Phase Structure of Quantum Improved Schwarzschild-(Anti)de Sitter Black Holes

The black holes are known to have thermodynamic properties including a characteristic temperature and an intrinsic entropy [1]. The thermodynamics of Schwarzschild black holes encounter unstable phase with negative heat capacity. The phase structure becomes more fascinating when we take into account a negative cosmological constant. For such Schwarzschild-Anti de Sitter (SAdS) black holes, the temperature has a minimal value which corresponds to the divergent point of heat capacity, giving the border line of unstable small black holes with negative heat capacity and stable large black holes with positive heat capacity. In particular, there is a first order Hawking-Page (HP) transition [2] between large (stable) SAdS black holes and thermal Anti de Sitter (AdS) at the critical temperature where both the black hole and thermal AdS have the same free energy. Above this critical temperature, the free energy of a large black hole is lower than the free energy of thermal AdS of equal temperature, causing the decay of thermal AdS into black holes. These thermodynamic aspects of black holes were originally discovered by using quantum field theory on a classical background spacetime, or by applying euclidean quantum gravity approach initiated in [3]. No quantum effects of gravity are considered in these results. It is thus interesting to revisit these issues in black hole thermodynamics, in particular the phase structure, taking into account recent advances in quantum theory of gravity.

Quantum effects of gravity are, in general, expected to play a significant role in the core region near the black hole singularity. This is indeed the case for asymptotically safe quantum gravity [4, 5, 6, 7, 8]. This approach considers the possibility of having an interacting quantum field theory of gravitation, originated from a non-Gaussian ultraviolet (UV) fixed point in the theory space, in the nonperturbative renormalization group (RG) framework. The existence of such a fixed point would permit having an RG trajectory in the theory space flowing to it and characterized by all the dimensionless couplings remaining finite when the UV cutoff is removed. In this way, the UV behavior of the theory becomes under control, and this provides a promising way to quantum gravity. An important result in this case is that the coupling “constants” depend on the energy scale along the RG flow. The essential issue in this approach is a suitable choice of identification of the energy scale with some length scale in the considered solutions. We call this quantum improvement. This allows us to understand how the quantum effects manifest themselves in the black hole geometry. Given a suitable identification, it turns out that quantum effects provide a “repulsive” force at the core of black hole which stabilizes the unstable phase of small black holes [9, 10]. For discussions of black holes in this context, see [11, 12, 13, 14] and references therein.

The main purpose of this paper is to study the thermodynamic phase structure of quantum improved static black holes in the asymptotically safe quantum gravity. As the mass decreases, the temperature of small AdS black holes reduces to zero, instead of rapidly increasing to infinity. Consequently, as we will show, the quantum effects create a local maximal in all asymptotically flat and (A)dS black holes. The small black holes become stable and, therefore there exist finite size zero temperature remnants. We would like to suggest that the Planck size cold remnants could be a candidate for dark matter.¹¹1 The possibility of “cold remnants” was already pointed out in Ref. [10]. In the context of the primordial black holes (PBHs) as dark matter (see, e.g., Refs. [15, 16, 17] and references therein), only those with initial mass larger than $10^{15}$g have so far been considered, since any PBHs with smaller mass $\lesssim 10^{15}$g would have already evaporated away by now. The asymptotic safe quantum improvement provides an interesting mechanism that can make even much smaller PBHs free from Hawking evaporation, and thus create stable black hole remnants. It is interesting to study the astrophysical implications of such stable Planck size black hole remnants. In the asymptotic safety there are a number of papers arguing that the gravitational effects reduce the strength of interactions of fermions and scalars (not black hole remnants), thereby making them candidates of dark matter, see, e.g. Refs. [18, 19]. Apart from that, it also raises another interesting question: Is there a new phase transition, analogous to the HP transition, between stable quantum improved small black holes and thermal AdS? In order to clarify this problem, it is necessary to study the phase structure of quantum improved black holes. We will find rich structure in the case of AdS spacetime. Note that the importance of AdS black holes has been well recognized for some time in the context of AdS/CFT correspondence [20], and also HP-type of phase transition, so it is interesting to see how the results are modified by quantum effects [21].

Here we address these questions and find that for the quantum improved SAdS black holes, when the quantum effects are not significantly large, there is an intermediate unstable phase between stable small and large black holes. Such kind of sandwich structure indicates a possibility of the existence of a second order phase transition from small to large black holes or vice versa. The evidence of such a phase transition is the presence of a swallow-tail pattern in free energy with respect to temperature. The phase structure of a thermodynamical system is encoded in the associated free energy. Unfortunately due to the lack of quantum action, we are not able to compute the free energy via the partition function. However, by assuming the thermodynamic free energy is still valid for quantum improved black holes, we can study the phase structure of quantum improved SAdS black holes. We find that the analogous HP transition can indeed occur only at unphysical negative temperature. Moreover, a new interesting swallow-tail pattern does appear when the strength of quantum effects are smaller than the critical value and the associated second order phase transition can happen.

For the quantum improved asymptotically flat black holes, the heat capacity is independent of their spatial topologies, and thus they all share similar phase structures. When the cosmological constant is non-vanishing, we find that the heat capacity is invariant under sign change of the cosmological constant $\Lambda\to-\Lambda$ and the spatial topologies $\sigma\to-\sigma$. But, both the mass and temperature flip their signs. This means, for example, the phase structure of topological Schwarzschild-de Sitter (SdS) “black holes” is alike those of SAdS black holes. However, for the topological SdS black holes the mass is negative [22]. The solutions for $\sigma\neq 1$ and $\Lambda\geq 0$ correspond to cosmological solutions with positive mass parameter.

This paper is organized as follows. In the next section, we will briefly describe how to quantum improved black hole geometries in the asymptotic safety scenario, and discuss the first law of a quantum improved Kerr-(A)dS black hole as an example. Then, in Sec. III, we will analyze quantum effects on thermodynamic phase structure of asymptotically AdS static black holes with different horizon topologies. We compute the Hawking temperature, heat capacity, and free energy for quantum improved AdS black holes, and for each type of horizon topology, we study in detail whether quantum improved AdS black holes exhibit the HP type phase transition. We find that for the spherical horizon case, a new type of second order phase transition can occur, due to the effects of quantum improvement, whereas for the hyperbolic and the planar horizon case, there are no phase transition analogous to the HP. We perform similar analyses of quantum improved black holes with positive cosmological constant in sec. IV and with vanishing cosmological constant in sec. V. Section VI is devoted to summary and discussions. In Appendix A, we briefly summarize how to compute free energy for SAdS black holes in general relativity.

Let us first summarize some relevant results here in the asymptotically safe scenario for quantum generalization of the general relativity with cosmological constant, in the units $c=\hbar=1,G_{0}=l_{P}^{2}=1/M_{P}^{2}$. The action is given by

The asymptotically safe scenario proposes the Newton coupling $G(k)$ and cosmological “constant” $\Lambda(k)$ are energy dependent [4]. Assuming the cosmological constant is is sufficiently small, the associated renormalization group equations lead to the solutions for Newton coupling [9, 24, 25, 26, 27]

where $\omega>0$ is the inverse of the fixed point value. Actually the “dimensionful” cosmological constant $\Lambda$ is given by [26, 12]

where $\lambda_{*}$ is the fixed point value of the “dimensionless” cosmological constant and $\Lambda_{0}$ is the infrared value of the cosmological constant. It might appear that $\Lambda$ increases for larger $k$ or near the singularity. However we are only concerned with the thermodynamic behaviors of the black holes and the near-horizon area is the region we consider. The typical value of $\lambda*$ is very small like $0.1$ [23]. So our approximation can be justified.

How to identify the energy scale $k$ with a spacetime distance scale is an essential issue for constructing quantum improved solutions [24]. See also related work [28, 29, 30, 31, 32, 33]. The first law of black hole thermodynamics, particularly for Kerr-(A)dS black holes, requires that a consistent identification near the horizon should be a function of the horizon area $A_{h}$ [34]. The same consequence can be derived with the relation of entropy variation $\delta S=\delta A_{h}/4G$ [35] in which it is implicitly presumed from the outset that the running coupling $G$ is a function of the horizon area $A_{h}$. Therefore, a physically admissible identification for energy scale is a function of area with given radius. According to dimensional counting, a naturally suggested scale identification for Kerr-(A)dS black holes at horizon is

with a dimensionless parameter $\xi$. Strictly speaking, we may have $r_{h}$-dependence in the cosmological constant, but we neglect this because we are interested only in the case of small cosmological constant. This identification is the simplest and natural one without introducing additional dimensional parameters. The resulting running Newton coupling is

The black hole entropy can be computed by integrating the first law of thermodynamics which, up to an integrating constant $S_{0}$, gives

The temperature determined by the surface gravity $\kappa$ is

The quantum effects enlarge the size of extremal (zero temperature) state which can still exist in the static black holes with finite mass. The stable extremal black holes could be a candidate of dark matter. For non-rotating black holes $a=0$, the vanishing condition of the temperature (7) gives the extremal limit with

where $\pm$ corresponds to dS ($\Lambda>0$)/AdS ($\Lambda<0$) and $r_{\mathrm{zero}}^{2}=\tilde{\omega}G_{0}$ for $\Lambda=0$.

In this section we are going to analyze the quantum effects on the thermodynamical phase structure of the asymptotically AdS black holes for $a=0$ with different horizon topologies, characterized by the unit sectional curvature $\sigma$ of the horizon manifold, i.e. sphere ($\sigma=1$), flat ($\sigma=0$) and hyperbola ($\sigma=-1$):

(We also refer to the hyperbola ($\sigma=-1$) case as the topological black hole.) We also denote the AdS radius-squared as

The identification (4) and (5) for non-rotating black holes ($a=0$) reduces to

Note that for the static black hole metric (9), the horizon area is given merely in terms of the radial coordinate $r$, different from the Kerr-(A)dS case, and therefore any scale identification of the type $k=\tilde{\xi}/r^{p}$ with $p>0$ examined in [27] (with suitable dimension for $\tilde{\xi}$) is consistent with the black hole first-law discussed in [34]. Then, depending on the power $p$, the quantum improved geometry near the center $r=0$ will differ. In particular, for the identification (5) with $a=0$, for which $k\sim\xi/r$, the metric function behaves $f(r)\sim\sigma-(2M/\tilde{\omega})r+O(r^{2})$ near the center and therefore the resultant quantum improved geometry admits a weak singularity at the center. However, since the temperature and heat capacity are computed at the horizon and also the euclidean path-integral for the free energy is performed outside the horizon as shown in Appendix A, the behavior of the quantum improved geometry (i.e., whether it is regular or not) near the center does not appear to affect much the evaluation of these thermodynamic quantities. For this reason, we will stick to the scale identification (5), which is the simplest and natural choice without any additional dimensional parameters, in the rest of this paper, despite possible more general dependence.

The location of the horizon $f(r_{h})=0$ gives the mass as

We see from (3) and (11) that there is actually some dependence on the horizon radius $r_{h}$ in the cosmological constant, but this could be neglected for small $\lambda_{*}$ unless we consider very small $r_{h}$ region. Moreover the dependence on the horizon radius $r_{h}$ is smaller for larger $r_{h}$ in which we will be interested.

The mass parameter $M$ is always positive for both $\sigma=1$ and $\sigma=0$. However, for $\sigma=-1$ and the “horizon” radius smaller than AdS radius, i.e. $r_{h}<L$, the mass parameter is negative. It is straightforward to compute the Hawking temperature

and the heat capacity

The heat capacity, temperature and mass parameter have an interesting duality under transformation $L^{2}\to-L^{2}$ and $\sigma\to-\sigma$

For studying the black holes thermodynamics, it is convenient to use the dimensionless variable $\rho$

and dimensionless parameter $\zeta$

The divergent points of heat capacity play important role in studying the thermodynamic phase structure. This happens when the denominator of Eq. (14) vanishes. This leads to a cubic equation

and its discriminant is

In the following we will discuss the phase structure for black holes with different spatial topology.

The thermodynamic quantities for de Sitter black holes can be obtained simply from the corresponding results in Anti-de Sitter cases by replacing $L^{2}\to-L^{2}$, namely

Let us discuss the thermodynamic properties for three different $\sigma$ separately.

For this case, the heat capacity reduces to

which does not depend on the sign of $\sigma$. The divergent point of the heat capacity is then determined by the equation:

The only positive root is $r_{h}^{2}=(2+\sqrt{5})\tilde{\omega}G_{0}$.

The Schwarzschild black holes ($\sigma=1$) are always unstable in GR, and Fig. 5 shows that the quantum effects stabilize the small black holes. For $\sigma=-1$, the black holes solutions may exist for negative mass parameter. However, according to the duality, and the Fig. 5 (a), the associated Hawking temperature is negative. For $\sigma=0$, there are no black hole solutions.

This exhausts the quantum gravity effects in all possible cases.

We have studied the quantum effects of gravity on various black holes with and without cosmological constant. Our findings can be summarized as follows.

• •

  For SAdS black holes, there always exists one and only one divergent point in GR (boundary of stable and unstable phases). Quantum improvement, with small quantum-effect parameter $\omega$, increases one more divergent point and creates a new stable phase in the small mass region (consequently generates a new zero temperature state with finite size). If $\omega$ increases, the “distance” (in mass coordinate) between two divergent points reduces, and when $\omega$ is larger than the critical value, both divergent points and unstable phase disappear, see Fig. 1 (b). If $\omega$ is not too big, the black holes can have a second order phase transition.

• •

  For the topological AdS black holes, the mass parameter can be negative (black holes with negative parameter had been discussed in [22, 39]). The system is thermodynamically stable if the quantum effects are small. Otherwise, the small black holes become unstable when the quantum effects are big enough.

• •

  For the planar AdS black holes ($\sigma=0$), since the black holes is always stable in GR, the “repulsive” effects does not give an obvious change.

• •

  For the SdS black holes, there are no divergent point in GR (only unstable phase exists). The quantum improvement again creates a new stable phase in small mass region which then also generates a divergent point. If the quantum improvement is big enough, there are no black hole solution with positive Hawking temperature, see Fig. 4 (b).

• •

  There is a duality in black hole thermodynamics. The heat capacity is invariant under the transformation $L^{2}\to-L^{2}$ and $\sigma\to-\sigma$, but the mass and temperature change sign. The topological SdS black holes share various properties with SAdS black holes.

As listed above, we found that quantum improved Schwarzschild black holes with negative cosmological constant have a richer phase structure than their classical counter-parts, as well as those with positive and vanishing cosmological constant. For this, we have however assumed that the thermodynamic free energy is valid for the black holes quantum improved at the solution level. It would be interesting to investigate whether one can also compute the free energy by considering quantum improvement of the reduced action for euclidean black holes. In this respect, once such an action for the quantum improvement is formulated, it would also be interesting to study the phase structure of quantum improved AdS black holes from the perspective of the AdS/CFT correspondence.

In the cosmological context, we are more concerned with quantum improved Schwarzschild black holes with positive or vanishing cosmological constant. In general, primordial black holes of mass $\sim 10^{14}$g produced in the early universe are expected to be undergoing now their final stages of the Hawking evaporation. However, once the quantum effects are turned on, the evaporating process would stop at some point where their horizon size is still finite, but the Hawking temperature is vanishing as can be seen in Fig. 4 and 5. Thus, such primordial black holes eventually settle down to thermodynamically stable, zero-temperature remnants, rather than being completely evaporated away. Although they may typically be Planck-size objects, such remnants might possibly be a candidate of dark matter. It is worth pursuing such an intriguing possibility.

In this paper, we have chosen the simplest identification (4) to study the phase structure of the black holes. In this case, we find that the singularity at the center of the black holes may not be resolved. Unfortunately, in the Kerr black holes, the consistency of the thermodynamics requires that the scale $k$ should be a function of the horizon area, and naive extension of this away from the horizon does not allow such singularity resolution [34]. However, it is also possible to consider the possibility to include the mass parameter of the black hole in the identification. Indeed, with such dependence, it is possible that the singularity may be resolved [9]. It would be interesting to investigate what restriction emerges when we allow the mass parameter in the identification, and whether we may get the possibility that the singularity may be resolved.