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Regular black holes with sub-Planckian curvature

We construct a sort of regular black holes with a sub-Planckian Kretschmann scalar curvature. The metric of this sort of regular black holes is characterized by an exponentially suppressing gravity potential as well as an asymptotically Minkowski core. In particular, with different choices of the potential form, they can reproduce the metric of Bardeen/Hayward/Frolov black hole at large scales. The heuristical derivation of this sort of black holes is performed based on the generalized uncertainty principle over curved spacetime which includes the effects of tidal force on any object with finite size which is bounded below by the minimal length.


I. INTRODUCTION
It is widely believed that quantum gravitational effects would remove the curvature singularity of a black hole. Before a complete theory of quantum gravity could be established, people have extensively investigated various non-singular black holes at the phenomenological level, which usually are not solutions to the vacuum Einstein equations. Instead some exotic matter fields must be introduced which in general violate some energy conditions in general relativity.
Regular black holes are characterized by a finite Kretschmann scalar curvature [1][2][3][4][5][6][7][8][9][10][11][12]. In history, most of regular black holes proposed are spherically symmetric spacetime with an asymptotically de Sitter core, which includes the well known Bardeen black hole, Hayward black hole as well as Frolov black hole [1][2][3][4]. In Ref. [5], the authors analyzed the general conditions that would lead to a finite Kretschmann scalar and proposed an original black hole solution with an asymptotically Minkowski core. This kind of regular black hole is featured by an exponentially suppressing potential and a vanishing Hawking temperature at the final stage of evaporation, in contrast to the phenomenon that the Hawking temperature becomes divergent for classical black holes, as well to the phenomenon that the Hawking temperature takes a maximal value staying at the Planck energy level for semi-classical black holes where the quantum effects of gravity such as the generalized uncertainty principle (GUP) [13][14][15][16] or modified dispersion relations (MDR) are taken into account [17][18][19][20]. Thus, this regular black hole solution provides more realistic picture for the final stage of black hole evaporation and more reasonable behavior of the black hole remnant. Subsequently this solution has been generalized into a kind of regular black holes with different forms of the exponential potential in [21][22][23][24][25][26]. However, all this sort of regular black holes with Minkowski core suffers from a shortcoming. Although Kretschmann scalar curvature is finite everywhere, its maximal value depends on the mass of black hole. For instance, once the parameter α is given in [5], which is supposed to be fixed by the quantum effects of gravity, the maximal value is proportional to the square of the mass. It means that Kretschmann scalar curvature is not bounded above, but can exceed the Planck mass density (M 4 p ) easily by increasing the mass of the black hole, which of course is not reasonable or expectable from quantum gravity point of view. Or in another word, it implies that the metric for all previous regular black holes with Minkowski core makes sense only for black holes with small mass at the Planck scale, or for the final stage of black hole evaporation. Remind that the regular Bardeen/Hayward/Frolov black holes do not suffer from this weak point because their metric near the center are asymptotically de-Sitter, its maximal value of Kretschmann scalar curvature is always mass independent.
In addition, the regular black holes with asymptotically Minkowski core is constructed by introducing the exponentially suppressing potential, while the regular black holes with asymptotically de-Sitter core is constructed by introducing an effective cosmological constant near the center. At present it seems they have nothing to do with each other, except that at infinity both kinds of black holes approach the Schwarzschild metric.
The purpose of this paper is twofold. Firstly, we intend to construct a new sort of regular black holes with Minkowski core, getting rid of the shortcoming that the Kretschmann scalar curvature is not bounded by the Planck mass density, such that the metric is applicable to the black hole with arbitrarily large mass, and thus applicable to all the stage of black hole evaporation from the beginning to the end. Secondly, we intend to disclose a closer relation between regular black holes with Minkowski core and those with dS core. Specifically, we will demonstrate that with different choices of the potential form, this sort of black holes can reproduce the metric of Bardeen/Hayward/Frolov black hole at large scales. In this situation, these two kinds of black holes may have distinct behavior only at the late stage of evaporation. This paper is organized as follows. In next section we will present the general setup for this sort of regular black holes with spherical symmetry, and then in section three we find the condition that Kretschmann scalar curvature can be bounded above and then show a specific example in comparison with the solution proposed in [5]. In section four and five, we will demonstrate that for specifical choice of the potential from, the regular black holes have the similar behavior as Bardeen/Hayward black holes at large scales, respectively. Finally, we will argue how the consideration of generalized uncertainty principle over curved spacetime could lead to this sort of black holes in a heuristical manner. We point out that the effect of tidal force on any object with finite size bounded by the minimal length plays an essential role in this modification.

SPHERICAL SYMMETRY
We consider a static spherically symmetric black hole with a general form of the metric with F (r) = 1 + 2ψ(r).
Typically, if ψ(r) = −M/r, then it is nothing but the Schwarzschild black hole. Throughout this paper, for simplicity we will set G = 1, which implies if one intends to recover the correct dimension of any physical quantity, the unit with Planck length l p or Planck mass m p should be inserted appropriately. Now, in order to include the strong quantum effects of gravity which would modify the singularity behavior at the Planck scale, we assume F (r) would be a general form, with the requirement that it goes back to Schwarzschild black hole at asymptotical infinity. The location of the horizon r h is determined by g rr = F (r h ) = 0.
From Einstein field equations G µ ν = 8πT µ ν , one can derive the effective stress-energy tensor which is given by T µ ν = 1 4πr diag{ ψ r + ψ , ψ r + ψ , ψ + r 2 ψ , ψ + r 2 ψ }. Furthermore, it is straightforward to derive Kretschmann scalar curvature which is given by The Hawking temperature of the black hole and the luminosity are respectively given by Now we propose a sort of regular black hole with asymptotically Minkowski core may be constructed with an exponentially suppressing form of the gravitational potential, where ψ(r) is specified as with α 0 > 0, n > x ≥ 0 and n ≥ 1 1 .
1 In addition, all these parameters are understood as dimensionless, thus to keep the exponential factor is always dimensionless, appropriate dimensions with the power of Planck length l p is assumed.
In comparison with all previous regular black holes with Minkowski core, the key generalization in our formalism is that the exponential form may depend on the mass of black hole, which plays an essential role in suppressing the maximal value of Kretschmann scalar curvature to be sub-Planckian, and we will explicitly demonstrate this feature in next sections. Obviously, if x = 0, then it goes back to the regular black hole proposed by Li et.al. [5] and others in [21][22][23][24][25][26].

SCALAR CURVATURE
In this section we will construct a regular black hole with sub-Planckian Kretschmann scalar curvature by setting x = 1 and n = 2.
First of all, it is helpful to understand why all the previous regular black holes with x = 0 lead to a huge Kretschmann scalar curvature whenever the mass of black hole becomes large. For instance, for the regular black hole proposed by Li et.al. [5], the gravity potential is ψ = − M r e −α 0 /r 2 , and Kretschmann scalar curvature is given by which is finite everywhere. In particular, near the center r → 0, K → 0, which is in contrast to the standard Schwarzschild black hole in which K becomes divergent. However, one notices that once α 0 is fixed as it should be from the quantum gravity point of view, the maximum value of K is proportional to M 2 , which means this value may easily exceed the Planck mass density for large black holes, namely K > 1 with the unit of m 4 p . We show this behavior in the right plot of Fig.1. Of course this feature is not satisfactory if one insists that all the reasonable quantities should be sub-Planckian due to the quantum gravity effects.
On the other hand, if we assume that the exponential factor could be mass dependent, then such situation would change dramatically. As a matter of fact, if we set n = 2 and replace α 0 by α 0 M x , then one obtains K with the similar expression as Eq.(6), simply replacing α 0 by α 0 M x . A simple algebra shows that now the maximal value of K is proportional to M 2−3x /α 3 0 . Therefore, if we demand x ≥ 2/3, then the maximal value would inversely be proportional to the mass. It is this observation that leads us to propose such generalized model. Without loss of generality, we will set x = 1 in this section and compare this regular black holes with the previous one [5], which is obtained by setting x = 0.
Firstly, we present the basic properties of the regular black hole when we take x = 1.
From F (r h ) = 0, one obtains the relation between the location of outer horizon r h and the mass M as Then the radius of the horizon r h can be expressed as a function of mass where is the Lambert-W function with W (θ) ≥ −1 [27]. The above equation is quite similar to the one for x = 0, but now the variable θ = − α 0 2M , rather than θ = − α 0 2M 2 in [5]. Moreover, as pointed out in [5], a real W requires θ ≥ −e −1 such that we have the lowest bound for the mass of the black hole When this bound is saturated, the black hole is characterized by the minimal radius of the horizon r h = √ eα 0 , which may be treated as the remnant of the black hole evaporation, as we will elaborate it as below.
Now we compare the distinct behavior of Kretschmann scalar curvature for these two regular black holes, namely x = 0 and x = 1. We plot K(r) as the function of the radius for different masses M in Fig.1. As we expect, the maximal value of Kretschmann scalar curvature K max increases dramatically with the increase of the mass for x = 0, while for x = 1, it decreases with M indeed. We further find that the relation K max ∝ M 2 for x = 0, and K max ∝ 1/M for x = 1 can be justified by numerical analysis. Therefore, for x = 1 we may fix the parameter α 0 to have K max < 1 for M = M min , then it is guaranteed that Kretschmann scalar curvature is always bounded from above for arbitrary mass M , as we illustrate in the right plot of Fig.1.
Next it is also interesting to take a look at the thermodynamical behavior of these two regular black holes. In parallel with the analysis presented in [5], it is straightforward to derive the Hawking temperature T , the heat capacity C ≡ dM dT as well as the luminosity L as the function of the mass M The temperature as well as the luminosity has the same expression for cases x = 0 and x = 1, but we stress that since the variable θ has a different relation with the mass, they exhibit distinct behavior with the mass M , as illustrated in Fig.2. For x = 0, the temperature goes to zero at the final stage W → −1 (M = eα 0 2 or r h = √ 2α 0 ), and the temperature always while for x = 1, the temperature goes to zero at the final stage W → −1 as well (M = eα 0 2 or r h = √ eα 0 ), but reaches its maximal value at W = (−5+ √ 17)/2. We also find T max ∝ 1/α 0 and M | T =Tmax ∝ α 0 , implying that during the evaporation, the temperature may reach its maximal value at larger scale and then go down to zero in a longer time period if α 0 > 1.
Finally, we remark that for both black holes, the capacity is vanishing at the final stage of evaporation, which is obvious to see in Eq.(11) as W → −1. In addition, the capacity of both black holes undergoes a transition form C < 0 to C > 0 at T = T max .

HOLE AT LARGE SCALE
In this section we present the regular black hole corresponding to Bardeen black hole at large scales by appropriately choosing the parameter x with n = 2. Recall that for Bardeen black hole, the maximum of Kretschmann scalar curvature K is independent of the mass of black hole. Remarkably, as we analyzed in the previous section, if we set x = 2/3, then the maximum value of K is independent of mass M as well. By virtue of this observation, we intend to consider the regular black hole with x = 2/3 and n = 2. In this case, the radius of the horizon r h is given by The low bound for the mass of the black hole is The Hawking temperature T and the luminosity L maintain the same form as shown in Eq.(11), while the heat capacity C becomes Next we show that this regular black hole reproduces the Bardeen metric at large scales.
On one hand, for large r √ α 0 M 1/3 , the function F (r) in the metric behaves On the other hand, for the Bardeen regular black hole, the gravitational potential ψ(r) is specified as Therefore, at large scales the function F (r) behaves which is identical to the regular black hole with x = 2/3 at large scale, indeed.
Nevertheless, these two black holes have different cores. One has asymptotically Minkowski core, while the other has asymptotically de-Sitter core. Thus it is interesting to compare the behavior of Kretschmann scalar curvature as well as thermodynamical properties for these two regular black holes. For compactness, we intend to summarize their key differences as the following list • The features of Kretschmann scalar curvature K. For Bardeen black hole with the above potential, Kretschmann scalar curvature is given by We plot Kretschmann scalar curvature K as the function of the radius for these regular black holes in Fig.3. Obviously, for both black holes the maximum value of K is independent of mass M . The location of K max is always fixed at the center for Bardeen black hole, independent of the parameter α 0 , while for regular black hole with x = 2/3, K is always zero at the center, and the location of K max moves to larger radius with the increase of the parameter α 0 .
• The features of thermodynamics. The Hawking temperature T of Bardeen black hole is given by Moreover, to form a black hole with horizon, the mass of the black hole is bounded from below When this bound is saturated, the black hole is characterized by the minimal radius We plot the Hawking temperature T as the functions of M for two black holes in Fig.4, respectively. Both black holes are characterized by a vanishing temperature at the final stage of evaporation. However, for Bardeen black holes, we remark that there is no maximal value for the Hawking temperature and its capacity exhibits a monotonic behavior with the mass with C = dM/dT > 0.

HOLE AT LARGE SCALE
In a quite parallel way, we construct a regular black hole corresponding to Hayward black hole at large scales. With the general form of gravity potential as Eq. (5), we find the maximal value of Kretschmann scalar curvature K behaves as K ∝ M 2 (α 0 M x ) 6/n . Thus if we set n = 3, then x = 1 leads to a mass-independent curvature as well. In this section we demonstrate that this regular black hole has the same behavior as Hayward black hole at large scales indeed.
We start with the gravitational potential ψ(r) as Then Kretschmann scalar curvature is given by If we set α = α 0 M , then it goes to the curvature with x = 1. The maximal value of K is proportional to 1/α 2 0 . The radius of the horizon r h is given by The mass of the black hole is bounded as and thus the minimal radius of the horizon is r h = 3 2 α 0 e 1/3 . The Hawking temperature T is given by Similar to the analysis in previous section, we show that this regular black hole reproduces the Hayward metric at large scales. On one hand, for large r (α 0 M ) 1/3 , the function F (r) in the metric behaves as On the other hand, for Hayward regular black hole, the gravitational potential ψ(r) is specified as Therefore, at large scales the function F (r) behaves which is identical to the regular black hole with x = 1 and n = 3 at large scale, indeed.
As well, we compare the features of these two black holes as follows.
• The features of Kretschmann scalar curvature K.

For Hayward black hole,
Kretschmann scalar curvature is given by We plot Kretschmann scalar curvature K as the function of the radius for these regular black holes in Fig.5. Again, for both black holes the maximum value of K is independent of mass M . The location of K max is always fixed at the center for Hayward black hole, independent of the parameter α 0 , while for regular black hole with x = 1, K is always zero at the center, and the location of K max moves to larger radius with the increase of the parameter α 0 .
• The features of thermodynamics. The mass of Hayward black hole is bounded by The minimal radius of the horizon is r h = 3α 0 2 . The Hawking temperature T is given by We plot the Hawking temperature T as the functions of M for two black holes in Fig.6, respectively. Both black holes are characterized by a vanishing temperature at the final stage of evaporation. It is also interesting to notice that the mass dependent behavior of the temperature is quite similar for these two black holes, even at the final stage of the evaporation.

VI. THE REGULAR BLACK HOLE WITH GENERAL EXPONENTIAL FORM
In this section, we consider the generalized form of the black hole with the gravitational potential as specified in Eq. (5). In Tab.I, we summarize the dependent behavior of different quantities on α 0 with factors x and n.
The special case happens whenever x = n. It is obvious to see that the mass of the black hole is not bounded by α 0 any more, and the temperature becomes divergent when mass goes to zero, just like the Schwarzschild black hole. Thus, to form a regular black hole with remnant, we require that x < n. Moreover, if we expect the maximal value of Kretschmann scalar curvature K is bounded above with the increase of the mass M , we find x ≥ n/3. In particular, when x = n/3, K max is mass independent. Thus, we conclude that to form a regular black hole with sub-Planckian Kretschmann scalar curvature, we demand On the other hand, given the regular black hole with the gravitational potential as specified in Eq.(5), we may construct the corresponding regular black hole with asymptotically dS core which has the same behavior at large scales. The function F (r) in the metric reads as F (r) = 1 + 2ψ(r) = 1 − 2M r n x −1 which can be viewed as the generalization of Bardeen and Haywald black holes. In addition, we may give a remark about Frolov black hole and its correspondence. For Frolov black hole, the gravitational potential reads as At large scale, the black hole is identical to the regular black hole with potential ψ(r) = Finally, we briefly discuss the effective stress-energy tensor and the energy condition. The effective stress-energy tensor corresponding to Eq.(5) is given by The strong energy condition (SEC) is violated when −T 0 0 + T 1 1 + T 2 2 + T 3 3 < 0, namely r < α 0 nM x n+1 1 n . We notice that only for x = 0, the region with the violation of SEC is mass independent, while for other allowed values of x, the region becomes larger with the increase of the mass.

VII. HEURISTICAL UNDERSTANDING ON MODIFIED GRAVITY POTEN-TIAL
In this section we will present a heuristical understanding on the origin of such exponentially suppressing potential, closely following the strategy proposed in [5,28]. As pointed out in [28], when the quantum effects of gravity are taken into account, the same observed results such as COW phase shift in gedanken experiment [29] can be understood by two equivalent pictures. One is that the classical gravitational field strength remains unchanged by quantum object, but the usual Heisenberg's uncertainty principle obeyed by the object is forced to be a generalized one where z(p) could be a general function of the momentum operator which reflects the effects of gravity on this quantum object. Typically, the widely considered one in literature is z(p) = 1 + αp 2 , which leads to a quantum theory with the minimal observable length.
Alternatively, one can assume that the quantum objects still obey the usual Heisenberg's uncertainty relation, namely the quantum theory is retained, but introduce an effective gravitational field strength to count in the interacting effects of gravity and the quantum object. It turns out that in latter point of view, the space time with Schwarzschild metric will be modified to be one characterized by an effective Newton constant G = G/z. Namely, we obtain a modified Schwarzschild metric as [5,28] which becomes the starting point that we propose the modified metric for regular black holes in this paper. However, rather than considering the ordinary GUP associated with z(p) = 1 + αp 2 , we introduce a general exponential function with z(p) = e αp a , which has also previously considered in [30]. Obviously, when a = 2 and αp 2 1, this goes back to z(p) = 1 + αp 2 + .... In this sense, such an exponential form includes the non-perturbative effects of quantum gravity because in the expansion of weak momentum it recovers the quadratic form of the momentum.
Next, we need to evaluate the momentum uncertainty of any quantum object with m in the gravitational field. We know, for an observable quantity, p 2 ≥ p 2 . The key point is that whenever the quantum effects of gravity are taken into account, the quantum probe must be characterized by a position uncertainty bounded below by the Planck length, namely x ≥ 1. Therefore, it must experience the gravitational tidal force in curved spacetime, which leads to where E and t are the characteristic energy of the probe like a photon and the time in the process of detection, which is supposed to have a photon-particle collision. To avoid new particle pair production, E ≤ m is assumed. Thus, we have In particular, if we set α = α 0 /2 and a = 2, then the corresponding metric is nothing but the regular black hole corresponding to Hayward black hole. Or, if we set α = 2 3/2 α 0 and a = 4/3, then it is exactly the regular black hole corresponding to Bardeen black hole. Or, in general if we set α ∝ α 0 M x−n/3 and a = 2n/3, then leading to the potential given in Eq.(5).

VIII. CONCLUSION AND DISCUSSION
In this paper we have introduced an exponentially suppressing gravitational potential to construct a sort of regular black holes with asymptotically Minkowski core. In contrast to all previous regular solutions with Minkowski core, the Kretschmann scalar curvature is not only finite everywhere, but also bounded above by the Planck mass density, regardless of the mass of the black hole. Without doubt, from the viewpoint of quantum gravity, the spacetime with sub-Planckian curvature is more realistic and applicable to the whole process of evaporation. We think this is a dramatic improvement in comparison with the previous regular black holes with Minkowski core. In the thermodynamical aspect, all these regular black holes are characterized by a vanishing Hawking temperature at the late stage of evaporation and the remnant has a minimal mass at the Planck scale, which may be viewed as the candidate of dark matter. On the other hand, the more detailed detection of dark matter would potentially provide constraints on the choices of the parameter value α 0 in future. Furthermore, we have demonstrated that for specifical choice of the potential from, the regular black holes have the similar behavior as Bardeen/Hayward/Frolov black holes at large scales, respectively. Therefore, we have established a one-to-one correspondence between the regular black holes with Minkowski core and those with de-Sitter core. This correspondence provides us a scheme to construct new regular black holes with dS core as well.
Finally, we have provided a theoretical understanding on the origin of such exponentially suppressing potential appeared in the black hole metric in a heuristic manner. Two essential ingredients are taken into account. Firstly, any quantum object has a size larger than the minimal length at Planck scale due to the effects of gravity. Secondly, any object with a finite size must be affected by the gravitational tidal force. Moreover, the exponentially suppressing form of the gravity potential in the metric results from the generalized function z(p) with a novel exponential form of the momentum in GUP, which is in contrast to the ordinary one with the quadratic form of the momentum. This generalization may reflect the non-pertubative effects of quantum gravity. It is also this relation that paves a bridge between the effective metric of regular black holes with asymptotically Minkowski core with that with asymptotically dS core.
Of course, in this paper the regular black holes that we have constructed are static, we expect the Hawking radiation may be added in such that they can be extended to dynamical ones such as Vaidya-like solutions to investigate the formation and evaporation of regular black holes with Minkowski core, as demonstrated for the black hole with dS core in [2].
The whole picture about the black hole evaporation will help us to understand the final fate of black hole evolution and disclose the mystery of black hole information loss paradox.