Matter field and black hole horizon geometry

Abstract: This paper investigates the influence of matter fields on the geometry of black hole horizons within higher-order gravity theories. Focusing on five-dimensional Einstein-Gauss-Bonnet gravity at a critical coupling constant ($\alpha = -3/(4\Lambda)$), we demonstrate that while vacuum solutions permit horizons with arbitrary geometry, the introduction of a scalar field imposes constraints. Specifically, the scalar hair restricts the horizon to manifolds of constant scalar curvature, extending beyond the Thurston geometries (Sol, Nil, $SL_2R$) previously identified. We prove a uniqueness theorem for the scalar-coupled solutions, showing that the metric and scalar field must adopt specific forms, with the horizon geometry solely required to satisfy the constant curvature condition. Furthermore, analogous results are established in generic $F(R)$ gravity, where arbitrary horizons with constant scalar curvature emerge at critical couplings, exemplified by $F(R) = R + \lambda R^2 - 2\Lambda$ with $\lambda = -1/(8\Lambda)$. These findings highlight a probably universal feature: critical couplings in higher-order gravity enable unconstrained horizon geometries in vacuum, while matter fields introduce geometric restrictions. This work deepens understanding of the interplay between matter, higher-curvature corrections, and black hole horizon geometry.