Weak-Field Limits of Black Hole Metrics from the KMOC formalism: Schwarzschild, Kerr, Reissner-Nordstr\"om, and Kerr-Newman

TL;DR

This paper uses the KMOC formalism to derive weak-field limits of various black hole metrics from quantum scattering amplitudes, including Kerr-Newman, and verifies consistency with known solutions.

Contribution

It demonstrates how to reconstruct weak-field black hole metrics from scattering amplitudes, including electromagnetic and interference effects, within the KMOC formalism.

Findings

Reproduces Schwarzschild, Kerr, Reissner-Nordstrom, and Kerr-Newman metrics in the weak-field limit.

Includes electromagnetic and interference effects in the metric reconstruction.

Results align with previous derivations from minimal coupling amplitudes.

Abstract

The KMOC (Kosower-Maybee-O'Connell) formalism establishes a bridge between quantum scattering amplitudes and classical observables in gravitational systems. In this work, we show how the weak-field limits of the four classical black hole metrics - Schwarzschild, Kerr, Reissner-Nordstrom, and Kerr-Newman - can be reproduced within this formalism. Starting from three-point amplitudes with exponential spin structure for both gravitational and electromagnetic interactions, we compute four-point scattering amplitudes and extract the momentum impulse via the KMOC formula. Matching these results with geodesic motion in a general metric allows us to reconstruct the metric components to leading order in G, a, and Q^2. For the Kerr-Newman case, we include interference terms between gravitational and electromagnetic interactions, which produce a Q^2 a/r^3 contribution to g_{t\phi} that does not appear in the Kerr or Reissner-Nordstrom weak-field limits separately. Our results are consistent with those of arXiv:1907.00431 [hep-th], where the Kerr-Newman metric is derived from minimal coupling amplitudes using the KMOC formalism arXiv:1908.04342 [hep-th]. All results are verified through their consistency with the well-known full metrics, though we emphasize that the KMOC formalism as applied here reproduces only the weak-field expansions, not the complete non-linear solutions.