Nonlinear electrodynamics: The missing trigger for the formation of astrophysical

charged black holes in gravitational core collapse supernovae

Herman J. Mosquera Cuesta∗

Instituto Federal de Educacao, Ciencia e Tecnologia do Ceara,Avenida Treze de Maio, 2081, Benfica, Fortaleza/CE, CEP 60040-531, Brazil

(Dated: July 22, 2014)

Theorists of the general theory of relativity have since long contended that in nature there exists electricallycharged black holes (CBH), celestial objects which a distant observer would characterize by their mass andcharge. Notwithstanding, none astrophysical mechanism has been proved to self-consistently break up forlong the universal global charge neutrality of most cosmic systems. Foundational arguments from nonlinearelectrodynamics (NLED) provide a mechanism able to drive the formation of an astrophysical CBH after aphase transition in a massive proto-neutron star (P-NS) and the subsequent gravitational collapse of its core.Due to its repulsive action (nonlinear exponential grow of the initial field in a rotating P-NS caused by positivefeedback to itself) NLED allows, as compared to the gravitational timescale (∆Tgrav ≃ 1/

√GρNS & 10−4

s), to make it longer the timescale for Coulombian (electrostatic) neutralization (∆T ≃ λDebye/c . 10−20

s). With no NLED effects such neutralization would take place at the P-NS inner crust-upper mantle chargeinterface much earlier than the gravitational core collapse would take over. In such stalled state of chargeseparation held up by NLED, the aftermath of gravitational collapse of the positively charged inner core canbe an astrophysical CBH.

PACS numbers: 97.60.Jd , 97.60.Lf , 97.60.-s , 03.50.De , 04.70.-s , 04.40.Dg

General relativity (GR) and charged black holes.—It has since long been contended that Einstein equa-tions (EEs) must somehow be realized in nature, astatement based on their exact mathematical solu-tions. One of those describes the space-time (S-T) ofa Reissner-Nordstrom CBH, the metric of which is writ-ten (t, r, θ, φ Schwarzschild coordinates, signature +,-,-,-,units G, c=1, M,Q: mass, charge|∞ :: dΩ2 = r2dθ2 +r2 sin2 θdϕ2)

ds2 = (1− 2M

r+

Q2

r2)dt2 − dr2

(1− 2Mr + Q2

r2 )− dΩ2 , (1)

In spite of this superb theoretical argument, most as-trophysicists still pose the question on the nature andmechanism able to break up the otherwise eternal globalcharge neutrality characterizing any astronomical object.To the best of our knowledge, the debate on this puzzlehas not conclusively been shut off (for related works see[1]). The issue then remains a very open problem in rel-ativistic astrophysics. Notwithstanding see Ref. [? ]Nonlinear electrodynamics.— NLED is a theory for

describing electromagnetic interactions in a relativisticinvariance set up. Several approaches were envisioned:Heisenberg; Euler and Kochel; Euler; Heisenberg andEuler (added F 2-term); Weisskopf (added logarithmic-like term) [2], Born; Born and Infeld [3] (bounded theelectric field strength by giving to the electron a fi-nite radius), and Plebanski (robust framework, includingplasma physics) [4], to extend Maxwell electrodynamics(linear in Lorentz invariants F , G) so as to deal with di-vergences in analysis of electromagnetic (EM) phenom-ena (see Eq.(2)). Among those problems are the ionizedgas for which a naive (even a quantum mechanical) calcu-lation of the ground-state energy density yields infinity,the electric field of point charges (infinite self-energy), or

the catastrophic instability of the semi-classical Bohr’satomic model, in which the orbiting electron should in-escapably plunge onto the proton due to radiation reac-tion. Examples of Lagrangians read (G = 0, µ, b const.)

a)LHE = −1

4F +

µ

4F 2, b)LB

I =b2

2

[

−(

1 +F

b2

)1

2

+ 1

]

(2)Applications of NLED have been extensively studied inthe literature, extending from cosmological and astro-physical contexts [5], to nonlinear optics [6], high powerlaser technology and plasma physics [7], and the fieldnonlinear exponential grow due to chiral plasma insta-bility during the weak parity-violating electron-capture(chirality imbalance) process in core collapse SNe [8] [?].In many respects, the feature highlighted above can

be understood as if the dynamics of the EM field in avacuum were afforded with some sort of (dark energy)repulsive action or back reaction effect [9, 10], i.e. EMfield feedback to itself (see Eq.(4) next), which appearsdue to self-interaction of the electron, proton and EMfield amidst of (simplest atom semi-classical model), orquantum vacuum frictional effect [11]. The repulsive ac-tion is a fundamental property of the quantum vacuum[12], often overlooked. Onwards we consider it to be thekey piece to pave the pathway to conclusively work outthe since long GR puzzle: How to form a charged blackhole in an astrophysical process such as gravitational corecollapse of (electrically ever neutral) massive P-NS?Theoretical framework.— NLED can be formulated: a)

by realizing that the electric permittivity (ǫ0) and mag-netic susceptibility (µ0) can be functional of the elec-tric (E) and magnetic field (B), b) upon the Maxwellinvariant (F ) and its dual (G), e.g., the power series

2

L =∑∞

j,k=0 cj,kFjGk, or c) via a 4-dim effective the-

ory from strings, M-theory, or AdS/CFT correspondence.The simplest NLED theory is described by the action

S =

∫ √−g L(F,G) d4x :: F = FµνFµν :: G = Fµν

∗Fµν

(3)with Fµν ≡ ∇µAν − ∇νAµ, ∇ν covariant derivative(used as |ν below), ∗Fµν = ǫµνρσFρσ dual bivector,ǫαβγδ = 1

2√−g

εαβγδ: εαβγδ Levi-Civita tensor (ε0123=-

1). By extremalizing Lagrangian L(F (Aµ)), w.r.t. thepotentials Aµ yields (LFn = dnL

dFn , n int., G = 0) [4]

∇ν(LFFµν) = 0 → ∇µF

µν = Jν ≡ −LF 2

LFFµνF|µ . (4)

It describes the propagation of the field discontinuities as(

gµν − 4LFF

LFFµαF ν

α

)

kµkν = 0 . (5)

Hence, photons propagate on an effective metric func-tional of the background field Fµα, a geodesic 6= gµν onthe background S-T. The derivative of Eq.(5) gives

kν∇νkα = 4

(

LFF

LFFµβF ν

β kµkν

)

|α, (6)

showing that NLED brings in a field retarded self-energy

or backreaction force accelerating∣

∣

+

− the photon along its

path. (Astrophysical or cosmological consequences in[5]).NLED inherent repulsion.— A general L(F ) leads to a

perfect fluidlike energy-momentum tensor (E-M T)

Tµν =2√−g

δL(F )√−g

δgµν≡ Tµν = (ρ+p)vµvν−pgµν . (7)

The left-hand-side of Eq.(7) yields (F = 2(ǫ0E2 − B2

µ0))

Tµν = −4LFFα

µ Fαν − Lgµν . (8)

By equating terms in Eqs.(7, 8), one gets (recall thatMaxwell Lagrangian yields: ρ = 3p = 1

2 (E2 +B2))

ρ = −L− 4E2LF , p = L+4

3(E2 − 2B2)LF . (9)

In virtue of the Lagrangian and E-M T structure themagnetic fluid can be thought of as a collection of non-interacting fluids indexed by k = −, 0,+, each of whichobeys the equation of state (EoS) : pk =

(

4k3 − 1

)

ρk [14].This means that there is room for the EoS to exert nega-tive pressure. i.e. reverting its action to push outwards.Let us have a look on other Lagrangians exhibiting

repulsive force (EM field positive feedback to itself):a) an interesting one is based on a truncated Laurent

series (α, β, µ are coupling constants) [14]

L = α2F 2 − 1

4F − µ2

F+

β2

F 2. (10)

That way, one obtains EoS describing ordinary radiationρ1 = −α2F 2 = −4α2B4

s1R8 :: p1 = 5

3ρ1 :::: ρ2 = 14F =

Bs

21R4 :: p2 = 1

3ρ2, plus fluids exerting repulsive action

ρ3 =µ2

F=

µ2

2B2s

R4 :: p3 = −7

3ρ3 (11)

ρ4 = − β2

F 2= − β2

4B4s

R8 :: p4 = −11

3ρ4 . (12)

b) or extending the standard LBI Eq.(2) to the form [14]

L = −γ2√

1 + βF − α2F 2, b)p+ ρ =γ2F (1 − 4α2γ2F )

3ρ.(13)

One can check for such a property by noticing thatEq.(13-b) hints at a field transition value F ≡ Ftrans,so that ρ + p is positive for F < Ftrans, while ρ + p isnegative (violation of strong energy condition) for val-ues larger than Ftrans! (see details in [14]). This way,Lagrangian (13) enters the set producing repulsive dy-namics. Further, E-M T conservation preserves Gausslaw: B = Bs

R2

NS

, a law often called for in high energy

astrophysics to estimate the B-field strength of nascent,glitching pulsars [10, 15], e.g. Eq.(11) in [8], or aftera P-NS structural rearrangement, usually a catastrophicphase transition [10, 16], which inevitably leads to theformation of a black hole.[19]Pulsar charge separation state stalled by NLED.— It is

decidedly attractive this EoS feature of producing nega-tive pressure, since such property can allow, following theonset of the P-NS phase transition (PT), to keep stalledthe P-NS charge separation state, preventing the overlay-ing crust to plunge onto the core, while its gravitationalcollapse can take over, whose dynamics is described by[21] (c stands for core, of radial coordinate rc at collapse

time tc, and A2 = 1− 2Mr + Q2

r2 )

drc

dtc= −A2(rc)

H(rc)

[

H2(rc)−A2(rc)]

1

2 , (14)

with H(rc) =MMc

− M2

c+Q2

2Mcrc, Mc core rest mass. At this

stage the characteristic timescale for Coulombian neu-tralization can grow longer in virtue of conservation ofthe large magnetic helicity associated to the B-field pos-itive exponential grow via self-feeding [8], so that thegravitational core collapse can proceed first. (Bunch ofastrophysical mechanisms for the PT to happen havebeen envisaged [10, 16–18, 20]. Yet a huge amount ofwork has been done to realistically characterize the struc-tural configuration of static, rotating and collapsing NSs[17, 18, 20, 22]). This astrophysical stage is of fundamen-tal incidence for, according to workers in field, it is theprelude of the formation of a CBH [21]. Indeed, the PTmay transiently produce a hybrid star or a quark star[16], before inevitably producing a second SN explosiondriven by the just formed CBH.

3

Vacuum induced magnetization.— In classical electro-dynamics [24] magnetization: magnetic dipole moment

per unit volume is defined by (E = 0 → F = −2B2

µ0

)

H = − ∂L

∂B=

B

µ0−mbr . (15)

On this prescription, the induced magnetization in thePT created vacuum interface, i.e. the response (mbr) tothe action of the pulsar dipole magnetic field, readsa) Born-Infeld in Eq.(2),

∂LBI

∂B= (

1√

1− 2B2

b2µ0

)B

µ0::: mbr|BI = (

1

2√

1− 2B2

b2µ0

)B

µ0

(16)

b) Heisenberg-Euler in Eq.(2) (with µ = 2α2

45(~/mc)3

mc2 ),(Note: this Lagrangian is used only to illustrate the pro-cedure, in the discussion below the Lagrangian of Ref.[11] used instead)

∂LHE

∂B=

B

µ0− 4µ

(

B2

µ0

)

B

µ0::: mbr|HE = 4µ

(

B2

µ0

)

B

µ0

(17)

c) extended Born-Infeld :: LF = − γ2

2 ( β−2α2F√1+βF−α2F 2

),

∂LB−IExt

∂B= −γ2

2

−4β − 16α2B2

µ0√

1− 2βB2

µ0

− 4α2

µ2

0

B4

B

µ0:::

mbr|B−I

Ext=

8α2γ2B2

µ0√

1− 2βB2

µ0

− 4α2

µ2

0

B4

B

µ0. (18)

Eq.(17) can be compared to Eq.(6) in Ref. [11] obtainedthrough a computation up to the first order in the fine

structure constant (α = e2

~c ≃ 1137 ). Thus, from Eqs.(16,

17, 18) the induced magnetization as functional F of theLagrangian defining the P-NS external field reads

mbr = F(

B

µ0

) ∣

∣

∣

∣

L

B

µ0. (19)

Meanwhile, in collapse theory some pre-SN stellar corescan achieve enough spin as to rotate near Keplerian equa-torial break-up frequency: ΩK ≥ ([ 23 ]

3GNMR3 )1/2, imply-

ing a period PK ∼ 0.6 s, after core bounce. Moreover,submillisecond PSRs spinning at Ω ≃ 1122 Hz have beendiscovered [28]. Thus, P −→ ΩR

c ≪ 1 indicates the (spin)range where vacuum magnetization is at work. Hence,by defining the P-NS by its m magnetic dipole moment,R radius and Bs surface B-field strength (Bs ≃ µ0m

4πR3 ::m = ‖m‖), the dipole B-field leading term reads [24]

B(r,t) ≃ µ0

4π

[

3r(m(t− rc ) · r)

r5− m(t− r

c )

r3

]

. (20)

The term t − rc in m accounts for retardation effects.

Eq.(20) states that at point r the induced magnetic mo-ment of the vacuum back reaction reads (its origin can be

traced back to Eq.(4): ∇µFµν = Jµ, Jµ = J

µind + J

µext =

−LFF

LFFµνF|ν , i.e. even if Jµ

ext = 0, the vacuum inducedcurrent stems from field feedback on itself (retarded self-energy))

dmbr(r,t) =

[

F(

B

µ0

) ∣

∣

∣

∣

B

I

,

∣

∣

∣

∣

H

E

,

∣

∣

∣

∣

B−I

Ext

]

B(r,t) dV (r, θ, φ),

(21)with dV = r2 sin θdrdθdφ, (r, θ, φ) and (x, y, z) spherical,and cartesian coordinates. Thus, at time t+ r

c the B-fielddBbr produced by dmbr(r,t) at the pulsar center r is

dBbr(0,t+r

c) ≃ µ0

4π

[

3r(dmbr(r,t) · r)r5

− dmbr(r,t)

r3

]

.(22)

This induced magnetization interacts with the P-NS spin-ning magnetic dipole moment by dissipating energy.As stressed above, (quantum) vacuum can ever be

thought of as an ordinary medium. [6] To this, classicalelectrodynamics dictates the rate at which energy is lost[24] (unit vector uz||Ωz :: Ω = 2π

P rotation frequency)

dEbr = −(

m(t+r

c)× dBbr(0,t+

r

c))

Ω · uz . (23)

By integration from the star radius to infinity, and aver-aging over several periods (P ), Eq. (23) yields

Ebr =

∫ ∞

R

∫ π

θ=0

∫ 2π

φ=0

〈dEbr〉P . (24)

Now, for the moment, let us focus on the study case ala Heisenberg-Euler using the full Lagrangian in Eq.(21).(For P-NS we showed in Ref.[23] that the Lagrangian inEq.(2) leads to p = 1

3ρ− ργ , with ργ = 163 c1B

4. For su-percritical fields ργ dominates, so that the EoS becomesnegative, i.e. the condition to provide repulsive dynamicsis reached). In connection to Eq.(24), Ref. [11] showed,after performing the analysis of the dissipation rate usingthe infinite series characteristic of the full Heisenberg-Euler Lagrangian, that for nearly overcritical B-fields(≃ 6× 1014 G) it reduces to

Ebr ≃ α

(

18π2

45

)

sin2 θ

µ0c

R4

B2cP

2B4

s , (25)

while a la Maxwell the energy dissipation rate reads [10]

EMaxw =

(

128π5

3

)

sin2 θ

µ0c3R6

P 4B2

s . (26)

A confrontation of these energy losses hints at funda-mental changes w.r.t. the method currently in use toestimate the B-field strength of pulsars [10, 15]. First,one can verify that the backreaction energy lost dependson B4

s , while the standard one grows as B2s . Then, the B-

field strength is inferred by assuming that the pulsar EMpower release is explained by the classical dipole model

4

[10, 15]. It can thus be conceded that in order to consis-tently infer the B-field strength of extremely magnetized,slow pulsars one should take into account the backreac-tion or vacuum frictional effects, otherwise such fieldswould be severely overstimated, as is the case for the so-called “magnetars” [11]. Let us now proceed to estimatethe B-field strength needed to delay the electrostatic neu-tralization process at the charged interface.

Making it longer the (+, -) neutralization timescale.—Let us first summarize the astrophysical situation underanalysis: a charged black hole (CBH) is to form. First, aPNS phase transition should take place [16, 18–20]. In a∼ 2.6 M⊙ supermassive PNS [22] it happens catching inthe crust mainly the swiftest relativistic electrons and theprecipitated protons in the core. NLED acts via a repul-sive action helping to avoid a quick neutralization, thusmaking longer the electrostatic timescale. Several formsof energy are relevant to this process: gravitational, ro-tational, magnetic, etc. Forming the CBH exhausts mostof those energies, except for the non extractable part asdiscussed in Ref.[25]. Dissipative effects are mainly elec-tromagnetic: vacuum friction and Maxwell radiation (nogravitational waves, nor plasma viscosity, etc). Becausethe gravitational timescale of collapse to form the CBH isnot modified, it is w.r.t. it that the timescale dictated byelectromagnetism must be compared to. The extractableenergy becomes the source of the supernovalike event fol-lowing the CBH formation, via vacuum polarization andpair creation which self-propels outward, while also con-sumes the total BH charge [21, 26, 27]. Finally, this su-pernovalike event should produce a late time bump in thelightcurve of the already expanding host SN. It is a keymatter to check for this signature in SNe data.

A typical neutron star has density ρNS = 5 × 1014 gcm−3, radius RNS ≃ 10 km, and mass 1.4 M⊙. The NStotal mechanical energy reads: ENS = Egrav + Espin +Emagn. Bearing in mind that NLED dictates the dynam-ics of the B-field permeating the charge interface, therebygenerating repulsive action to transiently avoid the neu-tralization, one can estimate how much longer can theelectrostatic timescale go on: ∆TNLED = ENS

EMaxw+Ebr,

by equating it to the timescale dictated by gravity:∆T grav = 1√

Gρ& 10−4 s. Such a relation can be cast

in the form

1√Gρ

=G

M2

NS

RNS+ 2

5MNSΩ2NSR

2NS +

B2

s

8π R3NS

α(

18π2

45

)

sin2 θµ0c

R4

B2cP

2B4s +

(

128π5

3

)

sin2 θµ0c3

R6

P 4B2s

(27)By solving for Bs this fourth order quadratic equationusing as fiducial period P ∼ 1 ms [28] and sin θ = [1, 12 ],the B-field strength at the charge separation interface is:Bs ≃ [∼ 3.5 × 1014− . 1015] G. This estimated B-fieldstrength at the charge interface accomplishes the condi-tion of validity of the (25) formula. Then, this timescalecould be made even more longer in virtue of either the

magneto-differential rotation [29] or the conservation ofthe large magnetic helicity (H =

∫

dxA ·B :: A vectorpotential) associated to exponential grow of the P-NSB-field caused by the large chiral imbalance of electrons(plasma instability) in the parity-violating weak processof deleptonization during the SN core collapse [8]. There-after, the gravitational collapse of the electrically-chargedcore can take over to produce a CBH.B-field amplification via differential rotation.— The

state-of-the-art in astrophysics is called for next, see[10, 15]. A newly-born NS may undergo vigorous con-vection during the first 30-60 s. If the P-NS spins dif-ferentially extremely fast (P . 1 ms) conditions are cre-ated for the α−Ω dynamo to get into action, which maysurvive depletion due to turbulent diffusion. In a dif-ferentially rotating P-NS, the poloidal (Hφ) and radial-dependent toroidal (Hr) B-fields are connected through

the relation [29]:dHφ

dt = Hr

[

r dΩdr

]

. At the initial stage:Hφ < H⋆

φ (poloidal B-field at the beginning of exponen-

tial grow), so that one can assume Hr

[

r dΩdr

]

= const.

This leads to the formation of multiple poloidal dif-ferentially rotating vortexes (v) governed by the law:dHr

dt = Hr:t⋆[

r dωv

dλ λ]

, with λ the vortex length scale. In

general, one can approximate:[

r dωv

dλ λ]

≃ α(Hφ −H⋆φ),

with Hr:t⋆ initial toroidal B-field. By assuming for thesake of simplicity that

[

r dΩdr

]

= A is a constant duringthe first stages, and taking H⋆

φ as a constant, one arrivesto the following equation:

d2

dt2(Hφ −H⋆

φ) = AHr:t⋆α(Hφ −H⋆φ) (28)

which leads to exponential grow of the B-fields, with

Hφ(t) = H⋆φ +Hr:t⋆e

√AαHr:t⋆ (t−t⋆) (29)

Hr(t) = Hr:t⋆ +H

3/2r:t⋆α

1/2

√A

[e√AαHr:t⋆ (t−t⋆) − 1](30)

Thus, both magnetic field (r, φ) components grow ex-

ponentially, ending up with ratio Hr(t)Hφ(t)

∼ 10−2 [8, 29,

30]. Hence, under collapse conditions, B-fields B ∼1017−18

(

P1ms

)

G may be generated as long as the differ-ential rotation is dragged out by the growing magneticstresses. For this process to efficiently operate the ra-tio: spin rate (P )/convection overturn timescale (τconv),the Rossby number (R0), should be R0 ≤ 1. Then, anordinary dipole Bdip ∼ [1012−1013] G can be built by in-coherent superposition of small dipoles of characteristicsize λ ∼ [ 13 − 1] km, so that a surface saturation strength

Bsat = (4πρ)1/2 λτconv

≃ 1016−17 G can be reached, asvery recently proved by [29, 30]. Indeed, in the dipole B-field scheme, this means that an induced magnetizationB ∼ 1020 G can be reached at the very km-scale deepinner core, catastrophically destabilizing it.Chiral plasma instability and large magnetic helicity

— Basic idea from Ref. [8].— In core collapse SN the

5

electron (e−) capture on protons leads to a right-to-lefthanded Fermi surface imbalance µR > µL, i.e. to anonzero (time-integrated) chiral e− chemical potential

µ5 = (µR−µL)2 > 0. Thus, the number of neutrons (n)

is equal to the number difference of right-to-left handede− (N5), so that n5 = µ5

3π2 (µ25+3µ2) ≃ ∆Nn, is the chiral

number density at low temperature, with µ ≡ (µR+µL)2

the chemical potential associated to the U(1) vector-like particle number, and n5 the e− chiral density, and∆Nn = (0.1−1) fm−3 is the n number due to e− captureat the P-NS (1 km size-scale) core. Using natural units(~, c = 1): ∆Nn = (0.1 − 1)Λ3, where Λ = 200 MeV isthe QCD energy scale. Thence, the well known charac-teristic e− chemical potential at the P-NS core: µ . Λimplies that µ5 ∼ Λ. In the above arguments was implicitthat the state with chemical potential µ5 is unstable, andquickly decays by converting its energy into a magneticfield a cause of the chiral plasma instability. Hence theB-field can be derived from energy conservation: e− en-ergy density from the chiral asymmetry, equals to theB-field pressure

∆E =1

4π2(µ4

5 + 6µ25µ

2) ≡ 1

2∆B2

inst , (31)

which leads to Bmax ∼ Λ2 ∼ 1018 G!

Meanwhile, magnetic helicity, which is a MHD invari-ant, guarantees that

d

dt

[

N5 +α

πH]

= 0, N5 =

∫

n5dx , (32)

with N5 the global chiral charge of electrons, and H isthe magnetic helicity, which can be computed as: ∆H =−π

α∆N5 ∼ − 1αVNSΛ

3, with V = 4π3 R3

core the volumeof the NS core. Such large helicity ensures for long thestability of the super strong (P-NS core) magnetic field.

Conclusion.— At the phase transition interface, mag-netic fields this high surely drive the P-NS to collapseto form a CBH, triggering a sort of second SN: a giantexplosion inside a SN. The signature of this vacuum ex-plosion in the light curve of the host SN can be similarto that from r-process heavy n-rech nuclei decay due tothe P-NS crust abundance of neutrons, which is blownoff after the CBH formation. This should produce a latetime bump or re-brightening in the light curve of thehost already expanding SN. This picture may find properrealization in many astrophysical contexts, especially inmodels of gamma-ray bursts (GRBs), including binarysystem-driven GRBs, in which the very central engine hasto be (at least) a Reissner-Nordstrom black hole, whichcan then afford vacuum polarization and a la Schwingerpair creation and the full relativistic hydrodynamics andlight curve evolution characterizing GRBs.

CAPES/ICRANet Program support is acknowledgedfor the Sabbatical Fellowship 0153-14-1 (2014)

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[31] With regard to this tantalizing issue, in an earlier paper[Mosquera Cuesta etal., Phys. Rev. D67 (2003) 087702]a mechanism inspired in brane-world physics was intro-duced which allows for mass disappearance (electrons,rather that protons, leaking) from the brane to the bulkproducing an asymmetry in an otherwise endlessly global

neutral (+ , -) charge distribution lying on the brane, e.g.a star. As a result, an astrophysical charged black holemay come out by end of a supernova (SN) gravitationalcore collapse. This mass leaking mechanism might havealso been at work during the very early universe drivinga matter-antimatter primordial asymmetry.

[32] These authors conceded not having identified what mech-anism helps to enlarge the NS magnetic helicity, thoughthey stress that the original B-field gives a positive feed-back to itself, to grow exponentially, being this last theactual chiral plasma instability. In our understanding,this field nonlinear enlargement is a prove that NLEDis doubtlessly at action inside just-born pulsars (seeEq.(4)). Besides, it is clear that an amplification of theB-field via magneto-differential rotation is concomitantwith this chiral plasma instability.