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Kerr-Newman from Minimal Coupling

Nathan Moynihan

TL;DR

The paper provides a unified, on-shell derivation of all four-dimensional Kerr-Newman and Reissner-Nordström black hole solutions at 1PN by relating classical observables to leading singularities of minimally coupled three-point amplitudes. It shows that rotating spacetimes arise from a spin-factor deformation of these amplitudes and that the Kerr-Newman metric corresponds to the on-shell analog of the Janis-Newman algorithm via an infinite-spin limit. By separating spin-independent and spin-dependent pieces and applying Holomorphic Classical Limit and Generalised Expectation Value normalization, the work recovers RN and KN metrics and clarifies how spin effects enter the classical potential and impulse. The approach provides a compact, universal framework for classical gravity from quantum amplitudes and points to extensions to higher orders, non-minimal couplings, and non-conservative dynamics with potential relevance for gravitational-wave physics.

Abstract

We show that at 1PN all four-dimensional black hole solutions in asymptotically flat spacetimes can be derived from leading singularities involving minimally coupled three-particle amplitudes. Furthermore, we show that the rotating solutions can be derived from their non-rotating counterparts by a spin-factor deformation of the relevant minimally coupled amplitudes. To show this, we compute the tree-level and one-loop leading singularities for a heavy charged source with generic spin s. We compute the metrics both with and without a spin factor and show that we get both the Kerr-Newman and Reissner-Nordström solutions respectively. We then go on to compute the impulse imparted to the probe particle in the infinite spin limit and show that the spin factor induces a complex deformation of the impact parameter, as was recently observed for Kerr black holes in a recent paper by Arkani-Hamed et al. We interpret these observations as being the on-shell avatar of the Janis-Newman algorithm for charged black holes.

Kerr-Newman from Minimal Coupling

TL;DR

The paper provides a unified, on-shell derivation of all four-dimensional Kerr-Newman and Reissner-Nordström black hole solutions at 1PN by relating classical observables to leading singularities of minimally coupled three-point amplitudes. It shows that rotating spacetimes arise from a spin-factor deformation of these amplitudes and that the Kerr-Newman metric corresponds to the on-shell analog of the Janis-Newman algorithm via an infinite-spin limit. By separating spin-independent and spin-dependent pieces and applying Holomorphic Classical Limit and Generalised Expectation Value normalization, the work recovers RN and KN metrics and clarifies how spin effects enter the classical potential and impulse. The approach provides a compact, universal framework for classical gravity from quantum amplitudes and points to extensions to higher orders, non-minimal couplings, and non-conservative dynamics with potential relevance for gravitational-wave physics.

Abstract

We show that at 1PN all four-dimensional black hole solutions in asymptotically flat spacetimes can be derived from leading singularities involving minimally coupled three-particle amplitudes. Furthermore, we show that the rotating solutions can be derived from their non-rotating counterparts by a spin-factor deformation of the relevant minimally coupled amplitudes. To show this, we compute the tree-level and one-loop leading singularities for a heavy charged source with generic spin s. We compute the metrics both with and without a spin factor and show that we get both the Kerr-Newman and Reissner-Nordström solutions respectively. We then go on to compute the impulse imparted to the probe particle in the infinite spin limit and show that the spin factor induces a complex deformation of the impact parameter, as was recently observed for Kerr black holes in a recent paper by Arkani-Hamed et al. We interpret these observations as being the on-shell avatar of the Janis-Newman algorithm for charged black holes.

Paper Structure

This paper contains 12 sections, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Scattering Amplitudes and Spin Operators
  3. Tree-Level Leading Singularity
  4. One-Loop Leading Singularity
  5. Classical Potential
  6. Spin-Independent Potential
  7. Spin--Orbit Potential
  8. Infinite Spin Limit
  9. Discussion
  10. Integral Transforms
  11. Impulse Fourier Transform
  12. Elliptical Integrals

Figures (4)

  • Figure 1: Gravitational probe of charged, spinning particles
  • Figure 2: Tree Level Diagram
  • Figure 3: LS Triangle Diagram
  • Figure 4: Diagrams contributing to the classical potential at order $G$ and $\alpha$.