Kerr Black Holes as Elementary Particles

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.

• The paper demonstrates that applying a Newman-Janis complex deformation to Schwarzschild metrics produces Kerr black holes analogous to spinning elementary particles.
• It employs amplitude formalism to derive classical impulses, linking gravitational interactions with gauge theory through the double copy principle.
• Its findings offer a framework for reconciling quantum mechanics and general relativity by exploring minimal coupling in high-energy scattering processes.

Kerr Black Holes as Elementary Particles

The paper "Kerr Black Holes as Elementary Particles" presents a nuanced exploration of the theoretical framework that equates certain properties of Kerr black holes with elementary particles within the context of scattering amplitudes. The authors, Arkani-Hamed, Huang, and O'Connell, draw upon the notion of complex deformation of metrics, notably demonstrated by Newman and Janis, to elucidate the underlying principles that conjoint the characteristics of Kerr solutions with particles that exhibit quantifiable spin.

Key Aspects and Methodology

The research primarily investigates the relationship between spinning particles and classical gravitational solutions. The Newman-Janis approach to obtaining the Kerr solution from a complex deformation of the Schwarzschild solution serves as the pivotal technique allowing the authors to draw parallels in classical and quantum domains. This method involves a complex shift, $z \rightarrow z + i a$, applied to a static solution, specifically targeting electromagnetic field configurations akin to a rotating charge distribution—a solution termed as $\sqrt{\text{Kerr}}$.

The authors expand on the amplitude formalism to bridge the classical and quantum interpretations. They analyze the impulse imparted to a test particle by a heavy spinning particle through three-particle amplitudes while incorporating minimal coupling. The amplitude of interest is naturally aligned with on-shell notions of minimal coupling, which emerges prominently in the high-energy limits of massive scalar coupling to photons and gravitons. These are expressed and studied within the formalism established in previous works, notably by Arkani-Hamed and collaborators.

Numerical Results and Claims

The paper reports that the classical impulse due to Kerr black holes, a central observable, seamlessly emerges from these gravitationally-coupled spinning particle models via an equivalence to electromagnetic cases. The shift observed in the Coulomb potential directly correlates with the spin factor exponentiation in scattering amplitudes, which intriguingly adheres to the so-called "double copy" principle. This method allows the gravitational solutions to be deduced from gauge theory solutions, reinforcing the connection between gravity and gauge interactions at a classical level.

Implications and Future Directions

The authors assert that their results deepen the understanding of how classical black hole properties, such as those inherent in Kerr solutions, can be derived from quantum mechanical scattering amplitudes, particularly those involving particles with large intrinsic spin. This insight potentially advances the broader quest for reconciling quantum mechanics with general relativity, given that it offers a cohesive framework to conceptualize particles traditionally treated as purely classical entities.

The paper’s approach could suggest further investigations into the implications of complex deformation practices in other spacetime solutions, such as Reissner-Nordstrom or alternate scenarios in gauge theories like Taub-NUT spaces. Moreover, the findings could stimulate further exploration of the interplay between different theoretical constructs, particularly in understanding the full landscape of gravitational interactions through a particle physics lens. The double copy conjecture remains a tantalizing avenue for further study, especially in expounding upon interpretative difficulties that arise in non-linear gravitational interactions.

Conclusion

In sum, the paper provides a rigorous and detailed examination of Kerr black holes and their mapping to elementary particles through the lens of modern theoretical physics. By leveraging amplitudes, complex shifts, and classical solutions, the authors present a compelling narrative that elucidates a profound connection within the intricate tapestry of particle interactions and gravitational phenomena. Such endeavors underscore the importance of interdisciplinary techniques in advancing theoretical physics, thereby paving the way for novel insights into longstanding cosmological mysteries.