot of attention (see [50–54] and references therein). Additionally, Stuchlík and Schee have demonstrated that

models that produce the weak-ﬁeld limit of Maxwell are considered relevant, as opposed to those that do not provide

the correct enlargement of black hole shadows in the absence of charges [39]. In particular, the existence of axially

symmetric non-linear charged black holes (at least transiently) has been studied [55], indicating neutrinos as good

probes thanks to their bountiful production in any astrophysical context. As a consequence, it would be interesting, in

∗acarleo@unisa.it

†lambiase@sa.infn.it

‡ali.ovgun@emu.edu.tr

arXiv:2210.11162v2 [gr-qc] 6 Feb 2023

principle, to investigate the nature of electromagnetism (linear or not), due to diﬀerent signatures in certain neutrino

phenomena, such as neutrino oscillations, spin-ﬂip and r-processes. The eﬀect of non-linear phenomena on the BH

shadow, BH thermodynamics, deﬂection angle of light and also wormholes have been investigated too [56–69]. In the

context of primordial physics, instead, NLED, when coupled to a gravitational ﬁeld, can give the necessary negative

pressure and enhance cosmic inﬂation [70] and some models also prevent cosmic singularity at the big bang [71–75] and

ensure matter-antimatter asymmetry [76]. The reason to consider NLED in the primordial Universe comes from the

assumption that electromagnetic and gravitational ﬁelds were very strong during the evolution of the early universe,

thereby leading to quantum correction and giving birth to NLED [77,78]. Recently, the non-linear electrodynamics

has been also invoked as an available framework for generating the primordial magnetic ﬁelds (PMFs) in the Universe

[79,80]. The latter, indeed, is a still open problem of the modern cosmology, and although many mechanisms have been

proposed, this issue is far to be solved. Seed of magnetic ﬁelds may arise in diﬀerent contexts, e.g. string cosmology

[81], inﬂationary models of the Universe [82,83], non-minimal electromagnetic-gravitational coupling [84,85], gauge

invariance breakdown [83,86], density perturbations [87], gravitational waves in the early Universe [88], Lorentz

violation [89], cosmological defects [90], electroweak anomaly [91], temporary electric charge non-conservation [92],

trace anomaly [93], parity violation of the weak interactions [94]. The current state of art points to an unexplained

physical mechanism that creates large-scale magnetic ﬁelds and seems to be present in all astrophysical contexts.

They might be remnants of the early Universe that were ampliﬁed later in a pregalactic period, according to one

idea. To create such large-scale ﬁelds, super-horizon correlations can only still be created during inﬂationary epochs.

However, it is still unclear how the electromagnetic conformal symmetry is broken. Diﬀerent theoretical techniques

have been taken into consideration for this, most notably non-minimal coupling with gravity, which by its very nature

broke conformal symmetry ([95] and reference therein). In a minimal scenario, electromagnetic conformal invariance

can also be overcome. In this instance, the major goal is to modify the electromagnetic Lagrangian to a non-linear

function of F.

= (1/4)Fµν Fµν , as done in [79,80,96].

Since all NLED models signiﬁcantly depend on scale factors (dimensionless or not), which may cause overlaps

with other physics observables, it is obvious that determining the presence of non-linear phenomena is not free

of uncertainty. 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 [97]. The Blandford-Znajeck (BZ) process [98–103] and the (very recent) magnetic reconnection

mechanism [104,105] are the two diﬀerent energy extraction techniques used today, along with a revised version of

the original Penrose process [106] called magnetic Penrose process [107–109]. Among them, the BZ mechanism is still

the most widely accepted theory to explain high energy phenomena [110,111] (even if there are still open questions

in certain models or combinations [112–114]). It involves a magnetic ﬁeld generated by the accretion disk, whose

ﬁeld lines are accumulated during the accretion process and twisted inside the rotating ergosphere. Charged particles

within the cylinder of twisted lines can be accelerated away from the black hole, composing the jets. A characteristic

feature of this mechanism is that the energy loss rate decays exponentially. This has been conﬁrmed in a good fraction

of observations (X-ray light curves) of GRBs [115]. Furthermore, black holes with brighter accretion disks have more

powerful jets implying a correlation between the two. Even if accretion onto a black hole is the most eﬃcient process

for emitting energy from matter it is not able to reach the energy rate of the GRBs, while other energy extraction ways

such as the Hawking radiation give predictions on temperature, time-scale and energy rate highly in conﬂict with the

observations [116]. Numerical models of black hole accretion systems have signiﬁcantly progressed our understanding

of relativistic jets indicating two types of jets, one associated with the disc that is mass-loaded by disc material and

the other associated directly with the black hole [117]. In the ﬁrst case, however, jets with high Lorentz factors are

not supported. The BZ process, which produces highly relativistic jets by electromagnetically extracting black hole

spin energy, remains the most astrophysically plausible mechanism to do so and is in good agreement with direct

observations [118]. In this sense, understanding the general relativistic magnetohydrodynamic (GRMHD) model of

the bulk ﬂow dynamics near the black hole (where relativistic jets are formed) is essential to study the central engine.

In this paper, in order to determine if non-linear eﬀects may change the rate of energy extraction and the magnetic

ﬁeld conﬁguration surrounding a (non-charged) black hole encircled by its magnetosphere, we will investigate the

Blandford-Znajek mechanism in the context of the NLED framework.

The layout of the paper is as follows: in Sec. II we derive, for the ﬁrst time, the general version of energy ﬂux up to

second order in the spin parameter. Sec. III is devoted to computing and solving the magnetohydrodynamic problem

in Kerr-Schild coordinates, searching, in particular, for separated (monopole and paraboloid) solutions. In Sec. IV

we give some estimates of the energy extraction w.r.t. standard BZ mechanism. We study primordial magnetic ﬁelds

from (minimally coupled) NLED for diﬀerent non-linear models in Sec. V, while discussion and conclusions are drawn

in the Sec. VI. In this work, we adopt natural units G=c= 1 and for simplicity set M= 1 in order to handle

adimensional quantities (r,a,...). The negative metric signature (+,−,−,−)is also adopted.

II. NON-LINEAR MAGNETOHYDRODYNAMICS

In this section, following [98] and [119], we derive the energy extraction rate for a spinning, non-charged black hole

in presence of stationary, axisymmetric, force-free, magnetized plasma and an externally sourced magnetic ﬁeld. In

the Kerr-Schild coordinate 1, the axially symmetric spacetime line element is

ds2=1−2r

Σdt2−4r

Σdrdt −1 + 2r

Σdr2−Σdθ2−sin2θhΣ + a21 + 2r

Σidφ2

+4ar sin2θ

Σdφdt + 2a1 + 2r

Σsin2θdφdr, (II.1)

where Σ := r2+a2cos2θ,∆ = r2−2r+a2. The metric determinant is g:= |det(gµν )|=−Σ2sin2θ. We consider now

a general electromagnetic Lagrangian governing the surrounding plasma and call it LNLED ; it is generally a function

of the two invariants X:= (1/4)Fµν Fµν and G:= (1/4)Fµν F∗µν , where, called Aµ= (Φ,−A)the four-potential,

Fµν =∂µAν−∂νAµis the electromagnetic ﬁeld strength tensor and F∗µν =1

2Fαβαβµν is its dual (is the anti-

symmetric Levi-Civita tensor ). Clearly, Maxwell theory is recovered when LNLED =−X. The energy-momentum

tensor, in absence of magnetic charges, is

TEM

µν := 2

√−g

δLNLED

δgµν =−L(X)gµν +LXFµρFνσ gρσ ,(II.2)

where with LXwe indicate the derivative of Lw.r.t. X. In principle, the total energy-momentum tensor should

also take matter contribution into account, i.e. Ttot

µν := TEM

µν +TMAT

µν , but in the free-force approximation the latter

disappears [119]. This leads to

∇νTtot

µν ≈ ∇νTEM

µν = 0,(II.3)

together with the generalized Maxwell equations

√−g∂µh√−gLXFµν i=−Jν,(II.4)

∂µF∗µν = 0,(II.5)

with Jν= (ρ, J)the four-current density. Since the plasma is assumed ideal, the electric ﬁeld in the particle frame,

E0, is zero. However, the presence of an external magnetic ﬁeld leads to a non-zero electric ﬁeld E, but the ideal MHD

approximation implies that E·B= 0, i.e. G= 0, from which [119]

∂θAt

∂θAφ

=∂rAt

∂rAφ

=: w(r, θ),(II.6)

where we introduced the function w(r, θ). With this notation, the electromagnetic tensor is

Fµν =√−g





0wBθ−wBr0

−wBθ0−BφBθ

wBrBφ0−Br

0−BθBr0





(II.7)

which automatically satisﬁes (II.4). The radial energy and angular momentum ﬂux, as measured by a stationary

long-distance observer, are given by

F(r)

E:= Tr

t, F (r)

L:= −Tr

φ.(II.8)

1Unlike the classic Kerr coordinates, the Kerr-Schild ones ensure ﬁniteness of the electromagnetic ﬁeld on the horizon. Notice that here

we use a diﬀerent metric signature than [119] and that in [98] simpler Kerr coordinates are used.

Therefore

F(r)

E=−LXFtθFθφgrφ +FtθFθrgrr −F2

tθgrtgθθ,

and hence

F(r)

E=LXh2B2

rwrw−a

2r+wBrBφ∆isin2θ, (II.9)

while the angular momentum ﬂux is F(r)

L=F(r)

E/w. On the horizon, r+:= 1 + √1−a2, Eq. (II.9) reads as

FE(θ) := −2L(r+)

XB2

rwr+(ΩH−w) sin2θ, (II.10)

where L(r+)

X:= LX(r+, θ)and ΩH:= a/(2r+)is the angular velocity of the horizon. Apart from the factor LX, these

relations are equal to the linear (Maxwell) case. However, although the change is minimal, the physical consequences

could be decisive. Indeed, FE(θ)>0not only if 0< w < ΩH, but also if LX<0at the horizon. Moreover, since LX

is a function of X, and 2

X=1

2hB2

r(1 −w2) + B2

θ(1 −w2) + B2

φi,(II.11)

the energy ﬂux will depend not only on the radial magnetic ﬁeld Br, but in general also on the other two components,

namely Bθand Bφ. The power extracted (energy rate) is

PNLED := ZZ dθdφ√−gFE(θ) = 4πZπ/2

dθ√−gFE(θ).(II.12)

In order to evaluate PN LED , we need to solve MHD equations and ﬁnd the expressions for Br,Bθand Bφ. This is

not an easy task, being quite laborious already in the standard Maxwell theory. As a ﬁrst approach, we can certainly

proceed with a perturbative series expansion in powers of a, as originally done in [98]. Since typically one assumes

w= Ω/2, then FE∝a2so a Schwarzschild solution (i.e. a= 0) is ﬁne to obtain an expression for PN LED good up

to second order in the spin parameter. It is clear that such a relation would be accurate only in the regime a1.

Since we want to completely solve the magnetohydrodynamic equations, instead of Eq. (II.3), we use the (equivalent)

set of equations Fµν Jν= 0, coming from free-force approximation. Only two equations are independent, and they

give

Jr=−µ(r, θ)Br, Jθ=−µ(r, θ)Bθ, Jφ=−µ(r, θ)Bφ+Jtw(II.13)

where we deﬁned µ:= −Jθ/Bθ=−Jr/Br. The above equations are formally equivalent to those of [98] and seem not

to depend a priori on the speciﬁc NLED model. However, when coupled to Maxwell equations, diﬀerence with the

linear theory appears clear. Indeed, in order to ﬁnd the explicit expression for µand Jt, from Eqs. (II.4), we get the

following set of equations:

∂rhsin2θLXBθ∆wY + 4r2w−2rai+∂θhsin2θLX2rBφ−wBrYi=−JtΣ sin θ

∂θhsin2θLX(2rwBr+ ∆Bφ−aBr)i=−JrΣ sin θ

∂rhsin2θLX(2rwBr+ ∆Bφ−aBr)i=JθΣ sin θ

∂rhsin2LXθ2rawBθ−a2Bθ+∆Bθ

sin2θi+∂θhsin2θLXaBφ−Br

sin2θi=−JφΣ sin θ .

(II.14)

Together with Eqs. (II.13) and in a very similar way to [98], they lead to 3

µ=−d

dAφhsin2θLX∆Bφ+ 2rwBr−aBri (II.15)

2X=1

2|B|2− |E|2

3Notice that our deﬁnition for Bφdiﬀers from that of [98] by a factor √−g(as assumed in [119]) and we use Kerr-Schild coordinates.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Non-linearElectrodynamicsinBlandford-ZnajeckEnergyExtractionA.Carleo,1,2,3,G.Lambiase,1,2,yandA.Övgün4,z1DipartimentodiFisica,UniversitàdiSalerno,ViaGiovanniPaoloII,132I-84084Fisciano(SA),Italy2INFN,SezionediNapoli,GruppocollegatodiSalerno,Italy3INAF,OsservatorioAstronomicodiCagliari,ViadellaScienz...

展开>> 收起<<

Non-linear Electrodynamics in Blandford-Znajeck Energy Extraction A. Carleo1 2 3G. Lambiase 1 2yand A. Övgün4z 1Dipartimento di Fisica Università di Salerno Via Giovanni Paolo II 132 I-84084 Fisciano SA Italy.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Non-linear Electrodynamics in Blandford-Znajeck Energy Extraction A. Carleo1 2 3G. Lambiase 1 2yand A. Övgün4z 1Dipartimento di Fisica Università di Salerno Via Giovanni Paolo II 132 I-84084 Fisciano SA Italy

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: