Scaledependent rotating BTZ black hole
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Abstract
This work presents a generalization of the rotating black hole in two plus one dimensions, in the light of scaledependent gravitational couplings. In particular, the gravitational coupling \(\kappa _0\) and the cosmological term \(\varLambda _0\) are not forced to be constants anymore. Instead, \(\kappa \) and \(\varLambda \) are allowed to change along the radial scale r. The effective Einstein field equations of this problem are solved by assuming static rotational symmetry and by maintaining the usual structure of the line element. For this generalized solution, the asymptotic behavior, the horizon structure, and the thermodynamic properties are analyzed.
1 Introduction
To formulate a consistent and predictive quantum theory of gravity (QG) is one of the mayor challenges for the community seeking a unified description of the known fundamental interactions. Currently, at least 16 major approaches to quantum gravity have been proposed in the literature (see [1] and references therein), but none of these approaches have reached the goal in a completely satisfactory way.

Black holes (BHs):
Black Holes are objects of paramount importance in gravitational theories [2]. They allow to study gravitational systems at the transition between a quantum and a classical regime as for example through the the famously predicted Hawking radiation [3, 4]. BHs are thus excellent laboratories to investigate and understand several aspects of general relativity at the transition between a classical and quantum regime [5].

\(2+1\) dimensions: It can be expected that the features of a successful solution of the problem of quantum gravity are universal for gravitational theories of different dimensionality. Since gravity in \(2+1\) dimensions is mathematically less involved than in \(3+1\) dimensions, this lower dimensional theory is a good toy model if one aims to understand the underlying mechanisms of full quantum gravity in \(3+1\) dimensions. Apart from this motivation by quantum gravity, the study of gravity in \(2+1\) dimensions is of interest because of its deep connection to ChernSimons theory [6, 7] and because of its applications in the context of the AdS/CFT correspondence [8, 9, 10, 11, 12]. Within this lower dimensional gravitation theory the black hole solution found by Bañados, Teitelboim, and Zanelli (BTZ) [13, 14] plays a crucial role.

Scale dependence (SD): Before actually attacking the whole problem of QG with all its different, and up to now limited, realizations, one can begin with a more modest approach and concentrate on generic common features, which are expected from such a theory. One feature which is shared by most of the candidate theories for quantum gravity (actually by most quantum field theories) is that they predict a scale dependence of the coupling constants in the corresponding effective action. Luckily there is a well defined formalism which allows to deduce background solutions from a given effective action. We will follow those techniques which have been previously probed with a variety of problems [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In this paper we aim to study the dominant effects such a scale dependence could have on the BTZ black hole in the Einstein Hilbert truncation of the effective action of gravity in 2+1 dimensions. By using a well defined method which is based on the variational principle one can explore leading local effects of quantum gravity on a rotationally symmetric spacetime in a source free region (like BTZ), even without the knowledge of the exact underlying theory.
This paper is organized as follows: after this introduction, we present the action and the classical BTZ solution in the next section. Then, the general framework of this work is introduced in Sect. 3. The scale dependence for a rotating BTZ black hole is presented in Sect. 4. The bevaviour of the Ricci scalar, the asymptotic spacetime as well as the thermodynamics is investigated in Sects. 5 and 6 respectively. The discussion of this result and remarks are shown in Sect. 7. The main ideas and results are summarized in the conclusion Sect. 8. Note that throughout the paper we will use natural units with (\(c= \hbar = k_B = 1\)).
2 Classical BTZ solution with \(J_0 \ne 0\)
3 Scale dependent couplings and scale setting
4 Scale dependence BTZ solution with \(J_0 \ne 0\)
4.1 Solution
One observes that the lapse function f(r) presents two real valued horizons after the inclusion of nonzero angular momentum, just like the classical case. However, the location of those two horizons changes due to the inclusion of scale dependence. Thus, for non vanishing \(J_0\), there are two horizons independent of the presence (\(\epsilon \ne 0\)) or absence (\(\epsilon =0\)) of scale dependence. One remembers that for vanishing angular momentum, there is only a single horizon for the BTZ black hole which also gets shifted to lower values if one allows for scale dependence \(\epsilon >0\) [19]. In the scale dependent case there does not exist any finite \(\epsilon \) value for which the black hole becomes extremal. This will be discussed in more detail in Sect. 6. However, if one considers the limit \(\epsilon \rightarrow \infty \), the lapse function approaches that of an extremal black hole.
It is important to note that, some relevant quantities, such as the black hole radius \(r_H\), depend on the scale dependence parameter \(\epsilon \). However, the asymptotic spacetime for \(r\rightarrow \infty \) does not show this dependence. This important fact will be discussed in more detail in Sect. 5.
4.2 Horizon structure
5 Invariants and asymptotic spacetimes
This section discusses different asymptotic limits. In particular, we will focus on the asymptotic line element and the behavior of the the Ricci scalar R.
5.1 Asymptotic line element
5.1.1 Behaviuor when \(r \rightarrow 0\)
5.1.2 Behaviuor when \(r \rightarrow \infty \)
When studying the subleading corrections one has to be carefull with the two competing limits \(\epsilon \rightarrow 0\) and \(r \rightarrow \infty \), which can not be commuted. In this context we note that the naming of the integration constants (\(J_0, M_0, \dots \)) and thus of their physical interpretation was based on the classical limit \(\epsilon \rightarrow 0\).
5.2 Asymptotic Invariants
5.2.1 Behaviuor when \(r \rightarrow 0\)
5.2.2 Behaviuor when \(r \rightarrow \infty \)
6 Thermodynamic properties
The (numerical) knowledge of the horizons allows to study the thermodynamic properties of the scale dependent rotating black hole solution (25).
6.1 Hawking temperature
We notes that indeed the curves with (\(\epsilon \ne 0\)) and without scale dependence (\(\epsilon =0\)) coincide at the same minimal mass \(M_0=J_0/\ell _0\).
6.2 BekensteinHawking entropy
7 Discussion
Effective quantum corrections can be systematically introduced to the BTZ black hole by assuming a scale–dependent framework. This implies nontrivial deviations from classical black hole solutions. In this work, one of the integration constants (\(\epsilon \)) of the generalized field equations is used as a control parameter, which allows to regulate the strength of scale dependence, such that for \(\epsilon \rightarrow 0\), the wellknow classical BTZ background is recovered. This article discusses the BTZ black hole taking into account angular momentum in the context of scale dependent couplings. A solution of the corresponding field equations is presented and compared it with three different known cases: the classical case (\(\epsilon =0\)) without angular momentum, the classical case (\(\epsilon =0\)) with angular momentum, and the scale dependent case (\(\epsilon \ne 0\)) without angular momentum.
The new scale–dependent solution has some interesting features, for instance the lapse function increases rapidly when \(r \rightarrow \infty \) (which is present in the classical case) but now the effect is deeper, see Fig. 2 and compare the black curve (\(\epsilon = 0\)) with red curve (\(\epsilon = 1\)). By comparing Eq. (5) with Eq. (47) and with Eq. 44, we observe the deviation given by the scale–dependent framework respect to the classical solution. It is remarkable that when we are close to the origin the lapse function suffers a shift, while when we are far from the origin it shows a decrease by a factor of \(1 / \epsilon r\). In both cases the solution is affected.
Furthermore, according to Fig, 3, the outer horizons decrease when \(\epsilon \) increases. The effect of the scale dependent approach is thus that it produces smaller horizons, when compared to the usual case. Interestingly this decrease does not come with a change of the critical mass, where the two outer horizons merge.
An analysis of the Ricci scalar reveals that a singularity appears at \(r\rightarrow 0\) which is absent in the corresponding classical BTZ solution. Indeed, the BTZ black hole has a constant scalar, according to Eq. (51), whereas in the scale dependent case (\(\epsilon \ne 0\)) the singularity at \(r=0\) is always present according with Eq. (52). This is a consequence of the scale–dependent scenario.
Regarding the Hawking temperature, it is interesting that the scale dependent formula and the corresponding classical counterpart, coincide, under the replacement \(G_0 \rightarrow G(r_H) = G_0/(1+ \epsilon r_H)\) (23). It is further remarkable that the extreme black hole condition is also maintained and, therefore, the Hawking temperature is equal to zero when \(M_0 ^{\text {min}}=J_0/\ell _0\), independent of the strength of scale dependence \(\epsilon \). Moreover, we note that in presence of scale–dependent couplings the temperature is lowered with respect to the classic BTZ solution for large values of \(M_0\). Whereas when \(M_0\) is close to zero (for \(J_0 = 0\)) and when \(M_0\) is close to \(M_0 ^{\text {min}}\) (for \(J_0 \ne 0\)), the classical and the scale dependent solution are very similar. One notes that the BekensteinHawking entropy is increased by the scale dependence \(\epsilon \ne 0\) and that for large values of \(M_0\) the solutions with and without angular momentum match for a given value of \(\epsilon \), but they differ for different values of \(\epsilon \). Throughout the numeric analysis we also have used a relatively “small” value of \(\epsilon \), a choice which can be motivated by the assumption of relatively weak quantum effects provoking scale dependence at the level of the effective action (15). Lets mention in this context that the integration constant \(\epsilon \) can be made dimensionless for example by defining \(\epsilon = {\bar{\epsilon }} M_0\), in which case the graphical and analytical results with respect to \({\bar{\epsilon }}\) would have to be rescaled correspondingly.
Finally, lets comment on the ansatz (22). This type of ansatz also works for the spherically symmetric case. However, inspired by the ideas presented by Jacobson [53] it was possible to show that, for spherically symmetric static black holes, this type of ansatz is actually a consequence of a simple Null Energy Condition (NEC) [19, 20, 21].
This condition allows the avoidance of pathologies such as tachyons, instabilities, and ghosts [54, 55, 56]. Further, the NEC plays a crucial role in the Penrose singularity theorem [57]. However, a straight forward implementation of a generalized NEC to the rotating BH was not achieved, since the appearance of angular momentum reduces the symmetry of the problem. One would first have to generalize the arguments given in [53] to the rotational symmetry, before one can try to build an argument deriving the ansatz (22), as a consequence of some kind of NEC. Thus, at this point the use of the ansatz (22) is well justified, since it agrees with the NEC for vanishing rotation and since it further implements the structure of the line element for the case of the classical (not scaledependent) counterpart.
8 Conclusion
In this work we have studied the scale dependence of the rotating BTZ black hole assuming a finite cosmological term in the action. After presenting the models and the classical black hole solutions, we have allowed for a scale dependence of the cosmological “constant” as well as the gravitational coupling, and we have solved the corresponding generalized field equations with static circular symmetry. We have compared the classical solutions distinguishing two different cases, i.e. with and without angular momentum, with the corresponding scale dependent solution for same values of angular momentum. In addition, the horizon structure, the asymptotic spacetime and the thermodynamics were analyzed. In particular, the analysis of the Hawking temperature allowed to find a extremal black hole which coincides with the classical counterpart.
Notes
Acknowledgements
We wish to thank Prof. Maximo Bañados for some illuminating comments. The author A.R. was supported by the CONICYTPCHA/ Doctorado Nacional/201521151658. The author B.K. was supported by the Fondecyt 1161150 and Fondecyt 1181694.
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