Gravitational solitons, hairy black holes and phase transitions in BHT massive gravity

Hairy black holes and gravitational solitons in three dimensions for the new (BHT) massive gravity theory are considered at the special case when there is a unique maximally symmetric solution. Following the Brown-York approach with suitable counterterms, it is shown that the soliton possesses a fixed negative mass which coincides with that of AdS spacetime regardless the value of the integration constant that describes it. The soliton is then regarded as a degenerate ground state labeled by a modulus parameter. The Euclidean action is shown to be finite and independent of modulus and hair parameters for both classes of solutions, reproducing the hairy black hole free energy. Modular invariance implies that the gravitational hair becomes determined by the modulus parameter. Cardy formula is shown to agree with the semiclassical entropy provided the modulus parameter of the ground state is spontaneously fixed, suggesting that the hairy black hole is in a broken phase. Indeed, it is found that the critical temperature T-c = (2 pi l)(-1) characterizes a first order phase transition between the static hairy black hole and the soliton which, due to the existence of gravitational hair, can take place in the semiclassical regime.