The Dirac Electron Consistent with Proper Gravitational and Electromagnetic Field of the Over-rotating Kerr-Newman Black Hole Solution

: We consider the Dirac electron as a nonperturbative particle-like solution consistent 1 with its own Kerr-Newman (KN) gravitational and electromagnetic ﬁeld. We develop the earlier 2 models of the KN electron regularized by Israel and López, and consider the non-perturbative 3 electron model as a bag model formed by Higgs mechanism of symmetry breaking. The The 4 López regularization determines the unique shape of the electron in the form of a thin disk with a 5 Compton radius reduced by 4 π . In our model this disk is coupled with a closed circular string 6 which is placed on the border of the disk and creates the caused by gravitation frame-dragging 7 string tension produced by the vector potential of the Wilson loop. Using remarkable features 8 of the Kerr-Schild coordinate system, which linearizes the Dirac equation, we obtain solutions 9 of the Dirac equation consistent with the KN gravitational and electromagnetic ﬁeld, and show 10 that this solution takes the form of a massless relativistic string. Parallelism of this model with 11 quantum representations in Heisenberg and Schrodinger pictures explains remarkable properties 12 of the stringy electron model in the relativistic scattering processes. 13


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One of the main points of confrontation between Gravity and Quantum theory 17 is the structure of elementary particles, which are considered in quantum theory as 18 structureless, like a point-like electron in Dirac theory, but must be represented as an 19 extended field model in configuration space for compatibility with the stress-energy 20 tensor of Einstein's equations. 21 A revolutionary step towards unification quantum with gravity was taken in super- 22 string theory, which represented particles as extended strings. Gravitational black holes 23 (BH) have been considered as candidates for elementary particles repeatedly since 1980, 24 and since the 1990s, they have also attracted attention in the theory of superstrings. 25 However, as one of its founders, John Schwartz, noted,"... Since 1974, superstring 26 theory has ceased to be regarded as particle physics... " and "... a realistic model of 27 elementary particles still seems a distant dream ..." [1]. 28 Meanwhile, a renewed interest to relationships between black holes and elementary 29 particles has been obtained recently in the works [2][3][4][5]. 30 Formation of BHs is related with gravitational effect of frame-dragging. In the 31 rotating Kerr-Newman BH solution, with parameters J, m, a corresponding to spin, mass 32 and Kerr's rotational parameter a of elementary particle, spin creates a giant over- 33 rotating dragging of space, which is directed along of direction of rotation, leading to 34 a new important effect, formation of the closed Wilson loop, which never was used in 35 particle physics before. 36 In contrast to considered earlier cases of the Schwarzschild or Reissner-Nordström gravity, the characteristic scale of the KN gravity is essentially increases, because it is determined by radius of the Kerr singular ring which corresponds to the reduced Compton wave length of the particle.

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This fact, established already in the first models of an electron based on the Kerr 38 geometry [6-12] was remarkable itself, because it was known, but was not timely es-39 timated as one of the first evidences of the correspondence between KN particle and 40 quantum theory. The gigantic ratio between the spin and mass values for elementary particles in KN geometry violated the generally accepted concept of the weakness of gravity, based on the earlier estimations of gravitational radius of the Schwarzschild solution r g = 2Gm.
(2) the KN bubble is formed of the Higgs field, which is in a superconducting vacuum state.

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The thin shell of bubble is replaced by a domain-wall solution, which is described by the For r e = e 2 /2m disk is very thin, and r e /a = α corresponds to the fine structure const.
interpolates between the superconducting (and supersymmetric) internal vacuum state 66 and the external exact gravitational KN solution.

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The obtained by C.Lopez bubble source of the KN geometry [10] (see Fig.2    In these coordinates, metric of the KN solutions is [7] where η µν is flat metric of the auxiliary Minkowski space, and H is the scalar function which for the KN solution takes the form The KN vector potential is given as The field k µ (x) forms a Principal Null Congruence (PNC), k µ k µ = 0, shown on Fig.1.

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In terms of BH geometry this field shows a local direction of dragging the frame, that Kerr's congruence can be represented as an electromagnetic radiation which propa-105 gates (with twist) from infinity towards the Kerr ring, penetrates it, and coming out on 106 the other sheet of the Kerr geometry goes out again to infinity. In Cartesian coordinates 107 x µ ∈ M 4 , the form k µ dx µ shows local direction of frame-dragging.

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In the Kerr angular coordinates PNC is presented in [7] by the form The relation between Cartesian coordinates and Kerr's angular coordinates is the follow- The incoming PNC is directed to the Kerr ring. Rays lying in equatorial plane (cos θ = 0) focus on the Kerr singular ring. Other incoming rays, passing trough the ring, turn into out-going rays propagating on another (say "negative") sheet of the Kerr space. Thus, the Kerr solution in the KS form describes two different sheets of space-time with two different congruences and two different metrics on the same Minkowski background x µ ∈ M 4 . Working with outgoing Kerr field corre-111 sponding to retarded potentials, we choose sign plus in (8), and following [7] we take The Kerr theorem. 114 Kerr theorem defines two fields of PNC, k + (x) and k − (x), in terms of Penrose's twistor theory [29][30][31]. Kerr theorem presents two complex analytic solutions Y ± of the equation where F is quadratic holomorphic function of the projective twistor coordinates are the null Cartesian coordinates of the auxiliary Minkowski space x µ ∈ M 4 .

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In the class of quadratic in Y functions F(T A ), the Kerr theorem gives two analytic solutions Y ± (x µ ), of the equation (10), which correspond to two projective spinor coordinates which are antipodically conjucate and the corresponding Weyl spinors ξ˙α and η α define two antipodal fields of the principal null directions

Shape of the KN bag model and Wilson loop 119
The Löpez boundary of the bubble, where the KN space can be matched continuously with the flat internal metric η µν , is unambiguously determined by the Kerr-Shild metric form (3), as the surface where H = 0. Setting H KN = 0 we obtain that gives us the "classical" electron radius.

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Since r is the Kerr radial coordinate, we obtain that the bag boundary represents 121 indeed an oblate ellipsoidal surface -a thin disk of the radius a, which is about the 122 reduced Compton wave length, and the thickness of the disk r e , which is equal to 123 classical electron radius. One sees that degree of oblateness of the disk is r e /a = 1/137 124 that corresponds to the fine structure constant α. 125 Therefore, the Kerr-Newman spin parameter a leads to a strong deformation of the 126 shape of the bag model, and this deformation of the bag leads to the appearance of a 127 relativistic string at the sharp edge of the KN disk (see Fig.4).

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The existence of this string is evidenced by the Wilson loop of the vector potential From (5) and (6) we obtain that vector-potential of the regularized KN solution takes its maximal value in the equatorial plane (cos θ = 0) at the bag border r = r e , This potential is tangent to the bag border r = r e , and for the fixed time t = const., it forms the closed Wilson loop C : φ K ∈ [0, 2π], so that the loop integral W(C) = P exp e C A max µ dx µ , gives the following incursion of the potential Integration gives δφ = 4πma, and using relation J = ma we obtain Definiteness of potential requires δφ = 4π J, leading to quantum condition J = 1 2 .
compatible with metric and with in-going Kerr congruence k − µ . This process shows that disk-like source of KN field has two faces: one from the 149 side of the in-going fields k − µ , and the other from the side of the out-going fields k + µ .

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These two sides are related with reverse sign of the disk rotation a → −a, and change 151 the orientation angle φ k → −φ k for the incoming field.

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The corresponding string-like structure, was suggested in [28] as an orientifold  The full orientifold world-sheet is formed as a folded string on the doubled interval 159 σ ∈ [0, 2π], and contains the sum of the left and right modes X = X L (τ + σ) + X R (τ − σ). 160 The orientifold string is left-right symmetric in the static representation, t = const., 161 which in quantum theory is called as Heisenberg picture, however the symmetry Ω is 162 broken on the rotating disk.

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Superconducting vacuum state of the Higgs field inside the Bag leads to equations which shows that inside the superconductor current I µ is pushed out, I µ = 0, and is 177 concentrated in a surface layer with a depth of penetration δ, [36]). Potential of the KN where we tuck also into account the change of charge sign by the transition a → −a.

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Applying these solutions to the out-going vector field A + µ (r + e ) on the boundary 194 r = r + e , which is dragged by gravitational field of the Kerr congruence, forming the 195 closed Wilson loop C + : t = const. on the border r = r + e we obtain: 196 1) incursion of the potential A + µ along the loop C + is controlled by the Higgs phase χ + , and integration of the equations I + µ = 0 ⇒ χ + , µ +eA + µ = 0 gives 2) similarly, the out-going potential A + µ , acting on the boundary r − gives and therefore, the phases of the Higgs fields (t + + aφ  Out-going Kerr congruence propagates from the both sides of the disk r + and r − towards direction +∞. The in-going congruence propagates from −∞ towards the disk and focus at the both sides of the disk r + and r − .

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The transition from out-going picture to in-going is connected with the replacement 228 r → −r, that in the coordinate transformation (7) corresponds to the replacement 229 ρ + → ρ − , changing in the direction of rotation a → −a, and in the orientation angle φ K .

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The Kerr disk is located at the scattering boundary t = 0, which corresponds to  The Dirac operator in the charged and curved space-time is defined by the replacement where B = 1 2 ∇ µ γ µ can be represented in the form Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 December 2020 doi:10.20944/preprints202012.0758.v1 and is canceled because |g| = 1.

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As a consequence, γ-matrices of the auxiliary Minkowski space can be used in the Dirac 247 equation.

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The same argument leads also to the linearization of the electromagnetic field by 249 the use of the Kerr-Schild coordinates in the KN solution, [12].

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In the considered earlier analogous physical model of the rotating KN disk-like source, the in-going and out-going Kerr congruences are controlled by two related phases of the Higgs field, χ + = −χ − , and the momentum p µ of the string solution must be completed by an "internal" angular momentum of two semi-strings p s = p s+ + p s− , associated with rotation of the KN disk under its evolution in time, According (14), the spinors ξ˙α and η α have different helicities with respect to helicity Integrating the Ginzburg-Landau equations for the out-going phase of the Higgs 267 field and r = r + , we obtained χ + | r + = 2m(t + aφ), which for J = ma = 1/2 gives 268 p s µ | r + = (2m, ∂ φ K ). The corresponding vector potential is eA µ = (2m, eA φ K ).

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For the boundary r = r − , we have the opposite sign of charge, which corresponds

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To simplify notations we will omit further the index K in the Kerr angular coordinate φ K .  Heisenberg picture for t = const..
For any ξ˙α, η α and m = 0, the first equation is identically satisfied when and the second equation is identically satisfied when Spinors |u p >= ξ˙α η α are normalized as <ū p |u p >= 2m.

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In the Heisenberg picture presenting the KN string at fixed time t = const. we have:

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The potential energy of the semi-strings tension is determined by the Wilson loop at the boundaries r ± , eA 0 = ±2m, eA φ = ±2ma, and the full energy of the semi-strings is cancelled, as it was shown when integrating 294 the BPS equations for the DW-AntiDW (breather) source of the KN solution, [17].

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In the Weyl representation for matrices γ µ , the out-going and in-going fields are 296 ordered in time, and the fields with negative frequencies do not arise. In the Schrodinger picture the plane waves and in the Kerr-Schild coordinates are described by wave function [47] where −px = −p µ x µ = p 0 x 0 − px, and x µ = (t, x), p µ = (p 0 , p), = p 0 = + p 2 + m 2 .
In the rest system, = m, p = 0, functions ψ p and u p are connected by unitary 300 transformation U = e −iHt , where H = m is the Hamiltonian of the system. 301 We consider U as operator acting on a state vector |u p >= ξ˙α η α , in the static Heisenberg picture, while the plane wave represents the state vector |ψ p > in the dynamic Schrodinger picture.

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In the Schrodinger picture the string turns out to be asymmetric: