Nled and formation_of_astrophysical_charged_b_hs_03_june_2014
1.
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 Educa¸c˜ao, Ciˆencia e Tecnologia do Cear´a,
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 electrically
charged black holes (CBH), celestial objects which a distant observer would characterize by their mass and
charge. Notwithstanding, none astrophysical mechanism has been proved to self-consistently break up for
long the universal global charge neutrality of most cosmic systems. Foundational arguments from nonlinear
electrodynamics (NLED) provide a mechanism able to drive the formation of an astrophysical CBH after a
phase 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 positive
feedback 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 charge
interface much earlier than the gravitational core collapse would take over. In such stalled state of charge
separation held up by NLED, the aftermath of gravitational collapse of the positively charged inner core can
be 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, a
statement based on their exact mathematical solu-
tions. One of those describes the space-time (S-T) of
a 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
= r2
dθ2
+
r2
sin2
θdϕ2
)
ds2
= (1 −
2M
r
+
Q2
r2
)dt2
−
dr2
(1 − 2M
r + Q2
r2 )
− dΩ2
, (1)
In spite of this superb theoretical argument, most as-
trophysicists still pose the question on the nature and
mechanism able to break up the otherwise eternal global
charge neutrality characterizing any astronomical object.
To the best of our knowledge, the debate on this puzzle
has 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 relativistic
invariance set up. Several approaches were envisioned:
Heisenberg; Euler and Kochel; Euler; Heisenberg and
Euler (added F2
-term); Weisskopf (added logarithmic-
like term) [2], Born; Born and Infeld [3] (bounded the
electric ﬁeld strength by giving to the electron a ﬁ-
nite radius), and Plebanski (robust framework, including
plasma 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 ionized
gas 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’s
atomic 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)LH
E = −
1
4
F +
µ
4
F2
, b)LB
I =
b2
2
− 1 +
F
b2
1
2
+ 1
(2)
Applications of NLED have been extensively studied in
the literature, extending from cosmological and astro-
physical contexts [5], to nonlinear optics [6], high power
laser technology and plasma physics [7], and the ﬁeld
nonlinear 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 a
vacuum were aﬀorded with some sort of (dark energy)
repulsive action or back reaction eﬀect [9, 10], i.e. EM
ﬁeld feedback to itself (see Eq.(4) next), which appears
due to self-interaction of the electron, proton and EM
ﬁeld amidst of (simplest atom semi-classical model), or
quantum 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 the
key piece to pave the pathway to conclusively work out
the since long GR puzzle: How to form a charged black
hole in an astrophysical process such as gravitational core
collapse 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 Maxwell
invariant (F) and its dual (G), e.g., the power series
2.
2
L =
∞
j,k=0 cj,kFj
Gk
, 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) d4
x :: F = Fµν Fµν
:: G = Fµν
∗
Fµν
(3)
with Fµν ≡ ∇µAν − ∇νAµ, ∇ν covariant derivative
(used as |ν below), ∗
Fµν
= ǫµνρσ
Fρσ dual bivector,
ǫαβγδ
= 1
2
√
−g
εαβγδ
: εαβγδ
Levi-Civita tensor (ε0123=-
1). By extremalizing Lagrangian L(F(Aµ)), w.r.t. the
potentials Aµ yields (LF n = dn
L
dF n , n int., G = 0) [4]
∇ν (LF Fµν
) = 0 → ∇µFµν
= Jν
≡ −
LF 2
LF
Fµν
F|µ . (4)
It describes the propagation of the ﬁeld discontinuities as
gµν
− 4
LF F
LF
Fµα
F ν
α kµkν = 0 . (5)
Hence, photons propagate on an eﬀective metric func-
tional of the background ﬁeld Fµα
, a geodesic = gµν on
the background S-T. The derivative of Eq.(5) gives
kν
∇ν kα = 4
LF F
LF
Fµβ
F 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 = −4LF F α
µ Fαν − Lgµν. (8)
By equating terms in Eqs.(7, 8), one gets (recall that
Maxwell Lagrangian yields: ρ = 3p = 1
2 (E2
+ B2
))
ρ = −L − 4E2
LF , p = L +
4
3
(E2
− 2B2
)LF . (9)
In virtue of the Lagrangian and E-M T structure the
magnetic ﬂuid can be thought of as a collection of non-
interacting ﬂuids indexed by k = −, 0, +, each of which
obeys the equation of state (EoS) : pk = 4k
3 − 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 = α2
F2
−
1
4
F −
µ2
F
+
β2
F2
. (10)
That way, one obtains EoS describing ordinary radiation
ρ1 = −α2
F2
= −4α2
B4
s
1
R8 :: p1 = 5
3 ρ1 :::: ρ2 = 1
4 F =
Bs
2
1
R4 :: p2 = 1
3 ρ2, plus ﬂuids exerting repulsive action
ρ3 =
µ2
F
=
µ2
2B2
s
R4
:: p3 = −
7
3
ρ3 (11)
ρ4 = −
β2
F2
= −
β2
4B4
s
R8
:: p4 = −
11
3
ρ4 . (12)
b) or extending the standard LB
I Eq.(2) to the form [14]
L = −γ2
1 + βF − α2F2, b)p + ρ =
γ2
F(1 − 4α2
γ2
F)
3ρ
.(13)
One can check for such a property by noticing that
Eq.(13-b) hints at a ﬁeld transition value F ≡ Ftrans,
so that ρ + p is positive for F < Ftrans, while ρ + p is
negative (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 Gauss
law: 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 after
a P-NS structural rearrangement, usually a catastrophic
phase transition [10, 16], which inevitably leads to the
formation 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 the
onset of the P-NS phase transition (PT), to keep stalled
the P-NS charge separation state, preventing the overlay-
ing crust to plunge onto the core, while its gravitational
collapse can take over, whose dynamics is described by
[21] (c stands for core, of radial coordinate rc at collapse
time tc, and A2
= 1 − 2M
r + Q2
r2 )
drc
dtc
= −
A2
(rc)
H(rc)
H2
(rc) − A2
(rc)
1
2
, (14)
with H(rc) = M
Mc
−
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 of
the large magnetic helicity associated to the B-ﬁeld pos-
itive exponential grow via self-feeding [8], so that the
gravitational core collapse can proceed ﬁrst. (Bunch of
astrophysical mechanisms for the PT to happen have
been envisaged [10, 16–18, 20]. Yet a huge amount of
work 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 the
prelude of the formation of a CBH [21]. Indeed, the PT
may transiently produce a hybrid star or a quark star
[16], before inevitably producing a second SN explosion
driven by the just formed CBH.
3.
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 the
PT created vacuum interface, i.e. the response (mbr) to
the action of the pulsar dipole magnetic ﬁeld, reads
a) Born-Infeld in Eq.(2),
∂LB
I
∂B
= (
1
1 − 2B2
b2µ0
)
B
µ0
::: mbr|B
I = (
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)
∂LH
E
∂B
=
B
µ0
− 4µ
B2
µ0
B
µ0
::: mbr|H
E = 4µ
B2
µ0
B
µ0
(17)
c) extended Born-Infeld :: LF = −γ2
2 ( β−2α2
F√
1+βF −α2F 2
),
∂LB−I
Ext
∂B
= −
γ2
2
−4β − 16α2 B2
µ0
1 − 2β B2
µ0
− 4α2
µ2
0
B4
B
µ0
:::
mbr|B−I
Ext =
8α2
γ2 B2
µ0
1 − 2β B2
µ0
− 4α2
µ2
0
B4
B
µ0
. (18)
Eq.(17) can be compared to Eq.(6) in Ref. [11] obtained
through a computation up to the ﬁrst order in the ﬁne
structure constant (α = e2
c ≃ 1
137 ). Thus, from Eqs.(16,
17, 18) the induced magnetization as functional F of the
Lagrangian deﬁning the P-NS external ﬁeld reads
mbr = F
B
µ0 L
B
µ0
. (19)
Meanwhile, in collapse theory some pre-SN stellar cores
can achieve enough spin as to rotate near Keplerian equa-
torial break-up frequency: ΩK ≥ ([2
3 ]3 GN M
R3 )1/2
, imply-
ing a period PK ∼ 0.6 s, after core bounce. Moreover,
submillisecond PSRs spinning at Ω ≃ 1122 Hz have been
discovered [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 − r
c ) · r)
r5
−
m(t − r
c )
r3
. (20)
The term t − r
c 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
= Jµ
ind + Jµ
ext =
−LFF
LF
Fµν
F|ν, i.e. even if Jµ
ext = 0, the vacuum induced
current 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-ﬁeld
dBbr 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, classical
electrodynamics dictates the rate at which energy is lost
[24] (unit vector uz||Ωz :: Ω = 2π
P rotation frequency)
d ˙Ebr = − 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
d ˙Ebr P . (24)
Now, for the moment, let us focus on the study case `a
la Heisenberg-Euler using the full Lagrangian in Eq.(21).
(For P-NS we showed in Ref.[23] that the Lagrangian in
Eq.(2) leads to p = 1
3 ρ − ργ, with ργ = 16
3 c1B4
. For su-
percritical ﬁelds ργ dominates, so that the EoS becomes
negative, i.e. the condition to provide repulsive dynamics
is reached). In connection to Eq.(24), Ref. [11] showed,
after performing the analysis of the dissipation rate using
the 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
B2
c P2
B4
s , (25)
while `a la Maxwell the energy dissipation rate reads [10]
˙EMaxw =
128π5
3
sin2
θ
µ0c3
R6
P4
B2
s . (26)
A confrontation of these energy losses hints at funda-
mental changes w.r.t. the method currently in use to
estimate the B-ﬁeld strength of pulsars [10, 15]. First,
one can verify that the backreaction energy lost depends
on B4
s , while the standard one grows as B2
s . Then, the B-
ﬁeld strength is inferred by assuming that the pulsar EM
power release is explained by the classical dipole model
4.
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 ﬁelds
would be severely overstimated, as is the case for the so-
called “magnetars” [11]. Let us now proceed to estimate
the 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 under
analysis: a charged black hole (CBH) is to form. First, a
PNS phase transition should take place [16, 18–20]. In a
∼ 2.6 M⊙ supermassive PNS [22] it happens catching in
the crust mainly the swiftest relativistic electrons and the
precipitated protons in the core. NLED acts via a repul-
sive action helping to avoid a quick neutralization, thus
making longer the electrostatic timescale. Several forms
of energy are relevant to this process: gravitational, ro-
tational, magnetic, etc. Forming the CBH exhausts most
of those energies, except for the non extractable part as
discussed in Ref.[25]. Dissipative eﬀects are mainly elec-
tromagnetic: vacuum friction and Maxwell radiation (no
gravitational waves, nor plasma viscosity, etc). Because
the gravitational timescale of collapse to form the CBH is
not modiﬁed, it is w.r.t. it that the timescale dictated by
electromagnetism must be compared to. The extractable
energy becomes the source of the supernovalike event fol-
lowing the CBH formation, via vacuum polarization and
pair 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 the
lightcurve of the already expanding host SN. It is a key
matter to check for this signature in SNe data.
A typical neutron star has density ρNS = 5 × 1014
g
cm−3
, radius RNS ≃ 10 km, and mass 1.4 M⊙. The NS
total mechanical energy reads: ENS = Egrav + Espin +
Emagn. Bearing in mind that NLED dictates the dynam-
ics of the B-ﬁeld permeating the charge interface, thereby
generating repulsive action to transiently avoid the neu-
tralization, one can estimate how much longer can the
electrostatic timescale go on: ∆T NLED
= 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
5 MNSΩ2
NSR2
NS +
B2
s
8π R3
NS
α 18π2
45
sin2 θ
µ0c
R4
B2
c P 2 B4
s + 128π5
3
sin2 θ
µ0c3
R6
P 4 B2
s
(27)
By solving for Bs this fourth order quadratic equation
using as ﬁducial period P ∼ 1 ms [28] and sin θ = [1, 1
2 ],
the B-ﬁeld strength at the charge separation interface is:
Bs ≃ [∼ 3.5 × 1014
− 1015
] G. This estimated B-ﬁeld
strength at the charge interface accomplishes the condi-
tion of validity of the (25) formula. Then, this timescale
could be made even more longer in virtue of either the
magneto-diﬀerential rotation [29] or the conservation of
the large magnetic helicity (H = dxA · B :: A vector
potential) associated to exponential grow of the P-NS
B-ﬁeld caused by the large chiral imbalance of electrons
(plasma instability) in the parity-violating weak process
of deleptonization during the SN core collapse [8]. There-
after, the gravitational collapse of the electrically-charged
core 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 may
survive 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 rdΩ
dr . At the initial stage:
Hφ < H⋆
φ (poloidal B-ﬁeld at the beginning of exponen-
tial grow), so that one can assume Hr rdΩ
dr = const.
This leads to the formation of multiple poloidal dif-
ferentially rotating vortexes (v) governed by the law:
dHr
dt = Hr:t⋆ rdωv
dλ λ , with λ the vortex length scale. In
general, one can approximate: rdωv
dλ λ ≃ α(Hφ − H⋆
φ),
with Hr:t⋆ initial toroidal B-ﬁeld. By assuming for the
sake of simplicity that rdΩ
dr = A is a constant during
the ﬁrst stages, and taking H⋆
φ as a constant, one arrives
to 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
3/2
r:t⋆ α1/2
√
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 P
1ms G may be generated as long as the diﬀer-
ential rotation is dragged out by the growing magnetic
stresses. 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, an
ordinary dipole Bdip ∼ [1012
−1013
] G can be built by in-
coherent superposition of small dipoles of characteristic
size λ ∼ [1
3 − 1] km, so that a surface saturation strength
Bsat = (4πρ)1/2 λ
τconv
≃ 1016−17
G can be reached, as
very recently proved by [29, 30]. Indeed, in the dipole B-
ﬁeld scheme, this means that an induced magnetization
B ∼ 1020
G can be reached at the very km-scale deep
inner core, catastrophically destabilizing it.
Chiral plasma instability and large magnetic helicity
— Basic idea from Ref. [8].— In core collapse SN the
5.
5
electron (e−
) capture on protons leads to a right-to-left
handed Fermi surface imbalance µR > µL, i.e. to a
nonzero (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 handed
e−
(N5), so that n5 = µ5
3π2 (µ2
5 +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−
capture
at the P-NS (1 km size-scale) core. Using natural units
( , c = 1): ∆Nn = (0.1 − 1)Λ3
, where Λ = 200 MeV is
the QCD energy scale. Thence, the well known charac-
teristic e−
chemical potential at the P-NS core: µ Λ
implies that µ5 ∼ Λ. In the above arguments was implicit
that the state with chemical potential µ5 is unstable, and
quickly decays by converting its energy into a magnetic
ﬁeld a cause of the chiral plasma instability. Hence the
B-ﬁeld can be derived from energy conservation: e−
en-
ergy density from the chiral asymmetry, equals to the
B-ﬁeld pressure
∆E =
1
4π2
(µ4
5 + 6µ2
5µ2
) ≡
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
π
H = 0, N5 = n5dx , (32)
with N5 the global chiral charge of electrons, and H is
the magnetic helicity, which can be computed as: ∆H =
−π
α ∆N5 ∼ − 1
α VNSΛ3
, with V = 4π
3 R3
core the volume
of the NS core. Such large helicity ensures for long the
stability 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 collapse
to form a CBH, triggering a sort of second SN: a giant
explosion inside a SN. The signature of this vacuum ex-
plosion in the light curve of the host SN can be similar
to that from r-process heavy n-rech nuclei decay due to
the P-NS crust abundance of neutrons, which is blown
oﬀ after the CBH formation. This should produce a late
time bump or re-brightening in the light curve of the
host already expanding SN. This picture may ﬁnd proper
realization in many astrophysical contexts, especially in
models of gamma-ray bursts (GRBs), including binary
system-driven GRBs, in which the very central engine has
to be (at least) a Reissner-Nordstrom black hole, which
can then aﬀord vacuum polarization and `a la Schwinger
pair creation and the full relativistic hydrodynamics and
light curve evolution characterizing GRBs.
CAPES/ICRANet Program support is acknowledged
for the Sabbatical Fellowship 0153-14-1 (2014)
∗
Electronic address: herman@icra.it
[1] R. Ruﬃni, L. Vitagliano, Int. J. Mod. Phys. D 12, 121
(2003); C. Cherubini, R. Ruﬃni, L. Vitagliano, Phys.
Lett. B 545, 226 (2002); R. Ruﬃni, L. Vitagliano, S.-S
Xue, Phys. Lett. B 559, 12 (2003)
[2] H. Euler, B. Kochel, Naturwissenchaften 23, 246 (1935);
H. Euler, Ann. Phys. Lpz. 5, 398 (1936); W. Heisen-
berg, H. Euler, Z. Phys. 98, 714 (1936); V. Weisskopf,
Kong. Dans. Videns. Selskab, Math-fys. Meddeltser 14,
6 (1936); V. F. Weisskopf, “On the Self-energy and the
Electromagnetic Field of the Electron”, Phys. Rev. 56,
72-85 (1939). See also the complete review by G. V.
Dunne, Int. J. Mod. Phys. A 27, 1260004 (2012) and
refs. thereof; G. V. Dunne, Int. J. Mod. Phys. Conf. Ser.
14, 42-56 (2012); arXiv:1202.1557 [hep-th]; R Battesti,
C Rizzo. (2013), “Magnetic and electric properties of a
quantum vacuum”, Rep. Prog. Phys. 76:1, 016401 (2013)
[3] M. Born, Nature (London) 132, 282 (1933); Proc. R. Soc.
A 143, 410 (1934). M. Born, L. Infeld, Nature (London)
132, 970 (1933); Proc. R. Soc. A 144, 425 (1934). J.
Schwinger, Phys. Rev. 82, 664 (1951)
[4] J.F. Plebanski, “Lectures on nonlinear electrodynamics”.
Monograph of the Niels Bohr Institute (Nordita, Copen-
hagen 1968)
[5] H. J. Mosquera Cuesta, G. Lambiase, JCAP 1103, 033
(2011); C. Corda, H.J. Mosquera Cuesta, Astropart.
Phys. 34, 587 (2011); H.J. Mosquera Cuesta, J.M. Salim,
M. Novello, arXiv:0710.5188 [astro-ph].; H.J. Mosquera
Cuesta, G. Lambiase, Phys. Rev. D 80, 023013 (20009);
H.J. Mosquera Cuesta and J.M. Salim, MNRAS 354,
L55 (2004; H.J. Mosquera Cuesta and J.M. Salim, ApJ
608, 925 (2004); H.J. Mosquera Cuesta, J.A. de Fre-
itas Pacheco and J.M. Salim, IJMP A21, 43 (2006); J-
P. Mbelek, H.J. Mosquera Cuesta, M. Novello and J.M.
Salim, Eur. Phys. Letts. 77, 19001 (2007); J.P. Mbelek,
H.J. Mosquera Cuesta, MNRAS 389, 199 (2008)
[6] D. H. Delphenich,“Nonlinear optical analogies in quan-
tum electrodynamics”, arXiv: hep-th/0610088 (2006)
[7] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78, 591
(2006). J. Lundin, G. Brodin, M. Marklund, Phys. of
Plasmas 13, 102102 (2006). E. Lundstrom, etal. Phys.
Rev. Lett. 96, 083602 (2006); T. Heinzl, “Strong-ﬁeld
QED, high-power lasers”, Int.J.Mod.Phys. A27,15 (2012)
[8] A. Ohnishi, N. Yamamoto, “Magnetars and chiral plasma
instability”, arXiv: 1402.4760 v1 (2014). See theory
of chiral plasma instabilities in Y. Akamatsu, N. Ya-
mamoto, Phys. Rev. Lett. 111, 052002 (2013)
[9] R. Ruﬃni, S-S. Xue, Phys. Lett. A 377, 2450 (2013)
[10] Thomas Gold, Nature 218, 731 (1968); P. Goldreich, W.
H. Julian, “Pulsar Electrodynamics”, Astrophys. J. 157,
869 (1969); Y. B. Zeldovich, I. D. Novikov, “Stars and
Relativity” (University of Chicago Press, Chicago 1971);
S. L. Shapiro, S. A. Teukolsky, “Black holes, white dwarfs
and neutron stars: The physics of compact objects” (Wi-
ley & Sons Inc., New York 1985); N.K. Glendenning,
“Compact stars: Nuclear physics, particle physics and
general relativity” (Springer, New York 1997); and T.
Padmanabhan, “Theoretical astrophysics, Vol. II” (Cam-
bridge University Press, Cambridge, England 2001)
[11] A. Dupays, C. Rizzo, D. Bakalov, and G. F. Bignami,
Eur. Phys. Lett. 82, 69002 (2008)
6.
6
[12] P. C. W. Davies, J. Opt. B 7, S40-S46 (2005)
[13] J. Hadamard, “Le¸cons sur la propagation des ondes et les
equations de l’Hydrodynamique” (Hermann, Paris 1903)
[14] M. Novello, J. M. Salim and A.N. Ara´ujo, Phys. Rev.
D 85, 023528 (2012); M. Novello, S.E. P´erez Bergliaﬀa,
J.M. Salim, Phys. Rev. D 69, 127301 (2004); V.A. De
Lorenci et al., Phys. Rev. D 65, 063501 (2002); V.A. De
Lorenci et al., Phys. Lett. B 482:134-140 (2000).
[15] J. M. Lattimer, M. Prakash, “The Physics of Neutron
Stars”, Science 304, 5670, 536-542 (2004); J. M. Lat-
timer, Ann. Rev. Nucl. Part. Sc. 62, 485-515 (2012); A.
K. Harding, D. Lai, Rept. Prog. Phys. 69, 2631 (2006). A.
P´erez Mart´ınez, H. P´erez Rojas, H. J. Mosquera Cuesta,
Eur. Phys. J. C 29, 111123 (2003)
[16] I.N. Mishustin, M. Hanauske, A. Bhattacharyya, L. M.
Satarov, H. Stoecker, W. Greiner, Phys. Lett. B 552, 1
(2003); G. F. Marranghello, T. Regimbau, J. A. de Fre-
itas Pacheco, Int. J. Mod. Phys. D 16, 313-318 (2007); G.
F. Marranghello, C. A. Z. Vasconcellos, J. A. de Freitas
Pacheco, Phys. Rev. D 66, 064027 (2002)
[17] J. Schaﬀner-Bielich, Nucl. Phys. A 804, 309-321 (2008)
[18] G. F. Marranghello, C. A. Z. Vasconcellos, M. Dillig, J.
A. de Freitas Pacheco, Int. J. Mod. Phys. B 17, 5191
(2003); C. A. Z. Vasconcellos, R. O. Gomes, V. Dex-
heimer, et al., e-print arXiv:1402.5624 v1 [astro-ph.SR]
(2014). A. R. Taurines, C. A. Z. Vasconcellos, M. Mal-
heiro, M. Chiapparini, Phys. Rev. C 63, 065801 (2001)
[19] G. E. Brown, H. A. Bethe, Astrophys. J. 423, 659 (1994)
[20] H. Heiselberg, M. Hjorth-Jensen, Phys. Rept. 328, 237
(2000), Pag.327
[21] R. Ruﬃni, L. Vitagliano, S.-S Xue, Phys. Lett. B 573, 33
(2003), and Refs. therein
[22] R. Belvedere, D. Pugliese, J.A. Rueda, etal., Nucl. Phys.
A883, 1 (2012); R. Belvedere, K. Boshkayev, J.A. Rueda,
R. Ruﬃni, Nucl. Phys. A921, 33 (2014)
[23] C. Corda, H. J. Mosquera Cuesta, Mod. Phys. Lett. A
25, 28, 2423 (2010)
[24] A. Sommerfeld, “Electrodynamics” (Academic Press,
New York 1952); J. D. Jackson, Classical Electrodynam-
ics, Chap. 5, pag.168 (Wiley & Sons Inc., New York
1975); L.D. Landau, E.M. Lifshitz, “Electrodynamics
of continuous media” (Pergamon, New York 1963); W.
Greiner, “Classical Electrodynamics” (Springer, 1st Edi-
tion 1998); L. D. Landau, E. M. Lifshitz, “The classical
theory of ﬁelds” (Pergamon, New York 1970)
[25] J. P. Pereira, H. J. Mosquera Cuesta, J. A.
Rueda, R. Ruﬃni, Physics Letters B (2014)
http://dx.doi.org/10.1016/j.physletb.2014.04.047
[26] T. Damour, R. Ruﬃni, Phys. Rev. Lett. 35, 463 (1975)
[27] C. Cherubini, A. Geralico, J. A. Rueda, R. Ruﬃni, Phys.
Rev. D 79, 124002 (2009)
[28] P. Kaaret, Z. Prieskorn, J. J. M. in ’t Zand, et al., As-
trophys. J. 657, L97 (2007)
[29] S. Moiseenko, G. Bisnovagti-Kogan, talk at Zeldovich
100th Aniversary Meeting, Minsk, Bielorus (2014)
[30] D. M. Siegel, R. Ciolﬁ, L. Rezzolla, “Magnetically driven
winds from diﬀerentially rotating neutron stars and ...”
e-print: arXiv:1401.4544 v2 (2014)
[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 bulk
producing 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 hole
may come out by end of a supernova (SN) gravitational
core collapse. This mass leaking mechanism might have
also been at work during the very early universe driving
a matter-antimatter primordial asymmetry.
[32] These authors conceded not having identiﬁed what mech-
anism helps to enlarge the NS magnetic helicity, though
they stress that the original B-ﬁeld gives a positive feed-
back to itself, to grow exponentially, being this last the
actual chiral plasma instability. In our understanding,
this ﬁeld nonlinear enlargement is a prove that NLED
is doubtlessly at action inside just-born pulsars (see
Eq.(4)). Besides, it is clear that an ampliﬁcation of the
B-ﬁeld via magneto-diﬀerential rotation is concomitant
with this chiral plasma instability.
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