Microscopic entropy of the Microscopic entropy of the

three-dimensional rotating black holethree-dimensional rotating black hole

of BHT massive gravityof BHT massive gravity

Ricardo TroncosoRicardo Troncoso

In collaboration withIn collaboration with

Gaston GiribetGaston Giribet

David TempoDavid Tempo

Julio OlivaJulio Oliva

arXiv:0909.2564 [hep-th]arXiv:0909.2564 [hep-th]

Microscopic entropy of the Microscopic entropy of the

three-dimensional rotating black holethree-dimensional rotating black hole

of BHT massive gravityof BHT massive gravity

Ricardo TroncosoRicardo Troncoso

In collaboration withIn collaboration with

Gaston Giribet, David Tempo and Julio OlivaGaston Giribet, David Tempo and Julio Oliva

Centro de Estudios Científicos (CECS) Centro de Estudios Científicos (CECS) Valdivia, Valdivia, ChileChile

BHT Massive BHT Massive GravityGravity

Field equations(fourth order)

Linearized theory: Massive graviton with two helicities (Fierz-Pauli)

E. A. Bergshoeff, O. Hohm, P. K. Townsend, PRL 2009

BHT Massive BHT Massive GravityGravity

Special case:

Reminiscent of what occurs for the EGB theoryfor dimensions D>4

Unique maximally symmetric vacuum [A single fixed (A)dS radius l]

Solutions of constant curvature :

• 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

• The field eqs. admit the following solutionThe field eqs. admit the following solution D. Tempo, J. Oliva, R. Troncoso, JHEP 2009

BHT massive gravity at the special BHT massive gravity at the special pointpoint

Depends on three parameters:

reduces to BTZ

Rotating Black Rotating Black holehole

• : the mass

• : the angular momentum

• : “gravitational hair” parameter

Does not correspond to any global chargegenerated by the asymptotic symmetries

• Conformally flat spacetime

• Asymptotically AdS

Rotating Black Rotating Black holehole

Only two global Only two global charges :charges :

• 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 Rotating black holehole

Angular velocity of :

Temperature :

Entropy :

Rotating black Rotating black holehole

The black hole fulfills :

Extremal case :Extremal case :

(due to rotation)(due to rotation)

degeneracy of states

Rotating black Rotating black holehole

The black hole fulfills :

Extremal case :Extremal case :

(due to gravitational (due to gravitational hair)hair)

single nondegenerate microscopic state

• 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 Rotating black holehole

Extremal case :Extremal case :

(rotation) (rotation)

Extremal case : Extremal case :

(gravitational (gravitational hair)hair)

• 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 First law ::

Gravitational hair, first law of Gravitational hair, first law of thermodynamics thermodynamics

and the ground stateand the ground state

Gravitational hair, first law of Gravitational hair, first law of thermodynamics thermodynamics

and the ground stateand the ground state

Dependence on the gravitational hair parameter :

entirely absorbed by a shift of the global charges

• 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 Gravitational hair, first law of thermodynamics thermodynamics

and the ground stateand the ground state

• Relaxed as compared with Brown-Henneaux

• Invariant under the same asymptotic symmetries: Two copies of the Virasoro algebra (Conformal group in 2D)

Relaxed asymptotic Relaxed asymptotic conditionsconditions

• 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 Relaxed asymptotic conditionsconditions

• The algebra of the conserved charges also acquires a central extension

Is it possible to compute the entropyIs it possible to compute the entropy

of the rotating black hole of the rotating black hole

of BHT massive gravity of BHT massive gravity

by means of Cardy’s formula ?by means of Cardy’s formula ?

• 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 Microscopic entropy of the rotating black holeblack 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

• Exact agreement with the semiclassical result

Microscopic entropy of the rotating Microscopic entropy of the rotating black holeblack hole

Hence Hence ::

• 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 Microscopic entropy of the rotating black holeblack hole

The ground The ground statestate

Extremal case : Extremal case :

(gravitational (gravitational hair)hair)

Extremal case :Extremal case :

(rotation) (rotation)

• As it has to be for a suitable ground state

• 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 Ending remarksremarks