1. Microscopic entropy of the three-dimensional rotating black hole of BHT massive gravity of BHT massive gravity Ricardo Troncoso Ricardo Troncoso In collaboration with Gaston Giribet David Tempo Julio Oliva Julio Oliva arXiv:0909.2564 [hep-th]

2. Microscopic entropy of the three-dimensional rotating black hole of BHT massive gravity of BHT massive gravity Ricardo Troncoso In collaboration with Gaston Giribet, David Tempo and Julio Oliva Centro de Estudios Científicos (CECS) Valdivia, Chile

3. BHT Massive Gravity Field equations (fourth order) Linearized theory: Massive graviton with two helicities (Fierz-Pauli) E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL 2009

4. BHT Massive Gravity Special case: Reminiscent of what occurs for the EGB theory for dimensions D>4 Unique maximally symmetric vacuum [A single fixed (A)dS radius l] Solutions of constant curvature :

5. The metric is conformally flat Asymptotically locally flat and (A)dS black holes Once suitably (double) Wick-rotated, describes: Gravitational solitons and wormholes in vacuum The rotating solution is found boosting this one T The field eqs. admit the following solution D. Tempo, J. Oliva, R. Troncoso, JHEP 2009 BHT massive gravity at the special point

6. Depends on three parameters: reduces to BTZ Rotating Black hole

7. : the mass : the angular momentum : “gravitational hair” parameter Does not correspond to any global charge generated by the asymptotic symmetries Conformally flat spacetime Asymptotically AdS Only two global charges :

8. depending on the range of M, a and b : the solution possesses an ergosphere and a singularity that can be surrounded by event and inner horizons. Rotating black hole Angular velocity of : Temperature : Entropy :

9. Rotating black hole The black hole fulfills : Extremal case : (due to rotation) degeneracy of states

10. Rotating black hole The black hole fulfills : Extremal case : (due to gravitational hair) single nondegenerate microscopic state

11. Extremality due to gravitational hair is stronger than extremality due to rotation is regarded as the ground state Lowest bound for the mass allowed by cosmic censorship Single nondegenerate microscopic state Rotating black hole Extremal case : (rotation) (gravitational hair)

12. No chemical potential can be associated with Variations of : reabsorbed by a shift of the global charges Deser-Tekin surface integrals: Rotating black hole possesses only two global charges: Reference background: massless BTZ black hole Absence of a global charge associated to : “gravitational hair” parameter. First law : Gravitational hair, first law of thermodynamics and the ground state

13. Gravitational hair, first law of thermodynamics and the ground state Dependence on the gravitational hair parameter : entirely absorbed by a shift of the global charges

14. First law is fulfilled: provided global charges (mass and angular momentum) are measured w.r.t. the extremal case that saturates the bound __________________________________________________ Stronger support to consider the extremal case as the ground state Gravitational hair, first law of thermodynamics and the ground state

15. Relaxed as compared with Brown-Henneaux Invariant under the same asymptotic symmetries: Two copies of the Virasoro algebra (Conformal group in 2D) Relaxed asymptotic conditions

16. Choosing the extremal case as the reference background: The only nonvanishing surface integrals for the rotating black hole are associated with the left and right Virasoro generators : The central charge is twice the one for GR : Relaxed asymptotic conditions The algebra of the conserved charges also acquires a central extension

17. Is it possible to compute the entropy of the rotating black hole of BHT massive gravity by means of Cardy’s formula ?

18. Strominger’s result for GR extends for the BHT theory Relies on Brown-Henneaux observation (80’s): This is currently interpreted in terms of the AdS/CFT correspondence Asymptotic symmetry group of GR: two copies of the Virasoro algebra, thus Consistent quantum theory of gravity: CFT in 2D We assume that the quantum theory for BHT massive gravity exists and it is consistently described by a dual CFT Microscopic entropy of the rotating black hole

19. Physical states form a representation of the algebra with If the CFT fulfills some physically sensible properties, the asymptotic growth of the number of states is given by Cardy’s formula Exact agreement with the semiclassical result Microscopic entropy of the rotating black hole Hence :

20. Left and right movers are decoupled: Equilibrium at different temperatures In the canonical ensemble: The semiclassical result for the entropy is easily recovered Microscopic entropy of the rotating black hole

21. The ground state Extremal case : (gravitational hair) Extremal case : (rotation) As it has to be for a suitable ground state

22. Entropy of the rotating black hole can be microscopically reproduced from Cardy’s formula Ground state: extremal case Computations can be extended perfectly well even for (not intended) Subtlety: for the configuration with suffers certain pathologies Remarkably, for the theory also admits a gravitational soliton Spacetime is regular everywhere: provide a suitable nondegenerate state naturally regarded as the ground state The black hole is conformally flat: solves the BHT field equations even in presence of the topological mass term Our results extend to this case. Ending remarks