Barrow black holes and the minimal length

Following Barrow's idea of fractal black hole horizon, we re-derive black hole entropy of static spherically symmetric black holes. When a black hole absorbs matter its horizon area will increase. Given the spherically fractal structure, we conjecture that the minimal increase of the horizon area should be the area of the smallest bubble sphere. From this, we find the black hole entropy has a logarithmic form, which is similar to that of Boltzmann entropy if we consider $A/A_{pl}$ as the number of microscopic states. We further calculate temperatures and heat capacities of Schwarzschild, Reissner-Nordstr{\"o}m(RN), and RN-AdS black holes. It is found that their temperatures are all monotonically increasing and the heat capacities are all positive, which means these black holes are thermodynamically stable. Besides, for RN-AdS black hole we find its heat capacity has Schottky anomaly-like behavior, which may reflect the existence of the discrete energy level and restricted microscopical degree of freedom.


Introduction
Bekenstein and Hawking found that in the framework of general relativity(GR) black hole entropy is proportional to the horizon area of a black hole, to be precise, it is S = A/4. In other theories of gravity, the Bekenstein-Hawking entropy can be modified by contributions from higher-order curvature terms and can be simply derived using the Wald formula. Even in GR, when some quantum effects are taken into account, the area law of black hole entropy can also be corrected.
The most commonly considered quantum effect is the generalized uncertainty principle (GUP).
GUP predicts the existence of a minimal length scale of the order of the Planck length, which can also be deduced from string theory and other tentative theories of quantum gravity [1][2][3][4][5][6]. There are more than one expressions for GUP, of which the most simple form is where l p is the Planck length and α is a positive constant. With the correction of the GUP, black hole thermodynamics can be significantly changed [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In general, black holes no longer evaporate completely, but leave behind a remnant at finite temperature. The black hole entropy will receive a logarithmic correction term proportional to ln A besides the leading Bekenstein-Hawking entropy The microscopic mechanisms behind gravitation and black hole entropy are yet to be fully understood. Recently, Barrow proposed a toy model for possible effects of quantum gravity by considering the fractal structure of horizon surface [21]. In this case, the area and volume of a black hole should be the sum of all the intricate structures. With these fractal structures, the entropies of a black hole and our Universe can be very large. Barrow's model has also been further extended to the study of dark energy [22,23] and black hole thermodynamics [24,25].
It is our concern in this letter that how the existence of a minimal length will affect the entropy of black holes with fractal structures. In this case, when a black hole absorbs a particle there will be a natural minimal increase in the horizon area, ∆A min . Following the idea of [8], we can derive the black hole entropy. According to the first law of black hole thermodynamics, we can further derive the temperature and heat capacity. We find that for black holes with fractal horizon these thermodynamic quantities are much different from that of standard black holes.
The paper is arranged as follows. We first simply review Barrow's fractal black hole and some of his results in section 1. In section 2 we re-derive black hole entropy based on this fractal structure.
We then calculate the temperature and heat capacity of some black holes with fractal structure. At last, we summarize our results and discuss the possible future study.

Barrow black holes with fractal structure
In this part, we give a brief introduction about Barrow's idea on the fractal structure of the black hole horizon. For details, one can refer to Barrow's paper. Barrow considered Schwarzschild black hole and imagined there are many smaller spheres attaching on the black hole horizon and then more smaller spheres attached to these spheres and so on. This is a fractal structure.
Suppose that at each step there are N spheres and the radius is λ times smaller than that of the sphere in the previous step. Let r 0 = r h , which is the Schwarzschild radius. If there is no cut off at some small finite scale, the actual surface area of the horizon and volume of the black hole are infinite series: Barrow discussed that when λ −2 < N < λ −3 the surface area will be infinite and the volume of the black hole is finite.
Besides, on the basis of the area law of black hole entropy Barrow further conjectured that the entropy can take the form of S ≈ A/A pl ≈ (A h /A pl ) (2+∆)/2 , where A and A h are the area of the fractal horizon and the standard horizon, A pl is the Planck area. 0 < ∆ < 1 with ∆ = 0 corresponding to the standard horizon and ∆ = 1 corresponding to the most intricate horizon.

The Entropy of Barrow black holes
In quantum gravity there is the idea of quantized spactime, which means the existence of the smallest finite length scale. Usually the Planck length l p is considered to be this scale. The GUP, which gives an apparent minimal length, is a realization of this idea. With this cut-off, Barrow black hole can have interesting thermodynamic properties.
Although l p puts the lower limit of length scale, the radius of the smallest sphere in the fractal structure need not exactly to be l p . In fact, according to the recurrence relation r n+1 = λr n it only needs, at some cut-off step n 1 , to satisfy In this case, the surface area should be where A h = 4πr 2 h is the original area of black hole horizon. Next we try to derive the black hole entropy by following the approach of the work [8].
dS dA where (∆S) min represents the minimal increase of entropy, the value of which is a constant ln 2 according to information theory. In the following we mainly focus on the calculation of the minimal increase of the surface area, (∆A) min .
Considering the fractal structure, when the black hole absorbs a particle the minimal increase of total area should be the area of the smallest sphere, namely where we have set Clearly, we have where we introduce A pl to obtain a dimensionless quantity in the logarithmic function. Below we will call this result as Barrow entropy for short.
Clearly, the Barrow entropy does not satisfy the usual area law, or the logarithmic correction to area law. However, this form makes one reminiscent of the well-known Boltzmann formula: ln Ω. If we make the correspondence A/A pl ↔ Ω and calibrate ln 2/c 1 to one 1 and set the integration constant to be zero, we can understand the Barrow entropy as Boltzmann entropy. In analogy to statistical mechanics, we can consider A pl as the area occupied by one microscopic state and therefore A/A pl is just the number of microscopic states of the black holes, Ω.
4 Temperature and heat capacity of Barrow black holes As a thermodynamic system, the thermodynamic quantities of black holes should satisfy the thermodynamic identity: On the basis of Barrow entropy, below we will further analyze the influence of the fractal structure on the thermodynamic properties of several black holes.

Schwarzschild black hole
If the metric of Schwarzschild black hole is not affected, we can derive the temperature where the subscript "B" is for Barrow for short.
Clearly, this temperature is proportional to the Schwarzschild radius, while in standard Schwarzschild black hole the Hawking temperature is inversely proportional to r h . Until now, we still do not know how black hole temperature depends on horizon radius through observation. We cannot directly rule out this possibility. Moreover, with this temperature, we find that the heat capacity of Schwarzschild black hole is always positive: Therefore, with this fractal structure Schwarzschild black hole can be thermodynamically stable.

Reissner-Nordstrom(RN) black hole
Barrow's idea can also be used for other static spherically symmetric black holes. We first take RN black hole as an example. The line element is with the the metric function The temperature corresponding to the Bekenstein-Hawking entropy is We then calculate the heat capacity at constant Q, which is a counterpart of the heat capacity at constant volume, C V , in P V T system.
Furthermore, the heat capacity should be (4.10) From Fig.1, we can see that the two temperatures both become zero in the extremal limit, r h = Q. As r h increases, the Barrow temperature is monotonically increasing, while the Hawking temperature first increases to a maximum and then decreases monotonically. Correspondingly, as is shown in Fig.2, the heat capacity for the RN black hole with fractal structure is always positive and finite, while for the standard RN black hole the heat capacity is positive only at a finite interval and turns negative after undergoing a divergent point.

RN-AdS black hole
For RN-AdS black hole, the metric function is where l represents the cosmological radius.
The mass function can be expressed according to the horizon radius, The standard Hawking temperature is (4.13) And the heat capacity is (4.14) As is shown in Fig.3, for different values of l this temperature can increase monotonically or have some extrema. In the former case, the heat capacity is always positive. In the latter case, the heat capacity is positive only in the interval where the slope of the temperature is positive. This configuration corresponds to the first-order phase transition between the smaller black hole and the larger one. Similar to the RN black hole, we can also obtain the Barrow temperature and the corresponding heat capacity, which are

Discussions and Conclusions
On the basis of Barrow's idea of the fractal horizon and the existence of a minimal length, we try to re-derive black hole entropy. The result relies on the choice of the minimal increase of horizon area, ∆A min . Due to the fractal structure, we think that the most natural choice should be the area of the smallest bubble sphere. In this way, we found that the entropy has a logarithmic form for static spherically symmetric black holes. This Boltzmann-like entropy may reflect the microscopic structure of black holes if we consider A/A pl as the number of microscopic states.
We assumed the laws of black hole thermodynamics still hold and further calculated the temperature and heat capacity of these black holes with fractal structures. First, the temperature is generally given by On the one hand, this result guarantees that the Barrow temperature has the same sign as that of the Hawking temperature. On the other hand, the factor r 2 h makes the Barrow temperature increase more quickly for large r h and leads to a monotonically increasing T B . Therefore, for the black holes we considered, This means these black holes are, at least locally, thermodynamically stable. Besides, the monotonicity of T B means that ∂T B /∂r h = 0, which implies that the heat capacities are continuous and have no divergent points. This indicates that the phase structures of these black holes are very simple.
There is an unexpected result for RN-AdS black hole with fractal structure. Its heat capacity exhibits a Schottky anomaly-like behavior, which has also been found and discussed in other gravitational system [26][27][28]. This can be attributed to the existence of discrete energy levels and restricted microscopic degrees of freedom. At least there are some low-lying energy levels separated from the remainder of the energy spectrum. At very low temperatures, the heat capacity increases rapidly with the temperature. At high enough temperature, the behavior of any thermodynamic system approaches that of its classical counterpart. In this case, k B T is much larger than the interval of adjacent energy levels, so the energy levels are quasi-continuous. In thermodynamic systems with finite energy, such as two-level system and dS black holes [28], the heat capacity should tend to zero in the high temperature limit. But the heat capacity of RN-AdS black hole with fractal structures has a nonzero value in this limit. This reveals that the total mass M of this black hole has no upper limit. As the temperature increases, the black hole can always absorb heat and become more energetic.
The presence of the discrete energy levels must be relevant to the fractal structure of the RN-AdS black hole. However, the Schwarzschild black hole and the RN black hole do not possess this property. We also want to know whether these interesting thermodynamic properties also exist for other black holes. It is of great interest to extend our current study to the higher-dimensional and more complicated spherically symmetric black holes, which may provide new insights toward a better understanding of the microscopic structure of black holes.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.