Conjecture on the integer coefficient n_{ΔJ} in the PN–spinning test-particle action matching

Establish whether the integer coefficient n_{ΔJ} in the lattice transformation relating post-Newtonian action variables at 1.5PN to spinning test-particle Kerr actions is exactly zero, i.e., prove that I_{ΔJ} = J_φ without any additive multiple of the spin-aligned action (J_ψ + s). Concretely, determine if n_{ΔJ} = 0 in the ansatz I_r = J_r + n_r (J_ψ + s), I_L = J_z + |J_φ| + n_L (J_ψ + s), I_{ΔJ} = J_φ + n_{ΔJ} (J_ψ + s), I_5 = n_s (J_ψ + s), for dynamics that smoothly interpolate between the spinning test-particle limit in Kerr spacetime and finite-mass-ratio binaries at 1.5PN.

Background

In the proposed dictionary between actions of spinning compact binaries at 1.5PN and spinning test particles in Kerr spacetime, the authors relate the PN action set (I_r, I_L, I_{ΔJ}, I_5) to the Kerr action set (J_r, J_z, J_φ, J_ψ) via an integer lattice transform. The matching conditions derived from equating the Hamiltonians determine n_s = 1, n_r = 0, and n_L = −1, but leave n_{ΔJ} unconstrained.

The specific relations considered are I_r = J_r + n_r (J_ψ + s), I_L = J_z + |J_φ| + n_L (J_ψ + s), I_{ΔJ} = J_φ + n_{ΔJ} (J_ψ + s), and I_5 = n_s (J_ψ + s), where J_ψ + s equals the spin component aligned with the orbital angular momentum (s_∥). Geometric arguments suggest that I_{ΔJ} and J_φ generate the same rotation (about the primary spin axis) even when the secondary spin is nonzero, motivating the conjecture n_{ΔJ} = 0.

Validating n_{ΔJ} = 0 would complete the integer mapping between the two action sets, ensuring consistent identification of the angle variables and fundamental frequencies across the PN and spinning test-particle limits, and strengthening the gauge-invariant dictionary for interpolation between these regimes.