A Double copies and the black hole electron

Because they evaporate, quantum micro black holes resemble unstable elementary particles: you can create them by smashing stable particles together, and then they will decay back into particles, as if it was all just another scattering process. Certainly in string theory, elementary particles and black holes lie on the same spectrum, the spectrum of string excitations: elementary particles at the light end, black holes at the heavy end.

There is also a fifty-year-old observation that electrons resemble black holes, specifically that a black hole with the mass, charge, and angular momentum of an electron, would have the same magnetic moment as an electron. The problem is that such an object would not actually be a black hole, because the electromagnetic force from the charge it carries would be stronger than the gravitational force from its mass. It would instead be a naked singularity. So the idea of the "black hole electron" has remained at the fringe of physics (Alexander Burinskii might be its most dedicated and capable proponent); although no less than Roger Penrose, in his book The Road to Reality (section 30,5), recommends this 2002 paper (by Ezra Newman) as explaining the coincidence.

Meanwhile, today Arkani-Hamed, Huang, and O'Connell have released this paper:
Nima Arkani-Hamed, Yu-tin Huang, Donal O'Connell
(Submitted on 24 Jun 2019)

Long ago, Newman and Janis showed that a complex deformation z -> z + ia of the Schwarzschild solution produces the Kerr solution. The underlying explanation for this relationship has remained obscure. The complex deformation has an electromagnetic counterpart: by shifting the Coloumb potential, we obtain the EM field of a certain rotating charge distribution which we term √ Kerr. In this note, we identify the origin of this shift as arising from the exponentiation of spin operators for the recently defined "minimally coupled" three-particle amplitudes of spinning particles coupled to gravity, in the large-spin limit. We demonstrate this by studying the impulse imparted to a test particle in the background of the heavy spinning particle. We first consider the electromagnetic case, where the impulse due to √ Kerr is reproduced by a charged spinning particle; the shift of the Coloumb potential is matched to the exponentiated spin-factor appearing in the amplitude. The known impulse due to the Kerr black hole is then trivially derived from the gravitationally coupled spinning particle via the double copy.
Note that this is "black holes as elementary particles", not "elementary particles as black holes".

This paper brings together two research programs. One is the "double copy" program, which obtains many properties of gravity from "two copies" of a gauge force like electromagnetism. The other is a recent extension of "spinor helicity formalism" for scattering amplitudes, to particles of all masses and spins. They then use this to explain the fact that the metric of a nonrotating (Schwarzschild) black hole, can be deformed in a certain way to give the metric of a rotating (Kerr) black hole.

Skimming the paper, their explanation seems to proceed as follows: They use the new formalism to show that scattering off a charged scalar can be deformed in that way, to describe scattering off a spinning charged particle. Then they use the double-copy relation, to show that, analogously, graviton scattering from a massive scalar, can be deformed to describe graviton scattering from a spinning massive particle. But in the classical limit, this second deformation is the deformation of Schwarzschild metric to become Kerr metric.

How could this relate to the case of the "black hole electron"? First let's be clear on how it's different. The electron has spin 1/2. Arkani-Hamed et al are working in a large-spin limit. In the stringy paradigm, that's high up the tower of excitations of the string, where you can find the black holes.

Also, the Kerr black hole has spin and mass but no charge. An electron has spin, mass, and also charge. A black hole with spin, mass, and charge is described by a modification of the Kerr metric, the Kerr-Newman metric. To obtain the Kerr metric, one deforms the Schwarzschild metric; to obtain the Kerr-Newman metric, one deforms the Reissner–Nordström metric, the metric of a charged nonrotating black hole; and Arkani-Hamed et al leave this case for future work. (Just to be clear, in the Schwarzschild-to-Kerr case, one starts with charge, and the double copy gives you gravitation. Here, the double copy must result in an object with gravitation and charge. This should still be doable, and will also involve beginning with just charge in some sense, but the details will have to be different somehow.)

What has grabbed my attention, is that Newman's 2002 explanation of the "black hole electron" coincidence, also involves a form of the Schwarzschild-to-Kerr complex deformation! So I am wondering if Newman 2002 can somehow be viewed as Arkani-Hamed et al 2019, but extrapolated to a "low-spin" limit, rather than a large-spin limit.
 

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