I had an exchange with Lubos Motl about this topic, in the comments here.(adsbygoogle = window.adsbygoogle || []).push({});

Very briefly, there is a 2004 paper in which the author (Kjell Rosquist) considers the old idea that the electron is actually a spinning, charged (i.e. Kerr-Newman) micro black hole. Using a purely classical model for the black hole, and treating the electron spin as the black hole's angular momentum, the author calculates that the "black hole electron" should have an electric quadrupole moment. An electric quadrupole indicates a charge distribution which deviates from spherical symmetry. As Lubos argues, this is group-theoretically impossible for a spin-1/2 object. I proposed that perhaps quantum corrections could generate an anomalous electric quadrupole, but he said no, the electron (e.g. in string theory) would still be an exact spin-1/2 doublet, even with corrections included.

He also observed that for an object with the mass and spin of an electron, the Kerr-Newman metric wouldn't be valid anyway, the higher-order corrections to the Einstein equation would dominate. Still, I assume that Rosquist is correct in saying that a standard Kerr-Newman black hole has an electric quadrupole moment (I haven't checked), so there's a question of what happens in string theory as you consider smaller and smaller black holes. As you vary mass, spin, and charge, where is the boundary between "Kerr-Newman objects" and "non-Kerr-Newman objects", and between objects with an electric quadrupole and objects without an electric quadrupole; and does the electron lie near either of these boundaries, or does it lie deep in "non-Kerr-Newman" and "non-quadrupole" territory?

I know this won't be straightforward, for a variety of reasons. First, I don't think there are any exact models of Kerr-Newman black holes in string theory, though there is a "Kerr/CFT duality". Second, it won't be easy to talk about electron-like objects; if we just stick to perturbation theory, you either have massless objects, or excited modes with Planck-scale masses. The light mass of an electron will come from something like Yukawa couplings to a Higgs condensate, and it is conceivable that esoteric nonperturbative considerations are very relevant for understanding the "boundary cases" that interest me.

So this post is a note for future reference. I'll add to it when I learn anything that illuminates the issues.

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# Electric quadrupoles of elementary particles and micro black holes in string theory

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