Regular black holes and its thermodynamics in Lovelock gravity
Abstract
In this work two new families of nonsingular or regular black hole solutions are displayed. These black holes behave as de Sitter space near its center and have a well defined AdS asymptotic region for negative cosmological constant. These solutions are constructed on a general ground through the introduction of a finite density of mass/energy. This removes the usual singularity of a black hole and also introduces a new internal geometry. The thermodynamic properties of these solutions are discussed as well.
1 Introduction
One of the most relevant predictions of General Relativity was the existence of black holes and nowadays there is substantial evidence that this is a usual phenomenon in nature. Now, despite the Schwarzschild black hole solution has been known for over a century it was not until the end of 60’s that was shown that, under general classical conditions, that the formation of a black holes is unavoidable provided the energy (density) within a region of the space surpasses a certain limit. Indeed, the matter in that region collapses until an (event) horizon is formed. However, under the same argument Penrose determined that the final stage of that gravitational collapse gives rise to a singularity as predicted by the existence of Schwarzschild and Kerr solutions.
One unwanted consequence of the presence of singularities is that they break predictability. Fortunately, it is certain that classical general relativity cannot be valid at all scales [1] and thus at Planck scales the description of nature must change drastically. It is precisely this new description which is expected to provide a tamed version of the singularities once quantum effects are considered. This scenario is supported by results in either String Theory or Loop Quantum Gravity. Results in LQG, for instance, determine that before matter can reach the Planck density, quantum (gravity) fluctuations actually generate enough pressure to counterbalance weight. For the physics of a black hole, this implies that the gravitational collapse stops before a singularity can be formed. Furthermore, this can be understood as the formation of a dense central core whose density is of the order of magnitude of the Planck density. These objects are called Planck stars [2]. Once one has embraced the idea that inside a black hole, instead of a singularity, a dense core exists, one has to propose model for it. The first approximation to do this is to treat the problem as a classical gravitational problem with an energymomentum density which condenses the quantum effects, in particular the existence of a pseudo repulsive force at the origin. In practice Planck stars can be studied as a geometry which far away from the core recovers a standard black hole solution, says Schwarzschild for instance, but whose center, although contains a dense core, can still be treated as a manifold. Moreover, the core of the geometry must approach, in a first approximation, a de Sitter space as such the geodesics diverge mimicking the repulsing force mentioned above. This kind of solutions are called nonsingular or regular black holes and in the context of this work can be considered synonyms to Planck star.
Lovelock gravity
One of the fundamental aspects of GR, which almost singles it out in three and four dimensions, is that its equations of motion are of second order and thus causality is guarantied. In higher dimensions, \(d>4\), however having second order equations of motion is a property of much larger families of theories of gravity. Among them Lovelock gravities have a predominant rôle [22, 23].
There are several known black hole solutions of Lovelock gravity in vacuum (\(T^{\mu \nu }=0\)). See for instance [23, 26, 27, 28, 29, 30, 31] and reference therein. However, there are not many known solutions in presence of matter fields, see for instance [32]. This is mostly due to the nonlinearities of any theory of gravity, which makes difficult, if not impossible, to solve analytically its equations of motion for an arbitrary matter field configuration. Indeed, only highly symmetric configuration can be studied analytically. For the case of our interest, classically regular black holes have been studied in [19, 20] within Einstein Gauss Bonnet theories.
In this work two new families of regular black holes will be displayed. These solutions share to belong to families of solutions which have a single locally AdS ground state. Moreover, these families have a single well defined asymptotically locally AdS region which approaches the ground state. These two new families correspond to the generalization of the Pure Lovelock solutions [28, 30, 32] and those discussed in [27] which have a nfold degenerated ground state.
During the next sections, first the general conditions to be satisfied by the mass density will be discussed. Next, it will be obtained the two families of solutions and analyzed their behavior. Finally their thermodynamics will be displayed.
2 A well posed mass definition
 1.
\(\rho \) must be a positive due the weak energy condition and a continuous differentiable function to avoid singularities. This implies that m(r) is a positive monotonically increasing function (\(m(r)>0\) \(\forall r\) and \(m(r_1) > m(r_2)\) if \(r_1>r_2\)) which vanishes at \(r=0\).
 2.\(\rho \) must have a finite single maximum at \(r=0\), the core, (\(\rho (0) > \rho (r)\) \(\forall r >0\)) and to rapidly decrease away from the core. This yields the conditionwith \(K>0\) proportional to \(\rho (0)\). The finiteness and snootness of \(\rho (0)\) forbid the presence of a curvature singularity at \(r = 0\) [4]. However, it must be noted that this is not enough to ensure a dS behavior near the center of the geometry, and thus additional conditions will be imposed in the next sections.$$\begin{aligned} \left. m(r) \right _{r\approx 0} \approx K r^{d1}, \end{aligned}$$(12)
 3.For the space to have a well defined asymptotic region, such as those to be studied, and to describe a physical object, \(\rho (r)\) must be such that m(r) be bounded for \(0< r <\infty \) , i.e., with a well defined limit for \(r \rightarrow \infty \). Therefore,for M some constant. Later it will be shown that M is proportional to the total mass of the geometry. This implies that :$$\begin{aligned} \lim _{r \rightarrow \infty } m(r) = {M}, \end{aligned}$$(13)$$\begin{aligned} \lim _{r \rightarrow \infty } \frac{d}{dr} m(r) = 0. \end{aligned}$$(14)
 4.
As mentioned above the idea of a regular black hole is to mimic the exterior of black hole. For this to happen the density \(\rho \) must be such that there is a radius \(r=r_*\) where is satisfied \(m(r_*) \approx M\) and \(\frac{d}{dr} m(r_*) \approx 0\). In general, one can also expected that for large masses that \(\ell _P \ll r_{*} \ll r_{+}\) be satisfied. This condition, however, is not satisfied for masses within the range of Plank scales but still the thermodynamics can be studied [5, 6].
3 First family of solutions: regular black holes in pure Lovelock theory
3.1 Global analysis
As mentioned above, any zero of f(r) defines an event horizon in the geometry. This fact significantly simplifies the analysis. Now, to proceed, the cases \(\varLambda >0\) and \(\varLambda <0\) will be discussed separately.
3.1.1 \(\varLambda < 0\) or negative cosmological constant
3.1.2 \(\varLambda >0\) or positive cosmological constant
3.2 Internal geometry
3.3 The non spherical symmetric solutions
4 Second family of solutions: regular black holes with nfold degenerated ground state
As mentioned above in general the Lovelock gravity might have more than a single constant curvature ground state, which makes those ground states unstable under dynamical evolution. One way to avoid this, observe Eq. (7), is by choosing the \({\alpha _p}\)^{3} such that \(P_l(x)\) becomes \(P_{l}(x) = (x \pm l^{2})^n\) creating \(n\)fold degenerated ground state of constant curvature \(\pm l^{2}\). From now on only the negative cosmological constant case will be discussed as this the case where a genuine asymptotic region exists. Still some comment on the positive cosmological constant case will be done when they are straightforward.
4.1 Horizons
4.2 Limits of this solution
5 The thermodynamics before the thermodynamics
After studying the properties of these solutions in a general framework one can proceed to analyze the thermodynamics of the black holes in the spectrum of solutions. As mentioned previously, the central concern in this work are only spaces with a well defined asymptotically locally AdS region, but still some general comments will be made as well. For instance, although the number of horizons, and their properties, depends on m(r), there is always a particular horizon which can be unambiguously cast as the horizon of the black hole. The associate zero will be denoted \(r_+\). This is even independent of the existence, for \(\varLambda > 0\), of an even outer (cosmological) horizon on the geometry.
It is utmost relevant to stress that the thermodynamics of black holes whose ground states are locally AdS or flat spaces differs of the thermodynamics of those whose ground state is a locally dS space. This is due to the obvious fact, mentioned already, that while for the latter there exists a well defined asymptotic region, for later there is no asymptotic region at all. It is well known that temperature and entropy can be defined independently of the existence of asymptotic region. However, the existence of conserved charges cannot, in particular the mass. In this way, a standard first law of thermodynamics, i.e. \(TdS = dE + \ldots \), can only be developed for spaces with a well defined asymptotic region. The lack of an asymptotic region, or the presence of a cosmological horizon, forbids such a kind of relations to exist. Moreover, in [34] was established that the presence of a black hole in geometry which also has a cosmological horizon can be interpreted as nonequilibrium thermodynamic system. That system, through the evaporation processes of both horizons, evolves into a de Sitter space. One can conjecture, for the solutions above, that the same can happen. That analysis, however, will be carried out elsewhere.
In what follows it will be analyzed the thermodynamics for asymptotically locally AdS spaces whose have a single well defined locally AdS ground state.
6 Thermodynamics of asymptotically AdS solutions
In this case the thermodynamics can be obtained following standard Wald’s prescriptions based on the adiabatic change of the Noether charges in the region \(]r_+ , \infty [ \) [35]. First, the mass of a solution can be determined in terms of the Noether charge associated with a timelike Killing vector in the asymptotic region. In this case above \(\partial _{t}\) in Eq. (8) is that Killing vector. In principle a second contribution to the Noether charge coming from action principle of the matters fields should be considered. However, and also in general, that second part of Noether charge usually becomes negligible at the asymptotic region. In the case at hand those additional parts of Noether charge will be assumed that do not contribute to the asymptotic value of the Noether charge.
6.1 Analysis of the first family of solutions
6.1.1 Charges
6.1.2 Temperature
6.1.3 Entropy
To pursue any further into an analytic analysis of the thermodynamics properties, for instance the evaporation of these black holes, would require to define the temperature as an function of the mass and additional charges of the solution, which in turn needs at least to consider a \(\rho (r)\) in particular. Unfortunately, this is not enough in general to obtain close expressions. Because of that in the next sections will be considered a particular case which will be analyzed through numerical methods.
6.2 Analysis of the second family of solutions
6.2.1 Charges
6.2.2 Temperature
6.2.3 Entropy
For large \(r_+\) this entropy always approach an area law. It can be noticed as well that for \(n=1\) the usual area law is recovered. This also occurs for \(\gamma =0\) for any value of n or d. It is direct to check that for \(\gamma =1\) and \(\gamma =0\) these functions are monotonically increasing functions of \(r_+\). Far more interesting is the fact that \(S_{\gamma =1}(r_{+})\) can be negative which imposes severe constrains in the space of parameters. As previously, \(r_+\) cannot be determined analytically and so to proceed a numerical approach would be required. This will be also considered for a next work.
7 Planck energy density
7.1 Structure of solutions
7.1.1 First family of solutions
7.1.2 Second family of solutions
7.2 Horizons

for \(M<M_{cri}\) f(r) has non zeros in the real numbers and thus the space is not a black hole. Moreover, due to the absence of singularities the space has a causal structure similar to a Minkowski space.

for \(M = M_{cri}\) f(r) has a double zero. This represents a regular extreme black hole with degenerate Killing horizon.

for \(M>M_{cri}\) f(r) has two zeros. This represents a regular black hole with both outer and inner horizons.
7.3 Temperature
Figures 3 and 4 display examples of the behavior of the temperatures, respectively. Figure 3a actually corresponds to both families of solutions since \(n=1\).
7.4 Heat capacity
In general, it can be observed a phase transitions at both \(r_*\) and \(r_{**}\). Going from \(C_Q > 0\) for \(r_+ > r_{**}\) to \(C_Q < 0\) for \(r_{*}< r_+ < r_{**}\). Finally \(C_Q > 0\) for \(r_+<r_{*}\). Moreover, one can notice as well that \(C_{Q}\) vanishes as \(T \rightarrow 0\).
8 Conclusions and discussions
In this work two families of regular black hole solutions have been discussed. Although each family is a solution of a different Lovelock gravity in d dimensions both share to have a single ground state, which is approached asymptotically by the solutions and is defined by a single cosmological constant. These theories correspond to the Pure Lovelock theory [28, 30, 32] and to the theory discussed in [27] which have a nfold degenerated ground state.
First it must be noticed that solutions have a minimum value of the parameter M, called \(M_{cri}\) above, to represent a black hole geometry. As expected, these solutions asymptotically, says for large r, are indistinguishable from the previously known solutions [27, 30] in vacuum. On the other hand, both families of solutions near the origin, for \(r \sim 0\), behaves as maximally symmetric spaces which can be fixed, unambiguously, to be of positive curvature, i.e. the parameters can be fixed such as the solutions approach de Sitter spaces, as required to model a regular black hole, at their origins.
Concerning the thermodynamics of the solutions it was shown that, although both families of solutions differ, in general terms their thermodynamics presents the same features. The associate temperatures of the horizons have, in general, a local maximum and a local minimum at \(r_*\) and \(r_{**}\) respectively. Indeed, one can notice that there can be three different values of \(r_+\) defining the same temperature. Because of this last, the heat capacity changes from positive for \(r_+ < r_{*} \) to negative at \(r_*< r_+ < r_{**}\) and finally to positive for \(r_{**} < r_+\), signing out two phase transitions of the system, from thermodynamically stable \((C_Q >0)\) into unstable \((C_Q <0)\) and viceversa. Therefore, in general, there is a range of the black hole radii, \( r_{*}< r_+ < r_{**}\), or equivalently of the masses, where the black holes become thermodynamically unstable. The existence of a phase transition for the vacuum solutions in [27] was known, and thus the phase transition at \(r_{**}\) could have been anticipated. However the existence of a second phase transition at \(r_+ = r_{*}\), and thus the existence a second stable range for \(r_+ < r_{*}\), is a new feature proper of these regular black holes.
In order to analyze the thermodynamics in detail it was considered as model the generalized Hayward energy density [7] defined in Eq,(48). Now, as mentioned in Sect. 2, the conditions to be satisfied by the mass density are quite general, and thus it can be quite interesting to explore other options in order to determine to what extent the arise of phase transitions is a model dependent feature. For instance, it would be quite interesting to study a generalization of the proposal in [5, 6]
Footnotes
Notes
Acknowledgements
This work was partially funded by grants FONDECYT 1151107. R.A. likes to thank DPI20140115 for some financial support.
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