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

Alfredo Perez, David Tempo, Ricardo Troncoso

TL;DR

This paper analyzes three-dimensional Bergshoeff–Hohm–Townsend (BHT) massive gravity at a special point with a unique AdS vacuum, focusing on gravitational solitons and hairy black holes. Using Brown–York charges and counterterms, it shows the soliton has a fixed negative mass identical to AdS and identifies its modulus as a degenerate ground-state parameter. The authors reveal a modular duality between the soliton and rotating hairy black holes, linking the hair parameter to the soliton modulus and applying Cardy entropy counting to argue for spontaneous fixing of the ground-state modulus, indicating a broken phase. They also establish a first-order phase transition between the static hairy black hole and the soliton at Tc = (2πl)^{-1}, with the transition analyzed semiclassically thanks to the hair-induced length scale. These results illuminate the interplay between ground-state structure, holographic duality, and semiclassical thermodynamics in BHT gravity.

Abstract

Hairy black holes and gravitational solitons in three dimensions for the new massive gravity theory proposed by Bergshoeff, Hohm and Townsend (BHT) 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 the one of AdS spacetime regardless the value of the integration constant that describes it. Hence, the soliton can be naturally regarded as a degenerate ground state labeled by a single modulus parameter. The Euclidean action, endowed with suitable counterterms, is shown to be finite and independent of modulus and hair parameters for both classes of solutions, and in the case of hairy black holes the free energy in the semiclassical approximation is reproduced. Modular invariance allows to show that the gravitational hair turns out to be determined by the modulus parameter. According to Cardy's formula, it is shown that the semiclassical entropy agrees with the microscopic counting of states provided the modulus parameter of the ground state is spontaneously fixed, which suggests that the hairy black hole is in a broken phase of the theory. Indeed, it is found that there is a critical temperature characterizing 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.

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

TL;DR

This paper analyzes three-dimensional Bergshoeff–Hohm–Townsend (BHT) massive gravity at a special point with a unique AdS vacuum, focusing on gravitational solitons and hairy black holes. Using Brown–York charges and counterterms, it shows the soliton has a fixed negative mass identical to AdS and identifies its modulus as a degenerate ground-state parameter. The authors reveal a modular duality between the soliton and rotating hairy black holes, linking the hair parameter to the soliton modulus and applying Cardy entropy counting to argue for spontaneous fixing of the ground-state modulus, indicating a broken phase. They also establish a first-order phase transition between the static hairy black hole and the soliton at Tc = (2πl)^{-1}, with the transition analyzed semiclassically thanks to the hair-induced length scale. These results illuminate the interplay between ground-state structure, holographic duality, and semiclassical thermodynamics in BHT gravity.

Abstract

Hairy black holes and gravitational solitons in three dimensions for the new massive gravity theory proposed by Bergshoeff, Hohm and Townsend (BHT) 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 the one of AdS spacetime regardless the value of the integration constant that describes it. Hence, the soliton can be naturally regarded as a degenerate ground state labeled by a single modulus parameter. The Euclidean action, endowed with suitable counterterms, is shown to be finite and independent of modulus and hair parameters for both classes of solutions, and in the case of hairy black holes the free energy in the semiclassical approximation is reproduced. Modular invariance allows to show that the gravitational hair turns out to be determined by the modulus parameter. According to Cardy's formula, it is shown that the semiclassical entropy agrees with the microscopic counting of states provided the modulus parameter of the ground state is spontaneously fixed, which suggests that the hairy black hole is in a broken phase of the theory. Indeed, it is found that there is a critical temperature characterizing 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.

Paper Structure

This paper contains 9 sections, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Soliton as a degenerate ground state
  3. Soliton mass and Euclidean action from the Brown-York approach with suitable counterterms
  4. Rotating hairy black hole: regularized Euclidean action and thermodynamics
  5. Modular invariance, solitons and microscopic entropy of the rotating hairy black hole
  6. Cardy formula and spontaneous fixing of ground state modulus parameter
  7. Hairy black hole-Soliton phase transition
  8. Gravitational hair, thermal fluctuations and phase transition in the semiclassical regime
  9. Summary and discussion

Figures (1)

  • Figure 1: Free energy as a function of the temperature for the soliton and the hairy black hole. Here $T_{c}=(2\pi l)^{-1}$.