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In general relativity, astrophysical black holes are simple objects, being described by just their mass and spin. These simple solutions are not exclusive to general relativity, as they also appear in theories that allow for an extra scalar degree of freedom. Recently, it was shown that some theories which couple a scalar field with the Gauss-Bonnet invariant can have the same classic general relativity black hole solutions as well as hairy black holes. These scalarized solutions can be stable, having an additional "charge" term that has an impact on the gravitational-wave emission by black hole binaries. In this paper, we overview black hole solutions in scalar-Gauss-Bonnet gravity, considering self-interacting terms for the scalar field. We present the mode analysis for the mono and dipolar perturbations around the Schwarzschild black hole in scalar-Gauss-Bonnet, showing the transition between stable and unstable solutions. We also present the time-evolution of scalar Gaussian wave packets, analyzing the impact of the scalar-Gauss-Bonnet term in their evolution. We then present some scalarized solutions, showing that nonlinear coupling functions and self-interacting terms can stabilize them. Finally, we compute the light-ring frequency and the Lyapunov exponent, which possibly estimate the black hole quasinormal modes in the eikonal limit.

Scalar modes, spontaneous scalarization and circular null-geodesics of black holes in scalar-Gauss–Bonnet gravity