Closed-form separation constant for angular equation on the C-metric

Determine a full closed-form expression for the separation constant λ that arises from separating variables in the massless scalar Klein–Gordon equation on the nonrotating C-metric (allowing for charge Q and nonzero acceleration a), beyond the currently known small‑a perturbative expansion.

Background

In the C-metric background, separating the massless scalar field equation yields radial and angular equations coupled through a separation constant λ. For a=0 (spherical symmetry), λ reduces to the familiar ℓ(ℓ+1)+1/3. However, for the general accelerated case a≠0, only perturbative results in a are available (e.g., series expansions), and no exact closed form is known.

The authors’ analysis of quasinormal modes and greybody factors in the eikonal limit relies on asymptotic information about λ. A closed-form expression for λ in the accelerated case would remove this limitation and enable analyses beyond perturbative regimes and potentially beyond the eikonal approximation.