Kerr Isolated Horizon Revisited: Caustic-Free Congruence and Adapted Tetrad

TL;DR

The paper develops an explicit Kerr spacetime description within the isolated-horizon framework using a non-twisting null congruence whose Carter constant $K$ depends on horizon angle. By connecting to the Kinnersley tetrad through targeted Lorentz transformations and using Killing–Yano structures, it constructs a parallel-propagated Newman–Penrose tetrad adapted to the horizon and extends it off-horizon in horizon-adapted coordinates. It provides three complementary evaluation strategies for the core quantities: a full analytic approach based on Kerr null geodesics with elliptic functions, a robust numerical method, and two near-horizon series expansions (radial and slow-rotation) with explicit coefficients. This yields a caustic-free, coordinate-covered, and physically transparent near-horizon Kerr description, accompanied by curvature scalars and initial data on characteristics, plus reproducible Mathematica notebooks. Overall, the work advances a practical, well-defined quasi-local Kerr IH framework that improves upon prior treatments and enables controlled perturbations of the exact Kerr geometry.

Abstract

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.

Figures (17)

• Figure 1: Transversal null geodesics ${\color{blue}{n}}^a$ of a Kerr black hole, with parameters ${{\color{olive}{r_\text{p}}}}=2$ and ${{\color{olive}{r_\text{m}}}}=1$, are shown (using a fictitious flat-space projection) in Kerr--Schild Cartesian coordinates for the Kinnersley tetrad and two non-twisting congruences. The grey line marks the rotation axis, the blue circle indicates the ring singularity, and the light blue surfaces represent illustrative portions of the inner and outer horizons. The geodesics are plotted in red, with lighter shading beneath the outer horizon and darker shading above it. Dashed geodesics represent nothing more then geodesics for different azimuthal angle. In the first row (a), the twisting geodesic congruence (unsuitable for isolated horizons) is shown. In the second row (b), the Carter constant is set to ${{\color{olive}{K}}}={\color{olive}{\mathsfit{a}}}^2$, producing caustics along the axis. In the third row (c), the improved choice ${{\color{olive}{K}}}\doteq{\color{olive}{\mathsfit{a}}}^2\sin^2{{\color{colorKN}{\theta_\text{p}}}}$ removes these axis caustics and yields geodesics that pass through the disc to the other sheet, apart from the obstruction at the ring singularity. For each row, the leftmost image gives a view along the axis, the rightmost image shows a perpendicular side view, and the central panels provide intermediate viewpoints along the camera path connecting the two.
• Figure 2: Spin coefficient ${\color{blue}{\mu}}$ given by tetrad \ref{['eq:rotated_tetrad']} for black hole with ${\color{olive}{M}} = 1$ and ${\color{olive}{\mathsfit{a}}} = 0.8$ in the Cartesian version of ingoing null coordinates \ref{['eq:cartesin_kerr_null_coor']}. The imaginary part is zero (as it should be for a non-twisting geodesic congruence) and the displayed real part is the expansion of the congruence (in the direction of ${\color{blue}{n}}^a$).
• Figure 3: Spin coefficient ${\color{blue}{\lambda}}$ given by tetrad \ref{['eq:rotated_tetrad']} for black hole with ${\color{olive}{M}} = 1$ and ${\color{olive}{\mathsfit{a}}} = 0.8$ in the Cartesian version of ingoing null coordinates \ref{['eq:cartesin_kerr_null_coor']}. This spin coefficient represents the shear of the congruence (in the direction of ${\color{blue}{n}}^a$). The plot shows the real part, the imaginary part is zero.
• Figure 4: Weyl scalar ${\color{blue}{\mathit{\Psi} }}_2$ given by tetrad \ref{['eq:rotated_tetrad']} for black hole with ${\color{olive}{M}} = 1$ and ${\color{olive}{\mathsfit{a}}} = 0.8$ in the Cartesian version of ingoing null coordinates \ref{['eq:cartesin_kerr_null_coor']}. The real (right) and imaginary (left) parts are plotted separately for the same quadrant (${{\color{colorKN}{\theta}}} \in \langle0, \uppi /2\rangle$), the imaginary part is mirrored in the plot.
• Figure 5: Weyl scalar ${\color{blue}{\mathit{\Psi} }}_4$ given by tetrad \ref{['eq:rotated_tetrad']} for black hole with ${\color{olive}{M}} = 1$ and ${\color{olive}{\mathsfit{a}}} = 0.8$ in the Cartesian version of ingoing null coordinates \ref{['eq:cartesin_kerr_null_coor']}. The real (right) and imaginary (left) parts are plotted separately for the same quadrant (${{\color{colorKN}{\theta}}} \in \langle0, \uppi /2\rangle$), the imaginary part is mirrored in the plot.