Kerr black holes as elementary particles

Abstract

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.

Kerr-Newman from minimal coupling

“…At tree level, there is only one possible diagram that we can consider Since there is no electromagnetic interaction, this will simply produce a purely gravitational interaction at order G, and has been calculated many times in the literature [1,27,28].…”

“…In order to derive the piece of the impulse that corresponds to the charged solution, we first note a useful identity [1] sinh…”

“…From far enough away, any black hole can be treated as a point particle, and as such can be given an effective one-body description. The proposed on-shell avatar of the no-hair theorem is that black hole solutions should be obtainable from minimal coupling, with deviations describing finite-size effects given by non-minimal deformations [1,27]. The construction of classical and quantum black hole metrics using loop amplitudes has been a fruitful endeavour, using everything from form factors [48,49] to unitarity based methods [13] and more recently with leading singularities [27,50,51].…”

“…[1,27,51]. Very recently, it was shown that this factorization, in the infinite spin limit, is the on-shell avatar of the Janis-Newman algorithm [1], which utilises a complex coordinate transformation of the Schwarzchild (Riessner-Nordström) solution leading directly to the Kerr (Kerr-Newman) solution [52,53]. We will show that the Kerr-Newman solution can be derived in precisely this way from Reissner-Nordström by simply attaching a spin-factor to the relevant minimally coupled three-point amplitudes.…”

From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist)

“…Even though, at this stage we resort to the existence of the quasi-isotropic gauge, this is ultimately the one (implicitly) chosen by the Fourier transform of the amplitude in the center of mass frame. Therefore, provided the matching discussed in [15][16][17][18] carries over to spin, as suggested in [19][20][21][22][23][24][25][26], the existence of this gauge is guarantee to all PM orders. As before, the exact form of the Hamiltonian is never needed, although it may be obtained and shown to agree with the existent literature, e.g.…”