This paper shows a new way to build an analytic description of a spinning black hole in a modified theory of gravity. The authors turn a technique called metric reconstruction into an analytic solver and use it to produce the first explicit spinning black-hole metric in parity-even cubic gravity at linear order in the theory’s coupling. The result is valid across the strong gravity zone and for generic subextremal spin, and it was checked against the modified field equations and independent predictions for horizon properties.

The construction follows three steps. First, the team solves a sourced, stationary Teukolsky equation. The Teukolsky equation is a differential equation for small changes in the curvature of a rotating black hole. Solving it mode by mode gives the curvature perturbations that encode how the black hole departs from the Kerr solution of general relativity. Second, they feed those curvature inputs into a set of Newman–Penrose transport equations. The Newman–Penrose formalism rewrites the problem using a lightlike basis (a “tetrad”), which turns the reconstruction into nested radial integrals that collapse to a single potential. Differentiating that potential and adding explicit source terms yields the metric correction. Third, boundary conditions—regularity at the horizon, behavior at large distance, and fixed mass and angular momentum—fix the remaining integration functions.

Concrete features of the calculation come from the chosen example, parity-even cubic gravity. There the radial integrals can be evaluated explicitly. The authors separate angular dependence with spin-weighted spherical harmonics and obtain radial functions made of rational terms, logarithms and dilogarithms. They build a horizon-regular solution using Green’s-function techniques in the Kinnersley tetrad (the standard tetrad for Kerr geometry). The final metric is analytic, keeps the full dependence on spin, and requires no slow-rotation or weak-field expansion.