Microscopic entropy of the three-dimensional rotating black hole

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

2. Microscopic entropy of the three-dimensional rotating black hole 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 E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL 2009 Field equations (fourth order) Linearized theory: Massive graviton with two helicities (Fierz-Pauli)

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

5. BHT massive gravity at the special point • The field eqs. admit the following solution • D. Tempo, J. Oliva, R. Troncoso, JHEP 2009 • 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

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

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

8. Rotating black hole • 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. Angular velocity of : Temperature : Entropy :

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

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

11. Rotating black hole Extremal case : (rotation) Extremal case : (gravitational hair) • 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

12. Gravitational hair, first law of thermodynamics and the ground state • 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 : • No chemical potential can be associated with • Variations of : reabsorbed by a shift of the global charges

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. Gravitational hair, first law of thermodynamics and the ground state • 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

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

16. Relaxed asymptotic conditions • The algebra of the conserved charges also acquires a central extension • The central charge is twice the one for GR : • 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 :

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

18. Microscopic entropy of the rotating black hole • 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

19. Microscopic entropy of the rotating black hole • 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 Hence : • Exact agreement with the semiclassical result

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

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

22. Ending remarks • 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.