Supported by the Serbian Science Foundation (171031)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We study the near horizon geometry of both static and stationary extremal Oliva Tempo Troncoso (OTT) black holes. For each of these cases, a set of consistent asymptotic conditions is introduced. The canonical generator for the static configuration is shown to be regular. For the rotating OTT black hole, the asymptotic symmetry is described by the time reparametrization, the chiral Virasoro and centrally extended

Article funded by SCOAP^{3}

The long-standing problem of the origin of black hole entropy is one of the most important open questions in contemporary physics. There are many proposals for interpreting the black hole entropy and the corresponding micro-states, such as: entanglement entropy [

Holographic duality [

A particularly interesting generalization is given in [

In this article, we analyse the near horizon limit of a black hole with soft hair known as the Oliva-Tempo-Troncoso (OTT) black hole [

We first analyse the static OTT black hole, which becomes extremal for the specific value of the hair parameter, and obtain the corresponding near horizon geometry. We then study the asymptotic structure of the near horizon geometry and obtain the asymptotic symmetry group.

We continue with a study of the rotating OTT black hole which can be made extremal in two different ways: either by tuning the hair parameter or the angular momentum. The solution obtained by tuning the hair parameter, surprisingly, leads to the same near horizon geometry as in the non-rotating case. The extremal OTT black hole with maximal angular momentum leads to a geometry with a richer structure. We conclude that the asymptotic symmetry is a direct sum of the time reparametrization, the Virasoro algebra and the centrally extended

Our conventions are the same as in Ref. [^{i}^{ij}^{i}^{i}^{i}_{m}^{m}^{ij}^{ij}^{i}_{k}^{kj}_{i}^{j}_{i}^{j}_{i}^{j}^{ijk}^{012}= 1, the Lie dual of an antisymmetric form ^{ij}_{i}_{ijk}X^{jk}

In the sector with a unique AdS ground state, the BHT gravity possesses an interesting black hole solution, the OTT black hole [^{(a)}_{i}^{(a)}^{ij}_{0}=1/16_{0} is a cosmological constant, and (_{1},_{2},_{3}) and (_{1},_{2},_{3}) are the coupling constants in the torsion and the curvature sector, respectively. In [^{i}^{i}^{i}^{i}^{i}_{m}L^{m}^{i}

By using the BHT condition that ensures the existence of the unique maximally symmetric background [

The usual construction of the canonical generator of the Poincaré gauge transformations, including diffeomorphisms and Lorentz rotations [^{m}_{ij}^{i}^{ij}_{i}_{i}_{ij}_{ij}_{G}

The construction of the canonical generator ^{i}_{i}_{mn}_{mn}^{μ}^{i}

The explicit form of the generator of Lorentz rotations, see [_{2}, which is of the form (

In general

In many cases the asymptotic conditions ensure the regularity of the Lorentz rotations generator and Γ_{2} = 0. However, it is worth noting that in the particular problem we are solving the contribution of the surface term of the Lorentz rotations generator is non-trivial, as we shall see in section

_{+} = _{−}. This condition is satisfied when ^{2}^{2}+4

We choose the triad fields in the simple diagonal form:
^{i}^{i}_{4}+2_{6}=0.

Let us consider the following asymptotic conditions for the metric in the region ^{−n} or faster. In accordance with (^{i}

Lorentz transformations that leave the asymptotic conditions invariant are

In terms of the Fourier modes _{n}_{0}(^{inφ}) and _{n}_{0}(^{imφ}), the algebra of the residual gauge transformations takes the form of a semi-direct sum of the Virasoro and the Kac-Moody algebras:

The gauge generator is not a priori well-defined because, for given asymptotic conditions, its functional derivatives may be ill-defined, as already mentioned in section _{i}

For the particular asymptotic conditions adopted in this paper, we conclude that the gauge generator is differentiable, so that there is no need for adding any surface term,

The conserved charges of the rotating black hole take the following form:

The rotating OTT black hole is a three-parameter solution, so that the extremal limit can be achieved in two different ways. The first is the same as in the non-rotating case, by requiring 4^{2}^{2} = 0. As a simple consequence, the resulting geometry is the same as if the black hole were non-rotating. This is not a surprising result if we note that both energy and angular momentum vanish in this case.

The second way to obtain an extremal black hole is to take

The horizon is located at

After changing the coordinates and taking the limit

The rotating OTT black hole for ^{2}, we obtain a near-horizon BTZ black hole geometry with two times smaller _{0} [_{0}, and it will lead to different values of the central charges. Thus, we are able to recover the results for the near-horizon BTZ black hole geometry from those of the OTT black hole, but not by simply taking

We consider the following asymptotic form of the metric
^{i}

The improved generator is given by
^{i}

The entropy of the extremal OTT black hole _{0} on the shell

It is well-known that the Virasoro algebra can be constructed as a bilinear combination of the elements of the Kac-Moody algebra. We apply this procedure, known as the Sugawara-Sommerfeld construction [

First we introduce the auxiliary operators

In theories with conformal symmetry, it is well-known that entropy can be reproduced by the Cardy formula. Sugawara-Sommerfeld construction includes an arbitrary parameter

There is an equivalent Cardy formula in which, instead of using the background values of the Virasoro zero modes, one uses the temperature. Thus, the required additional piece of information is the temperature of the dual CFT, which may be derived from the black hole thermodynamics.

We start from the first law of black hole thermodynamics
^{i}_{i} are potentials conjugate to ^{i}_{H} = 0, the first law implies that energy is a function of the conserved charges
_{L} is the left moving temperature.

The entropy, energy and angular momentum of the extremal OTT black hole are given by:
^{R} is undetermined, and that the left central charge is two times bigger than the Brown-Henneaux central charge

We investigated the near horizon symmetry of both static and stationary OTT black holes in the quadratic PGT. In the static case, the corresponding asymptotic symmetry is trivial, whereas in the stationary case, the set of consistent asymptotic conditions leads to a symmetry described by time reparametrization and the semi-direct sum of the centrally extended

The near horizon limit corresponds to deep infrared sector of the theory, which implies that only the soft part of the charge survives. This means that the corresponding charges represent the soft hair on the black hole horizon. Formula

Using the Sugawara-Sommerfeld construction, we build the Virasoro algebra as a bilinear combination of the