Hairy Black Holes with Arbitrary Small Areas

We obtained new hairy black hole solutions in Einstein-scalar theory, including asymptotic flat, de Sitter and anti-de Sitter black holes. The theory is inspired by Ref. 1, where traversable wormhole solutions from an Einstein-phantom scalar theory are constructed. In this work, we found new black hole solutions in an Einstein-normal scalar theory. Comparing with Schwarzschild metric, the hairy black holes have two interesting properties: i) the areas of the black holes are always smaller than the same mass Schwarzschild black holes; ii) A naked singularity with positive mass arises when the black hole mass decreases. The energy conditions for the black holes and naked singularities are checked. We found that, as hairy black holes, the null energy condition(NEC) and the strong energy condition(SEC) are hold, while the weak energy condition(WEC) is violated in the vicinity of black hole horizon. The naked singularity respects to all three energy conditions. We also investigate the quasinormal modes(QNMs) of the hairy black holes by a test scalar field. The results indicate that one can distinguish hairy black holes with the same mass Schwarzschilid black hole by their QNM spectra.