What tells gravity on the shape and size of an electron
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hat Tells Gravity on the Shape and Size of an Electron1 A. Burinskii Theor. Phys. Lab., NSI Rus. Acad. Sci., B. Tulskaya 52, Moscow, 115191 Russia Abstract—Gravitational field of an electron, fixed by experimental values of its mass, spin, charge and magnetic moment, is given by the metric of Kerr–Newman (KN) solution. Unexpectedly, this metric con tains a singular ring of the Compton radius, which should be regulated resulting in a weeak and smooth source. The consistent source takes the form of an oblate vacuum bubble, bounded by a closed string of the Compton radius. The bubble turns out to be relativistically rotating and should be filled by a coherently oscillating Higgs field in a false vacuum state. DOI: 10.1134/S106377961401016X 1
INTRODUCTION It is commonly recognized now that black holes are akin to elementary particles. The Kerr–Newman solution has gyromagnetic ratio g = 2 as that of the Dirac electron, and the experimentally observable parameters of electron determine its asymptotical grav itational field in accord with the Kerr–Newman solu tion. The spin/mass ration of the electron is extremely high, J/m ~ 1022 (we use the units G = c = ћ = 1), and the black hole horizons disappear, opening the naked Kerr singular ring of the Compton radius ~10–11 cm. It is very far from the expected pointlike electron of quan tum theory. Besides, quantum theory supposes a flat minkowskian background, and this singular region should be regularized by some procedure leading to a finite and smooth source of the KN solution with a flat metric in vicinity of the electron core. It is not a priory clear that such a source can be obtained, and the aim of this paper is to describe basic elements of the given in [1] electron model which is consistent with the external KN solution and the above mentioned quan tum requirements. STRUCTURE OF THE KN SOLUTION Metric of the KN solution has the form 2
mr – e /2 g μν = η μν + 2Hk μ k ν , H = , 2 2 2 r + a cos θ
(1)
where hμν is metric of auxiliary Minkowski space in the Cartesian coordinates x ≡ (t, x, y, z) ∈ M4, and kμ(x) ∈M4 is a lightlike vector field, forming a twisting congruence shown in Fig. 1. Coordinates r, θ and φK are Kerr’s oblate spheroidal coordinates (Fig. 2). The KN metric is singular at the circle r = cosθ = 0, which 1 The article is published in the original.
is branch line of the Kerr space into two sheets r+ for r > 0 and r– for r < 0, so that the field kμ(x) and the aligned with kμ metric and vector potential of the electromag netic (em) field, μ μ e α KN = Re k , r + i arccos θ
(2)
turn out to be twosheeted, taking different values on the different sheets of the same point x ∈ M4. Twosheetedness represents one of the main puzzles of the KN spacetime. For electron parameters, gravita tional field of the KN solution is concentrated very close to singular ring, forming a circular waveguide— analog of the closed relativistic string. It has been shown in [2, 3] that the KN solution in vicinity of the Kerr
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