Kerr Black Holes as Elementary Particles

TL;DR

The paper addresses why Kerr black holes behave like elementary particles by linking the Newman–Janis complex shift to the exponentiation of spin in the large-spin limit of minimally coupled three-point amplitudes. It shows that the electromagnetic field of the \sqrt{Kerr} solution corresponds to a shifted Coulomb potential, with the shift parameter identified as $a = s/m$, and that this spin-exponential reproduces the Kerr shift in position space. By applying the double-copy principle, the authors derive the gravitational Kerr impulse from its electromagnetic counterpart, validating the classical Kerr result as a double-copy of a spinning, minimally coupled system. The work establishes a direct on-shell, amplitude-based understanding of Kerr geometry, suggesting broader applicability to classical solutions and future explorations in $(2,2)$ signature formulations and beyond.

Abstract

Long ago, Newman and Janis showed that a complex deformation $z\rightarrow z+i a$ 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 $\sqrt{\rm Kerr}$. 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 $\sqrt{\rm Kerr}$ 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.

Figures (1)

• Figure 1: The exchange of a photon between a spin-$S$ and a scalar particle.