Odd-parity perturbations of geometrically regular black holes with hedgehog scalar hair

Abstract In this paper, we study linear perturbations of a geometrically regular black hole supported by a constrained $$\textrm{SO}(3)$$ SO ( 3 ) scalar triplet in the hedgehog configuration. Since the individual background scalars depend explicitly on the angular coordinates, the decoupling of polar and axial perturbations cannot be concluded from the spherical form of the metric or the energy–momentum tensor alone. To settle this point, we introduce the polar and axial metric and triplet amplitudes simultaneously and derive the corresponding projections of the radiative linearized Einstein–matter equations for $$\ell \ge 2$$ ℓ ≥ 2 . The polar equations contain only polar amplitudes, whereas the axial equations contain only axial amplitudes; hence, the two off-diagonal blocks vanish at linear order. The resulting odd-parity sector forms a closed Einstein-scalar system in which a tangent perturbation of the scalar triplet is coupled to the Regge–Wheeler metric amplitudes. We show that its scalar equation has a second-order hyperbolic principal part whenever $$K_Y\ne 0$$ K Y ≠ 0 , and that, for the nonlinear kinetic function supporting the background, $$K_Y>0$$ K Y > 0 throughout the black-hole exterior. The coefficient of the direct metric-scalar mixing is largest at the event horizon and decreases as $$r^{-4}$$ r - 4 at large distance, while the metric block in the vacuum limit reproduces the standard Regge–Wheeler equation and potential; the scalar kinetic coefficient simultaneously vanishes, so the full operator changes rank. These results establish linear parity decoupling and the consistency of the closed axial equations. The odd sector can furthermore be reduced to two coupled master variables, and the corresponding reduced quadratic action has positive kinetic and radial-gradient eigenvalues at every finite exterior radius for the regular branch with $$K_Y>0$$ K Y > 0 . The scalar normalization tends to zero asymptotically, however, and positivity of the full coupled potential operator is not established; hence we do not claim a complete mode-stability theorem.