Extraction of energy from a black hole in Einstein-Maxwell-scalar theory

Recently, it has been discovered that the nonlinear self-interaction of matter can induce energy extraction from black holes beyond superradiant instability. This process is closely associated with the occurrence of a dynamical first-order transition between different types of static black holes. To explore whether first-order phase transitions invariably lead to energy extraction, we have investigated the evolution of black holes in the Einstein-Maxwell-scalar model with a higher-order coupling. In this model, there are also dynamical first-order phase transitions between black hole solutions. Our findings indicate that energy can only be extracted from a small, stable hairy black hole in this model. However, this energy extraction is more closely related to the growth of the black hole horizon radius, rather than the dynamical transition between different types of black holes. This suggests that a dynamical first-order phase transition does not necessarily result in energy extraction.