Nonlinear electrodynamics: The missing trigger for the formation of astrophysical
charged black holes in gravitational cor...
L =
j,k=0 cj,kFj
, or c) via a 4-dim effective the-
ory from strings, M-theory, or AdS/CFT correspondence.
The simpl...
Vacuum induced magnetization.— In classical electro-
dynamics [24] magnetization: magnetic dipole moment
per unit volume...
[10, 15]. It can thus be conceded that in order to consis-
tently infer the B-field strength of extremely magnetized,
electron (e−
) capture on protons leads to a right-to-left
handed Fermi surface imbalance µR > µL, i.e. to a
nonzero (ti...
[12] P. C. W. Davies, J. Opt. B 7, S40-S46 (2005)
[13] J. Hadamard, “Le¸cons sur la propagation des ondes et les
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Nled and formation_of_astrophysical_charged_b_hs_03_june_2014

  1. 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. 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. 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. 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. 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: [1] R. Ruffini, L. Vitagliano, Int. J. Mod. Phys. D 12, 121 (2003); C. Cherubini, R. Ruffini, L. Vitagliano, Phys. Lett. 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