Kerr spacetime and scalar wave equation: Exact resummation of the renormalized angular momentum in the eikonal limit

TL;DR

This paper derives an exact resummation of the renormalized angular momentum for scalar waves in Kerr spacetime using hypergeometric functions, providing explicit series expansions and connecting it to the radial action via the Mano-Suzuki-Takasugi formalism.

Contribution

It introduces an exact resummation method for the renormalized angular momentum in Kerr spacetime, extending previous approximations and explicitly relating it to the radial action.

Findings

Resummation of the null geodesic radial action in Kerr spacetime using hypergeometric functions.

Explicit series expansions of the scattering angle up to fourth order in the Kerr parameter.

Proof of the relation between renormalized angular momentum and radial action via Mano-Suzuki-Takasugi formalism.

Abstract

We show that the null geodesic radial action for unbound orbits in the Kerr spacetime, and consequently the scattering angle, can be resummed in terms of hypergeometric functions, extending previous results [M.~M.~Ivanov, et al. arXiv:2504.07862]. We provide explicit expressions as series expansions in powers of the Kerr rotational parameter up the fourth order included. We finally use the Mano-Suzuki-Takasugi formalism to prove the relation between the renormalized angular momentum and the radial action highlighted in previous works.