Kerr isolated horizon revisited: Caustic-free congruence and adapted tetrad

We revisit the near-horizon description of the Kerr space-time in the isolated horizon formalism using a non-twisting null geodesic congruence and eliminate the coordinate and geodesic pathologies that arise when the Carter constant of motion is globally fixed to a single constant. Adopting instead a previously proposed choice of the Carter constant which depends on the polar angle on the horizon, we obtain an analytic construction of the Newman--Penrose tetrad adapted to isolated horizons together with horizon-adapted coordinates in which its defining properties are manifest. We compute the associated curvature scalars and provide initial data on characteristics for the isolated horizon. In addition to an analytical solution, derived by leveraging extensive results on Kerr null geodesics, we develop two complementary series expansions and outline a practical numerical recipe to make the construction readily usable. Relative to earlier treatments, our formulation avoids caustic-induced breakdowns and incomplete coordinate coverage while yielding a detailed description of the Kerr black hole in the isolated horizon approach.