Nonlinear scalarization of Schwarzschild black holes in Einstein-scalar-Gauss-Bonnet gravity

In this paper, we propose a fully nonlinear mechanism for obtaining scalarized black holes in Einstein-scalar-Gauss-Bonnet (EsGB) gravity which is beyond the spontaneous scalarization. Introducing three coupling functions f(φ) satisfying $f''(0) = 0$, we find that Schwarzschild black hole is linearly stable against scalar perturbation, whereas it is unstable against nonlinear scalar perturbation if the coupling function includes term higher than φ⁶. For a specific choice of coupling function f(φ)=α(φ⁴-βφ⁶), we obtain new black holes with scalar hair in the EsGB gravity. In this case, the coupling parameter α plays a major role in making different nonlinear scalarized black holes, while the other parameter β plays a supplementary role. Furthermore, we study thermodynamic aspects of these scalarized black holes and prove the first-law of thermodynamics.