Black Holes and Scalar Propagation in Three-Dimensional Einstein--Gauss--Bonnet Gravity

Abstract: We derive an analytic black-hole family in three-dimensional scalar--tensor Einstein--Gauss--Bonnet gravity with positive coupling. A single implicit equation determines the static circular metric and a scalar linear in time. We show that these solutions exhaust regular nonextremal exteriors with the stated AdS boundary conditions and a fixed nonzero coefficient of time in the scalar. The coupled metric and scalar perturbations have one propagating degree of freedom. On one branch, its kinetic coefficient is positive, its equation is hyperbolic throughout the exterior, and its bulk spatial energy is positive for perturbations of compact support. Scalar signals can cross the metric horizon outward, so exterior evolution needs information from the interior. For linear perturbations with the original metric and scalar boundary values fixed, nonzero compact initial master displacements with zero velocity can evolve only until their first contact with the AdS boundary. We derive this restriction from the original metric and scalar equations.