Are regular black holes from pure gravity classified within the same thermodynamical topology?

Black hole thermodynamics is an advanced field of study that investigates the intricate relationship between thermodynamic principles and black hole characteristics. This research interests emerged from the groundbreaking recognition that black holes exhibit entropy and temperature, parameters conventionally linked to thermodynamic systems Hawking:1975ParticleCreation ; Hawking:1976BHTherm ; Bardeen:1976FourLaws ; Gibbons:1976ActionIntegral . Despite significant advancements in exploring black holes, it seems that we still need further means to study them more thoroughly and clearly. To enhance our understanding and characterization of different black hole systems, the concept of thermodynamical topology has been effectively introduced Wei:2022dzw . This method views black hole solutions as topological thermodynamic defects, analyzing their topological numbers based on the asymptotic behavior of the generalized free energy, and classifies nearly all black hole systems into three distinct categories: +1, 0, and -1. This groundbreaking work has offered new insights into the fundamental nature of black holes and gravity. The topological approach introduced in Ref. Wei:2022dzw has gained significant popularity due to its wide applicability and efficiency of study. So far, this method has been applied to studies of rotating AdS black holes Wu:2023sue ; Wu:2022whe and Lovelock AdS black holes in AdS spacetime Bai:2022klw ; Liu:2022aqt . In addition, many other black holes have been studied, such as quantum BTZ black holes, Born-Infeld black holes, dyonic AdS black holes, and many more Wu:2024txe ; Chen:2023elp ; Chen:2024sow ; Fang:2022rsb ; Chatzifotis:2023ioc ; Wei:2023bgp ; Du:2023wwg ; Chen:2023ddv ; Rizwan:2023ivp ; Hazarika:2024dex .

The singularity problem of black holes, as a general prediction of general relativity, signifies the limitations of this theory HawkingLargeScale ; Senovilla:1998oua ; Penrose:1964wq . In order to address this issue, many fruitful approaches have emerged. Black holes that lack essential singularities but possess only coordinate singularities are termed regular black holes Lan:2023cvz . The study of regular black holes can be traced back to the pioneering work of Sakharov and Gliner, who proposed that essential singularities could be circumvented by substituting the vacuum with a vacuum-like medium characterized by a de Sitter metric Sakharov:1966aja ; Gliner:1966 .

The early model of regular black hole was developed by Bardeen by making the mass of the Schwarzschild black hole as a position-dependent function Bardeen:1968 , which is now referred to as the Bardeen black hole. Ayon-Beato and Garcia provided the first explanation of the Bardeen black hole Ayon-Beato:2000mjt that they are described by the Lagrangian with nonlinear electrodynamics source. The solution to the Einstein equations, coupled with the energy-momentum tensor associated with the magnetic field strength, corresponds to a self-gravitating magnetic monopole charge Ayon-Beato:2000mjt . Later, many other regular black hole solutions were obtained by making use of the nonlinear electrodynamics Ayon-Beato:1998hmi ; Bronnikov:2000vy ; Bronnikov:2000yz ; Ayon-Beato:2004ywd ; Dymnikova:2004zc ; Berej:2006cc ; Balart:2014jia ; Fan:2016rih ; Bronnikov:2017sgg ; Junior:2023ixh .

Recently, Bueno, Cano, and Hennigar introduced a novel method that presents the first instances of regular black hole solutions within purely gravitational theories, requiring no specific fine-tuning or constraints among the relevant parameters Bueno:2024dgm . Their work unveiled innovative families of regular black holes in spacetime dimensions of larger than five. These black holes introduce modifications to the conventional Schwarzschild black hole and belong to the class of quasi-topological gravities, a well-established category of metric theories of gravity with higher-curvature terms. Notably, these modifications are advantageous for investigating black hole solutions as their equations of motion simplify to second order when applied to static and spherically symmetric metrics Bueno:2024dgm ; Oliva:2010eb ; Myers:2010ru ; Dehghani:2011vu ; Ahmed:2017jod ; Cisterna:2017umf . Thus far, Ref. Bueno:2024dgm has inspired a series of remarkable studies utilizing its proposed approach Konoplya:2024hfg ; DiFilippo:2024mwm ; Konoplya:2024kih ; Ma:2024olw ; Bueno:2024qhh ; Zhang:2024ljd ; Estrada:2024moz ; Bueno:2024eig ; Bueno:2024zsx .

We carefully study the thermodynamics of regular black holes constructed using the method of Bueno, Cano, and Hennigar Bueno:2024dgm and found that they exhibit certain universal properties. We conjecture that regular black holes constructed by this method share some common thermodynamic features, classifying them as the same type of black hole. Is there a simple and intuitive way to illustrate this? Fortunately, through the analysis of the general case, we conclude that these black holes can reach zero temperature. By focusing on cases where the Hawking temperature has only one single zero point, our analysis in asymptotic limits shows that thermodynamic topology not only effectively describes their thermodynamic properties but also confirms, from a topological perspective, that they belong to the same category. Moreover, this category is actually $W^{0+}$ topological classification given in Ref. Wei24 , through which one can find that the the innermost and outermost black hole states are local thermodynamical stable and unstable. In particular, at low temperature limit, there could be the unstable large and stable small black holes appeared in pair. These result provide a clearer and more comprehensive understanding of the thermodynamic properties of these regular black holes, serving as the motivation for this paper.

The paper is organized as follows. In Sec. II, we review the construction of regular black holes using pure gravity methods. In Sec. III, we analyze their thermodynamical topology and draw our main universal conclusions that these regular black holes belong to one topological class. Two examples to validate these results, including specific details about the thermodynamical topology are given in Sec. IV. Finally, we summarize and discuss our results. The appendix provide some mathematical details.

Quasi-topological gravity is especially effective for analyzing black hole solutions, and they exist across all curvature orders in dimensions $D\geq 5$ Oliva:2010eb ; Myers:2010ru ; Dehghani:2011vu ; Ahmed:2017jod ; Cisterna:2017umf . This family of theories is extensive enough to serve as a foundational set of polynomial densities for expanding general relativity within the framework of effective field theory. As a result, they offer a robust framework for exploring the influence of higher-curvature corrections near black hole singularities. In this section, we would like to give a brief review for such quasi-topological gravity.

In quasi-topological gravity theory, the action is

where $\alpha_{n}$ are arbitrary coupling constants with dimensions of $\mathrm{length}^{2(n-1)}$, and $\mathcal{Z}_{n}$ are the quasi-topological densities Oliva:2010eb ; Myers:2010ru ; Dehghani:2011vu ; Ahmed:2017jod ; Cisterna:2017umf . Here $n_{\mathrm{max}}$ is an integer greater than or equal to 2, and under certain conditions, it can approach positive infinity. Then, let us consider the line element for the $D$-dimensional ($D\geq 5$) static and spherically symmetric black hole

To simplify the procedure, one can introduce Bueno:2024dgm

and series function

Then, the equations of motion derived from the action (1) can be expressed as

and its solution reads Bueno:2024dgm

where $\mu$ is an integration constant which is proportional to the ADM mass of the solution. Note that if $\alpha_{2},\alpha_{3},...,\alpha_{n}$ are given, then the metric function $f(r)$ can be obtained by solving Eq. (6).

Expanding this solution around $r=0$, one has

If $n_{\mathrm{max}}$ is finite, a curvature singularity occurs at $r=0$. However, as $n_{\mathrm{max}}\rightarrow\infty$, the metric near $r=0$ approaches that of a de Sitter universe with a cosmological length scale determined by $m/\alpha_{n_{\mathrm{max}}}$, which resolves the singularity. To ensure that the black holes with regular interiors exist as solutions to the full theory, the following conditions are sufficient,

Here, the condition that $\alpha_{n}$ is positive ensures the monotonicity of $h(\psi)$ for $\psi>0$, while the limit $\lim_{n\to\infty}(\alpha_{n})^{\frac{1}{n}}=C>0$ determines the radius of convergence of the series (4) as $\psi_{0}=1/C$. At $\psi=\psi_{0}$, the function $h(\psi)$ diverges. These conditions imply that $h$ is a smooth, bijective mapping from $(0,1/C)$ to $(0,\infty)$, ensuring the existence of an inverse function for $h(\psi)$. To clearly illustrate the divergence of $h(\psi)$, we show its behavior under various conditions in Fig. 1. Our calculations reveal that the divergence radius for the four cases is $\psi=1/C=1/\alpha$. These properties are crucial for the following analysis and will significantly influence the thermodynamic characteristics of the black holes under consideration, which will be discussed in detail subsequently.

Topological methods have emerged as a powerful tool for studying black hole thermodynamics, offering direct insights into their fundamental properties and enabling classification based on the topological invariants Wei:2022dzw . This approach provides a novel perspective on black hole physics, allowing us to understand their behavior from a topological viewpoint. Quasitopological gravity, an important class of gravitational theories discussed earlier, plays a key role in constructing various regular black hole solutions. Our goal is to further classify these solutions by using the framework of thermodynamic topology.

For black hole, the geometry near its event horizon is of critical importance. To analyze the horizon condition, one needs to examine $f(r)=0$. From Eq. (3), we find that

Here, $r_{+}$ denotes the largest root of the equation $f(r)=0$, which corresponds to the radius of the event horizon. The term $\psi_{+}$ indicates the value of $\psi(r)$ at $r=r_{+}$. Simultaneously, Eq. (6) gives

Since $h$ is closely related to the black hole metric, analyzing the behavior of $h$ is crucial for further studying the thermodynamic properties of black holes. Firstly, as discussed earlier, $h$ is a smooth, monotonically increasing function within its domain $(0,1/C)$. Secondly, when $\psi_{+}\rightarrow 0$, we have $r_{+}\rightarrow\infty$ and $h(\psi_{+})\rightarrow 0$; when $\psi_{+}\rightarrow 1/C$, $r_{+}\rightarrow\sqrt{C}$ and $h(\psi_{+})\rightarrow\infty$, as listed in Table 1.

Thus, $r_{+}$ should belong to the range $(\sqrt{C},\infty)$. The value of the lower bound $\sqrt{C}$ arises from the analytic domain of function $h$ itself, as shown in Fig. 1. Another point to note is that $\sqrt{C}$ obtained here is not necessarily the lower bound of $r_{+}$, which is because that Eq. (9) only indicates that $f(r_{+})=0$, while not implies $r_{+}$ is the largest root.

Drawing upon the conclusions from Ref. Bueno:2024dgm , one derives the thermodynamic quantities of the regular black holes as follows:

where $M$, $T$, $S$ are the ADM mass, Hawking temperature, and Wald entropy Wald:1993nt , respectively. The volume of the $D$-dimensional unit sphere reads

Here, we continue to analyze the general expression of the Hawking temperature. Substituting Eqs. (4) and (9) into Eq. (12), one will have

Here, we note that as $r_{+}\to\infty$ and $\psi_{+}\to 0$, the two series $\sum_{n=2}^{n_{\max}}\alpha_{n}\psi_{+}^{n-1}$ and $\sum_{n=2}^{n_{\max}}n\alpha_{n}\psi_{+}^{n-1}$ in Eq. (15) tend to zero. Consequently, we obtain

Clearly, since $D\geq 5$, the temperature $T$ will approach $0^{+}$ for large horizon radius. Conversely, as $r_{+}\rightarrow\sqrt{C}$, $\psi_{+}$ reaches its radius of convergence. Based on the analysis results in Appendix A, we conclude that when $r_{+}$ is sufficiently close to $\sqrt{C}$, the temperature $T$ can become negative. Thus, on the interval $(\sqrt{C},\infty)$, $T(r_{+})$ will have at least one zero and must have an odd number of zeros if the number of zero points is finite. Considering the complexity of multiple horizon scenarios, we mainly focus on the simplest case where $T(r_{+})$ has a single zero in the following discussion. It is worth noting that when $T(r_{+})<0$, $r_{+}$ will represent the inner horizon rather than the outer horizon, losing its original interpretation. To prevent this, $r_{+}$ gains a new lower bound $r_{+\mathrm{min}}$, given by $T(r_{+})=0$. This new lower bound is greater than the previously mentioned lower bound $\sqrt{C}$, establishing $r_{+\mathrm{min}}$ as the true lower bound of $r_{+}$.

In the previous section, we consider the thermodynamical topology for the general case and find that the total topological number vanishes. In this section, we will take two specific black hole solutions as specific examples to verify our above results.

In this paper, we analyzed the thermodynamical topology of regular black holes arising from pure gravity from a general perspective. We found that regular black holes constructed using the method from Ref. Bueno:2024dgm exhibit certain similar thermodynamic properties. We also observed that these black holes, with at most two horizons, have similar topological properties and the same topological number, $W=0$, as revealed by the asymptotic behavior of the constructed vector. In particular, they belong to $W^{0+}$ topological classification given in Ref. Wei24 , through which one can find that the the innermost and outermost black hole states are local thermodynamical stable and unstable. Given the general nature of our discussion, this result suggests that the thermodynamic topological property is intrinsically linked to the construction method of these regular black holes.

Specifically, we consider two representative examples of regular black hole solutions: one characterized by $\alpha_{n}=n\alpha^{n-1}$ and the Dymnikova black holes. We demonstrated the zero points of the free energy by constructing a vector field $\phi=\begin{pmatrix}\frac{\partial F}{\partial r_{+}},-\cot\Theta\csc\Theta\end{pmatrix}$. Through detailed calculations, we observed that for small $\tau$, there are no black hole states, leading to a vanishing topological number. However, for large $\tau$, two zero points of the constructed vector field emerge, corresponding to the regular black hole states. We then enclosed these zeros with a closed loop and counted the changes in the vector’s direction to determine the winding number associated with each zero point. Consequently, we obtained the overall global winding number, which yields a topological number of $W=0$. This result further supports our conclusion that the regular black holes (with at most two horizons in pure gravity) belong to the same topological class $W^{0+}$. Moreover, the small and large black hole states are stable and unstable indicated from their winding numbers, which strongly supports the thermodynamical properties of $W^{0+}$. Although we only provide a few specific examples, this does not affect the generality of our conclusion, as the choice of specific parameters does not alter the results of our previous discussion on the asymptotic behavior of $\tau$ and $\frac{\partial F}{\partial r_{+}}$.

On the other hand, our discussion has certain limitations. We only considered the case where the Hawking temperature has a single zero point and did not address more complex scenarios involving multiple horizons. If $\alpha_{n}$ is chosen as an appropriate function, such cases may arise, requiring more detailed study. Nevertheless, for the regular black hole solutions constructed by this method that have been identified so far, our findings remain applicable. Furthermore, the reasons behind the topological consistency of these black holes, and whether there are deeper underlying factors in their construction methods, merit further reflection and discussion.