Scaledependent polytropic black hole
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Abstract
In the present work we study the scaledependence of polytropic noncharged black holes in (3+1)dimensional spacetimes assuming a cosmological constant. We allow for scaledependence of the gravitational and cosmological couplings, and we solve the corresponding generalized field equations imposing the null energy condition. Besides, some properties, such as horizon structure and thermodynamics, are discussed in detail.
1 Introduction
The polytropic equations of state are appropriate in many situations in the context of General Relativity as well as in astrophysical problems. For example, it is well known that astrophysical objects as cores of stars, fully convective lowmass stars, white dwarfs, neutron stars and galactic halos can be modelled with matter which fullfills a polytropic equation of state [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In addition, polytropic equations of state have been considered in a cosmological context in order to model the matter content [12].
Recently, black hole (hereafter BH) solutions have been obtained considering polytropic equations of state [13]. In that work, the authors map the negative cosmological coupling with an effective pressure, demanding that it obeys a polytropic equation of state. After that, the matter content degrees of freedom are eliminated from the Einstein field equations and, finally, solutions matching polytropic thermodynamics with that of BHs are obtained.
The aforementioned solutions are obtained in the context of classical gravity. However, it is very well known that a more complete description quantum effects must be considered. As the full theory of quantum gravity is still lacking, a great variety of work has been devoted to getting some insight into the underlying physics (for an incomplete list, check [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and for a review see [33, 34]). Despite the fact that in that work the authors discuss different aspects of quantum gravity, most of them have the common feature that the resulting effective gravitational action acquires a scaledependence. This behaviour is observed through the couplings of the effective action: they change from fixed values to scaledependent quantities, i.e. \(\{G_0, \varLambda _0\} \mapsto \{G_k, \varLambda _k\}\), where \(G_0\) is Newton’s coupling and \(\varLambda _0\) is the cosmological coupling. Indeed, there is some evidence which supports that this scaling behaviour is consistent with Weinberg’s Asymptotic Safety program [35, 36, 37, 38, 39, 40, 41, 42]. In addition, the effective action assuming running couplings has been studied in threedimensional spacetimes in the context of BH physics in Refs. [43, 44, 45, 46, 47, 48], in four dimensions [49] and in the cosmological context [50]. In the aforementioned work, the corresponding scaledependent couplings take into account a quantum effect, namely, this approach admits corrections to both the classical BH backgrounds and the FLRW universe.
Then, inspired by this fact, the next step is to take advantage of the aforementioned approach to produce interesting BH solutions in new scenarios. The Van der Waals black hole [51] is a recent solution which, after identifying P with the cosmological constant \(\varLambda \), allow us to write down a BH equation of state \(P=P(V, T )\) and to compare it to the corresponding fluid equation of state [51]. We remark that this fact opened a window to investigate the analogy between BHs and certain fluids (for a recent review, see [52]).
In this work, inspired by this idea, we study how a polytropic BH in fourdimensional spacetime obtained in Ref. [13] is generalized to the case of scaledependent couplings, implemented and set at the level of an effective action.
The work is organized as follows: In Sect. 2 we introduce and summarize the polytropic BH solution, whereas Sect. 3 is devoted to a brief introduction of the scaledependent gravitational setting, which is employed in Sect. 4 to obtain the scaledependent BH solution. Along this section, the new solution is carefully studied with emphasys on horizons, asymptotics, singularities, thermodynamics and their comparison with the previously studied classical polytropic solution. Finally, some concluding remarks are given in Sect. 5.
2 Polytropic black hole solution

The metric (1) is a solution of the Einstein field equations \(G_{\mu \nu }+g_{\mu \nu }\varLambda _{0} = \kappa _0 T_{\mu \nu }\) with \(\varLambda _0<0\) and \(T^{\mu }_{\nu }=\text {diag}(\varrho ,p_{1},p_{2},p_{3})\).

The thermodynamics of the BH solution is matched with that of a polytropic gas after eliminating the matter degrees of freedom from the Einstein field equations.
It is worth mentioning that, in this case, the matter content does not depend on the constant K appearing in the polytropic equation of state for the dark energy content. However, this is not true in general because the result could be affected by the particular choice of the polytropic index n.
3 Scaledependent coupling and scale setting
4 Scaledependent polytropic black hole
4.1 Solutions
4.2 Horizons and black hole thermodynamics
As in the previous case, the temperatures coincide for small classical BH mass and a deviation is observed as \(M_{0}\) increases.
As a final comment, we note that the classical behaviour of both the temperature and the entropy is not considerably affected by the runnig. In this sense, in spite the metric changes with small values of the runnig (including topology changes), the thermodynamics remains robust because the horizon remains close to the classical one.
5 Concluding remarks
In this article, we have studied for the first time the scaledependence of a polytropic black hole in a spherically symmetric fourdimensional spacetime assuming a nonnull cosmological coupling. After presenting the model and the classical black hole solutions, we have allowed for a scaledependence of the gravitational as well as the cosmological coupling, and we have solved the corresponding generalized field equations by imposing the null energy condition. Besides, we have studied in detail the horizon structure, asymptotics and some thermodynamic properties.
As a mandatory remark, one should note that the scaledependent approach introduces an effective contribution to the energy momentum tensor thought \(\varDelta t_{\mu \nu }\). Moreover, in agreement with the classical solution, the Schwarzschild ansatz is preserved. Regarding the event horizon, it is important to mention that it is not analytical and therefore it is not possible to get an explicit expression for it. However, we are able to obtain a closed formula for the temperature and the entropy, writing those quantities in terms of the horizon radius. Note that, for small values of \(\epsilon \), the scaledependent solutions are in agreement with the classical black hole, but when \(\epsilon \) take large values, a strong deviation appears.
In addition, and following the philosophy of the scaledependent scenario, this novel solution (including the thermodynamic properties) should be quite similar to the classical counterpart. This is because we expect that the incorporation of quantum corrections slightly modifies the usual behavior, which is in agreement with Eqs. (54), (57) and (58).
As this is a general feature of several scaledependent black hole solutions studied in the past [44, 46, 49], we conclude that black hole thermodynamics is robust against this kind of deformations of the gravitational theory.
To conclude, some final comments are in order. First, we would like to point out that all of the results obtained here are independent of the proportionality constant K appearing in the polytropic equation of state. Even more, the choice \(n=1/3\) for the polytropic index and the particular fitting for the integration constants \(C_{1}\) and \(C_{2}\) in Sect. 2 lead to solutions which are unaffected by whether K depends on a certain scale or not. It can be shown that, for a different choice of the polytropic index and integration constants the matter sector must be modified incorporating the scaledependence on K. Second, we note that, although we have extended the polytropic solution given in [13], the same technique could be applied to different exact polytrope solutions with cosmological constant reported in the past (see, for example [59, 60]). Third, from the point of view of a possible astrophysical test of the solution here presented, an important feature to be studied is the socalled static radius (frequently known as the turnaround radius) [61, 62], which defines the equilibrium region between gravitational attraction and dark energy repulsion. Interestingly, this radius, computed for several polytropic solutions, can be related to the maximum allowable size of a spherical cosmic structure as a function of its mass [10, 62, 63, 64]. In this sense, astrophysical observations of largescale structures could indirectly shed light on possible bounds on the running parameter (which would enter as a new ingredient of the static radius), which would indicate deviations from general relativity.
These and other aspects, which are beyond the scope of the present work, are left for a future publication.
Footnotes
 1.
Note that, although the pressures are nonequal, we refer to a perfect fluid in the sense that \(\rho \) and p are related by \(\rho _{0}=p_{1}.\)
Notes
Acknowledgements
The author E.C. would like to acknowledge Nelson Bolivar for fruitful discussion. The author A.R. was supported by the CONICYTPCHA/Doctorado Nacional/201521151658. The author B.K. was supported by the Fondecyt 1161150. The author P.B. was supported by the Faculty of Science and Vicerrectoría de Investigaciones of Universidad de los Andes, Bogotá, Colombia. P. B. dedicates this work to Inés Bargueño–Dorta.
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