8/3/2019 Alexander Burinskii- Superconducting Source of the Kerr-Newman Electron

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arXiv:0910.5388v2

[hep-th]

8Nov2009

Superconducting Source of the Kerr-Newman Electron

Alexander BurinskiiTheoretical Physics Laboratory,

NSI, Russian Academy of Sciences,B. Tulskaya 52 Moscow 115191 Russia

Regular superconducting solution for interior of the Kerr-Newman (KN) spinning particle is ob-tained. For parameters of electron it represents a highly oblated rotating bubble formed by Higgsfield which expels the electromagnetic (em) field and currents from interior to domain wall bound-ary of the bubble. The external em and gravitational fields correspond exactly to Kerr-Newmansolution, while interior of the bubble is flat and forms a false vacuum with zero energy density.Vortex of the KN em field forms a quantum Wilson loop on the edge of the rotating disk-like bubble.

PACS numbers: 11.27.+d, 04.20.Jb, 03.65.+w

1.Introduction. Kerr-Newman (KN) solution for acharged and rotating Black-hole has g = 2 as that of theDirac electron and paid attention as a classical modelof electron coupled with gravity [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12]. For parameters of electron (in the unitsG = C = = 1) J = 1/2, m 1022, e2 1371 theblack-hole horizons disappear and the Kerr singular ringof the Compton radius a = /2m 1022 turns out to benaked. This ring forms the gate to a negative sheet ofthe Kerr geometry, significance of which is the old trou-ble for interpretation of the source of Kerr geometry. Theattempts by Brill and Cohen to match the Kerr exteriorwith a rotating spherical shell and flat interior [13] havenot lead to a consistent solution, and Israel suggested totruncate the negative sheet, replacing it by a thin (rotat-ing) disk spanned by the Kerr ring [2]. This model ledto a consistent source which was build from a very ex-otic superluminal matter, besides the Kerr singular ring

remained naked at the edge of the disk. Lopez [4] trans-formed this model to the model of an oblate rotating thinshell which covers the Kerr ring. By especial choice of theboundary of the bubble, r = r0 = e

2/2m (r is the Kerroblate spheroidal coordinate), he matched continuouslythe Kerr exterior with the flat interior, and therefore heobtained a consistent KN source without the KN singu-larity. However, like the other models, the Lopez modelwas not able to explain the origin of Poincare stress, anda negative pressure was introduces by Lopez phenomeno-logically. Meanwhile, the necessary tangential stress ap-pears naturally in the domain walls field models, in par-ticular, in the models based on Higgs [6]. It suggests that

a consistent field descriptions of this problem should con-tain, along with the KN Einstein-Maxwell sector, the sec-tor of Higgs fields and sector of interaction between theem field (coupled to gravity) and Higgs fields correspond-ing to superconducting properties of the KN source. Upto our knowledge, despite several attempts and partial

E-mail address: bur@ibrae.ac.ru

results, no one has been able so far to obtain a consistentfield model of the KN source related with Higgs field,and in this paper we present apparently first solution ofthis sort which is consistent in the limit of thin domainwall boundary of the bubble. The considered KN sourcerepresents a generalization of the Lopez model and in-corporate the results obtained earlier in [6, 7, 15, 16].Although we consider here only the case of thin domainwall, generalization to the case of a finite thickness isalso possible by methods described earlier in [6]. Thedescribed in [6, 16] KN source was a bag formed by apotential V(r) interpolating between the external truevacuum V(ext) = 0 and a false (superconducting) vac-uum V(in) = 0 inside of the bag. It was shown that cor-responding Higgs sector may be described by U(1)U(1)Witten field model with the given by Morris in [18] super-potential. It was shown also in [6, 7, 15] that consistencyof the Einstein-Maxwell sector with such a phase tran-

sition may be perfectly performed in the KS formalismwith use of the Gurses and Gursey ansatz [19]. How-ever, the problem with consistency appeared in [6] bythe treatment of interaction between the em and Higgsfield. In this short note we improve this deficiency whichallows us to present consistent solution.

2. Phase transition in gravitational sector. Following[6, 7, 15], for external region we use the exact KN solutionin the Kerr-Schild (KS) form of metric

g = + 2Hkk , (1)

where is metric of the auxiliary Minkowski back-ground in Cartesian coordinates x = (t,x,y,z). Elec-tromagnetic (em) KN field is given by vector potential

AKN = Ree

r + ia cos k, (2)

where k(x) is the null vector field which is tangent to avortex field of null geodesic lines, the Kerr principal nullcongruence (PNC). For the KN solution function H hasthe form

H =mr e2/2

r2 + a2 cos2 . (3)

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The used by Kerr especial spherical oblate coordinatesr,,K , are related with the Cartesian coordinates asfollows

x + iy = (r + ia)eiK sin , z = r cos . Vector field k

is represented in the form [1]

kdx = dr dt a sin2 dK . (4)

For the metric inside of the oblate bubble we use the KSansatz (1) in the form suggested by Gurses and Gursey[19] with function

H =f(r)

r2 + a2 cos2 .

If we set for interior f(r) = fint = r4, the Kerr singu-

larity will be suppressed.For exterior, r > r0, we use f(r) = fKN = mr e2/2

corresponding to KN solution. Therefore, f(r) describesa phase transition of the KS metric from true to falsevacuum (see fig.1).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.64

3

2

1

0

1

2

3

4

5

6x 10

4

flat interior

AdS interior

dS interior

rflat

rdS

rAdS

external KN field

x

x

x

r

f(r)

FIG. 1: Matching of the metric for regular bubble interiorwith metric of external KN field.

We assume that the zone of phase transition, r r0, isvery thin and metric is continuous there, so the point ofintersection, fint(r0) = fKN(r0), determines position ofdomain wall r0 graphically and yields the balance matterequation[7, 15],

m = mem(r0) + mmat(r0), mem(r0) =e2

2r0, (5)

which determines r0 analytically. Interior has constantcurvature, = 8/6. For parameters of electron a =J/m = 1/2m >> r0 = e

2/2m and the axis ratio of theellipsoidal bubble is r0/a = e2 1371, so the bubblehas the form of highly oblated disk.

3. Brief summary of the Higgs sector [6, 16]. The cor-responding phase transition is provided by Higgs modelwith two complex Higgs field and , two related gaugefields A and B, and one auxiliary real field Z. This isa given by Morris [18] generalization of the U(1) U(1)

field model used by Witten for superconducting strings[20].

The potential V(r) =

i |iW|2, where 1 = , 2 =

Z , 3 = , is determined by superpotentialW = Z( 2) + (cZ + m), where c, m, ,

are real constants. It forms a domain wall interpolatingbetween the internal (false) and external (true) vacua.

The vacuum states obey the conditions iW = 0 which

yield V = 0i) for false vacuum (r < r0): Z = m/c; = 0; || =

/c, as well asii) for true vacuum (r > r0) : Z = 0; = 0; = .

4. Interaction between the KN and Higgs fields. Weset B = 0 and use only the gauge fields A which in-teracts with the Higgs field (x) = |(x)|ei(x) having anonzero vev inside of the bubble, |(x)|r

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In particular, the tangent component at the edge is

A(str) = 2ma/e. Since J = ma we have A

(str) = 2J/e.

Setting J = n2 , n = 1, 2... we find out that vector po-tential forms on the edge of bubble a closed quantizedWilson loop

S =

eA

(str) d = 2n,

which has to be matched with angular periodicity of theHiggs field = 0 exp(i) and fixes its -dependence, exp{in}. For the time-like component of A inside

of the bubble, the r.h.s. of (6) states ,0 = eA(in)0 (r),

which determines the ,0 to be a constant correspondingto frequency of oscillations of the Higgs field, ,0 = =

eA(str)0 = 2m. Radial component of the KN field is a

full differential, and being extended inside the bubble,it is compensated by Higgs field in agreement with ther.h.s. of(6). Therefore, the Higgs field acquires the form

(x) = 0 exp{it i ln(r2 + a2) + in}. (9)

For exclusion of the region of string-like loop at equa-tor, the time and components of the gauge field havea chock crossing the boundary of bubble, which deter-mine a distribution of circular currents over the bubbleboundary.

5. Consistency. We find out that inside of the bub-

ble D = i[ + eA(in) ] 0. Together with the re-

sult that V(in) = 0, it leads to vanishing of the stress-

energy tensor of matter inside of the bubble, T(int) =

(D)(D)12g[(D)(D

)], and provides flatnessof the interior in agreement with our assumptions.

Therefore, the obtained superconducting solutionturns out to be consistent in the limit of the infinitelythin domain wall. It should be emphasized two impor-tant distinctions of this model from the typical soliton-like field models:

i) oscillations of the Higgs field with frequency = 2m,and ii) the appearance of quantum Wilson loop on theedge of KN disk.

The model can also be generalized for the domain wallsof a finite thickness ,, if the parameter r0 is much greaterthen . In this case, as it was considered in [6], one canuse the flat domain wall approximation, and the exact

field equations corresponding to finite region of phasetransition may be integrated, at least numerically.

Acknowledgements. Author thanks Theo Nieuwen-huizen for hospitality in ITP of Amsterdam Universityand discussions stimulating reconsideration of this oldproblem.

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