Quasinormal Modes and Thermodynamics of Regular Black Holes

By applying the dimensionless scheme, we investigate the quasinormal modes and phase transitions analytically for three types of regular black holes. The universal deviations to the first law of mechanics in regular black holes are proved. Meanwhile, we verify that second order phase transitions and Davies points still exist in these three models. In addition, we calculate their quasinormal modes in the eikonal limit by applying the light ring/quasinormal mode correspondence, and discuss the spiral-like shapes and the relations between the quasinormal modes and phase transitions. As the main result, we show that spiral-like shapes in the complex frequency plane are closely related to the parameterization, namely in some particular units the spiral-like shapes will emerge in the models, which may not be of the spiral behaviors reported by other authors. We also discover a universal property of regular black holes, i.e., the imaginary parts of their QNMs do not vanish for the extreme cases, which does not appear in singular black holes, such as the Reissner-Nordstr\"om and Kerr black holes, etc.


Introduction
Black holes (BHs) as the prediction of general relativity (GR) are of special properties, and expected to be the bridge between a gravitational theory and a quantum theory. Furthermore, the observation of gravitational waves from a binary black hole merger reported by the LIGO Scientific and Virgo collaborations [1] in 2016 provides a new window to study the BHs. Thus, the BH physics is now regarded as the core of modern physics. The quasinormal modes (QNMs) [2] as a complex frequency of damped oscillations from BH perturbations play an important role in the analysis of BH stability.
In particular, the gravitational waves are just the fundamental mode and thus carry the information of BHs.
The BH solutions of Einstein's equations are singular, which implies the breakdown of completeness of spacetimes and embodies the shortage of Einstein's GR. This singularity problem is unavoidable in Einstein's GR, which was proved by Penrose [3] and Hawking [4] in 1960s. As a result, it has been challenging to find such BH solutions that have no singularities or that are regular in other words. no spiral structures. In Sec. 3 we verify that the regularity and traditional first law of BH mechanics cannot exist simultaneously in a BH model. In other words, the first law of mechanics in any regular BH systems is broken, even though the regularity is involved only in the metric, such as the 4D EGB BH model. Then we consider in Sec. 4 our first regular BH associated with nonlinear electrodynamics in terms of the dimensionless scheme, and compute its QNMs in the eikonal limit. We show that the spiral structure does not exist in the unit of mass M , but it will emerge again if the charge Q is used as unit. Besides, we discover a novel phenomenon that the imaginary part of QNMs does not vanish as the horizon approaches to its extreme value. This does not appear in singular BHs and seems to be a universal property for all regular BHs considered in our current work. In Sec. 5 we turn to our second regular BH, i.e., the noncommutative Schwarzschild BH, and make a parallel discussion to that of the BH with a nonlinear electrodynamic source. We study in Sec. 6 the 4D EGB BH by following the same procedure as that for the above two regular BHs. We give our conclusions in Sec. 7 where some comments and further extensions are included.

Phase transition of 5D Myers-Perry black holes
For the 5D Myers-Perry BH (MP BH) with only one nonzero angular momentum a in the Boyer-Lindquist coordinates, the metric reads [25,26,22] where Σ = r 2 + a cos 2 ϑ and ∆ = r 2 + a 2 − µ. The physical mass and angular momentum are given in terms of mass parameter µ and angular momentum parameter a as follows, The area of the outer horizon, r + = µ − a 2 , can be computed by the integral, where σ is the induced metric obtained by the setting of t = const. and r = r + , and the surface gravity can be computed [27] via two Killing vectors that are associated with the time translation and the axisymmetry, Thus one can verify the first law of BH mechanics, Moreover, it can be proved by the semiclassical method [28,29,30] that the temperature without backreaction obeys the formula, T = κ/2π, namely, Therefore, the linear correspondence between the mechanic and thermodynamic variables is saved, i.e.
T ∝ κ and A ∝ S. The entropy is then obtained, or it takes the following form in term of M and J, The Smarr formula is JΩ + ST = 2M/3. By the definition of heat capacity, one has The Davies point can be found from the solution of the algebraic equation, 1/C J = 0, For the further discussions, it is convenient to work with the usage of dimensionless variables. First of all, one needs to introduce a factor l αi for each physical quantity, where the exponent α i is regarded as an index of scale factors, while l is an arbitrary positive real number. We are going to make a transformation for every variable, e.g. M → l α1 M and J → l α2 J, such that all relevant equations and physical laws are invariant. We observe that if the index of M is set to be unit, i.e. ind(M ) = 1, the indices of the other variables can be fixed, see Tab. 1.
Secondly, there are three ways to construct the dimensionless quantities since we have two independent parameters, M and J, in the components of the metric, see Eqs. (1) and (2). The first way is the normalization by M , the second by J, and the third by the mixture of M and J. Based on the indices in Tab. 1, we introduce a dimensionless quantity m by M = 3 2 mJ 2/3 , where the fraction 3/2 is introduced to normalize the Davies point to be unit, but generally it is not necessary. Therefore, we reconstruct the dimensionless formulations of the other variables that are of course associated with the initial variable M or J. For instance, the dimensionless heat capacity can be rewritten as C J /J or C J /M 3/2 , rescaled by the physical angular momentum is shown. The advantage of this dimensionless scheme will be remarkable for most of regular BHs whose horizons and the other variables cannot be obtained analytically, which will be shown obviously in the following models of regular BHs.

Quasinormal modes in the eikonal limit
In order to give the quasinormal modes in the eikonal limit, one can apply the light ring/QNM correspondence [22]. For the 5D MP BH, we have the equation for the photon sphere radius r c , where the sign ± corresponds to the corotating and counterrotating orbits, respectively, and solve the radius of circular null geodesics, Since two r c 's are larger than r + , we have to consider two cases for calculating QNMs, where one is the corotating and the other the counterrotating. The results are as follows,

Davies point as a saddle point of rescaled temperature
When the temperature, Eq. (6), is rescaled in terms of m, we can find that T 3 √ J reaches its maximum value temperatures are and where the components of zero orders are consistent with that of the Schwarzschild BH.
In fact, for singular BHs with n parameters α i except mass M , i ∈ [1, n], the Davies point must correspond to the saddle point of the temperature with respect to the BH mass (or rescaled mass variable). From the first law of black hole thermodynamics, where β i 's are the physical quantities of BHs rather than the temperature, one can derive which implies In other words, the Davies points as the roots of 1/C αi = 0 must be the saddle points of T with respect to M , i.e., the Davies points satisfy the equation, (∂T /∂M ) αi = 0. On the other hand, such a property of Davies points would be embodied in QNMs because QNMs are closely related to BH masses.
To identify whether the Davies points are a maximum or a minimum, one needs to observe the second derivative of temperature with respect to mass, i.e., The Davies points take the maximum value of temperature if ∂C αi /∂M > 0, while they correspond to the minimum value if ∂C αi /∂M < 0. Meanwhile, these two cases correspond to two different processes.
The former implies that the temperature increases at first if some amount of heat is given to a BH. After the BH crosses the Davies point, as the amount of heat increases, the temperature decreases, where the lost energy transforms to the Hawking radiation. The latter denotes a completely inverse process, that is, the BH radiates before the Davie point, after it crosses the critical point, the Hawking radiation stops, and then all the amount of heat given to the BH will show the increasing of temperature.
At the end of this section, we note that we have rescaled T by multiplying 3 √ J in order to obtain Eq. (16). Alternatively, we can also rescale the temperature by multiplying the factor √ M , However, the Davies point (m * = 1) is no longer the maximum of T √ M under this type of rescaling.
The reason is obvious, i.e., M is regarded as a variable but J a constant in the definition of heat capacity, and such a rescaling changes the function of T with respect to M . In other words, we have to avoid rescaling the temperature by using M in order to make the Davies point be located in the saddle point of normalized temperatures.

Regularity versus the first law of black hole mechanics
For a spherically symmetric BH, one can assume [31] its shape function as follows, where α is the abbreviation of parameters rather than mass, and then compute the surface gravity, If the traditional first law of black hole mechanics is valid, which means dM =κdA/(8π) + · · · , where A represents the area of BHs, one gets the surface gravity by an alternative way, . (26) If a BH satisfies the traditional first law, one has κ/ κ = 1, which leads to the following solution, namely, where ζ(r, α) is an arbitrary function of r and α. The shape function then takes the form, Since the term 2M/r cannot be subtracted anyway, r = 0 remains to be the singular point of f (r). On the other hand, this implies that the regularity and the traditional first law of BH mechanics cannot exist simultaneously in a BH model. In addition, if one rewrites the entropy from the first law as follows, it is easy to see that the linear relation S = A/4 no longer holds for the system whose first law breaks, i.e., ρ = 0. There will be an additional term δS in the entropy, S = A/4 + δS. In other words, the traditional first law of BH mechanics must be broken and the correction to the entropy, δS, will be nontrivial for regular BHs.
Therefore, a natural question is whether there are second order phase transitions, i.e., the Davies points, in regular BHs where the traditional first law of mechanics has been broken. If the Davies points appear, will the relationship between QNMs and phase transitions still exist? We shall study these issues in the three well-known regular BHs below.

Regular black holes generated by nonlinear electrodynamics
Let us now take our first example of regular BHs from Ref. [8], which is generated by nonlinear electrodynamics. The shape function reads where q stands for electric charge. Thereinafter, we call this model the Balart-Vagenas BH (BV BH).
This charged BH does not contain singularity, and has two event horizons, where W 0 (z) and W −1 (z) are Lambert's W functions. Because Lambert's W functions are not homogenous, we only need to rescale z to be dimensionless in order to have a dimensionless horizon. The scale factors of related variables are listed in Tab. 2.
Considering the characters of the model mentioned above, we introduce the rescaled parameters x and Q as follows, We then solve Q from Eq. (35), This equality will be frequently used as the formula satisfied by the dimensionless horizon in the following.
Moreover, we can find the horizon radius and charge of the extremal BV BH, We note that if the mass-to-charge ratio |M/q| is less than √ e/2, no horizons exist, but there are two

Geometric quantity and regularity
To verify the regularity of spacetime, we compute the Ricci scalar of the BV BH, which is positive and finite when the radial coordinate is from zero to infinity, i.e., the BV BH has a de Sitter core inside. On the two boundaries, x → 0 and x → ∞, the Ricci scalar vanishes, R(0) → 0 and R(∞) → 0, and it reaches its maximum value at x = 1/5, The contraction of two Ricci tensors takes the form, which is nonsingular on the two boundaries, R µν (0)R µν (0) and R µν (∞)R µν (∞) vanish. In addition, the Kretschmann scalar reads which also maintains nonsingular at the center and infinity, R ρ µνβ (0)R µνβ In summary, we have verified the regularity of the BV BH spacetime.
The first law of singular (traditional) black holes breaks down in regular black holes. To search differences between the first laws of singular and regular black holes, we investigate the first law of the BV BH. At first, we compute the area, Next, we give the surface gravity without backreaction, and derive the potential by integrating the electric field with respect to the radial coordinate, namely, As we mentioned in Sec. 3, it turns out that these mechanic variables do not satisfy the first law, i.e., dM = κdA/(8π) + φdq. This character of the BHs generated by nonlinear electrodynamics has already been noticed in Refs. [32,33]. By differentiating the area, we find where the coefficients of dA and dq can be computed explicitly, This shows that we cannot use the traditional first law of black holes to describe regular black holes and should rebuild a new first law of BH mechanics.
By rearranging the coefficients in Eq. (46), we can give the modified first law of BH mechanics, where the last two terms of the right hand side are corrected terms to the traditional first law, and the corresponding Smarr formula has the form, where Now we try to build the relationship among entropy, temperature and mass. At first, the equality, should hold for any thermodynamic systems, where E is total energy. Then, we can verify by the semiclassical method [28,29,30] that the temperature without backreaction still obeys the formula, Therefore, we conclude that the linear correspondence, S ∝ A, no longer holds.
Let us consider a deformation of entropy by following Ref. [34]. If the linear correspondence, E ∝ M , still holds, the first law becomes where In the integration of dS, we calculate the second part at first, and then have the total entropy, We plot the dependence of entropy deviation on area in Fig. 5, which represents a nonlinear relation.

Heat capacity and Davies points
Let us search whether there exists a second order phase transition in the BV BH which has a different first law from that of singular BHs. According to the indices of scale factors, we recast the temperature in the dimensionless form, where mass M is regarded as unit, then we obtain the dimensionless heat capacity in the unit of M , and plot the relation between the heat capacity and the rescaled parameter Q in Fig. 6. From Fig. 6, we find that there is still a Davies point at which a second order phase transition happens in the BV BH although its entropy related to area is different from that of singular black holes.
By solving the algebra equation, 1/C q = 0, we get the Davies point, at which the horizons of the BV BH can be calculated, Numerically, we have x * H1 ≈ 2.41 > x ext and x * H2 ≈ 0.51 < x ext , which implies that only x * H1 is physical. Alternatively, the temperature and heat capacity can also be rewritten in the unit of charge q, and of horizon x H with the help of Eq. (35). That is, using Q 2 = x H e 1/xH to replace Q 2 in Eqs. (57) and (61), we compute the temperature, and When x H = x ext , the temperatures in the two units vanish, i.e. T M = 0 and T q = 0, which is similar to the case of the extreme RN black hole. Moreover, since x H ≥ 1, considering the property of Lambert's W functions, we simplify the above temperatures to be

Quasinormal modes in the eikonal limit
To calculate the QNMs, we start with the equation of photon spheres [22], which takes the form for the BV BH, The radius of photon spheres cannot be solved analytically from the above equation. Thus we use the variables x and Q and simplify Eq. (66) to be It can be visualized in the x c − Q plane, see Fig. 7, where Q reaches its extreme values at x 0 = 5 + √ 13 /6 ≈ 1.43 > x ext = 1. For a positive charge, if Q + < Q + 0 ≡ Q + (x 0 ), the radius of photon spheres does not exist; if Q + = Q + 0 , there is one single root, x 0 ; if Q + > Q + 0 , there are two photon sphere radii, one is inner, r − c , and the other is outer, r + c . The real and imaginary components of QNMs can be found in the two units, M and q, respectively,

ΩM =
x c e 1/xc 3x c − 1 and The QNMs in the two different units are shown in Fig. 8. Note that the imaginary parts of QNMs disappear in the range of 1 < x c < x 0 ≈ 1.43. Does it mean that there are only normal modes of perturbation for the BV BH? We shall answer this question by analyzing the relationship between the photon sphere radius x c and the horizon radius x H below.   To investigate the dependence of temperature on QNMs, we have to represent the temperature and QNMs in a consistent way, namely, we replace x H in the temperature by x c via the following relation, which is dubbed "black hole-photon sphere cone" (BH-PS cone) and obtained by combining f (x H ) = 0 with the photon sphere radius, Eq. (67). This name comes from the similarity to the Dirac cone in form. The BH-PS cone is plotted in Fig. 9. Since an extremal BH has the minimal horizon radius and is surrounded by a photon sphere, only region II in Fig. 9 is physical. The BH-PS cone equation can be solved exactly, According to the property of Lambert's W functions [35], we find x H1 > 1 and 0 < x H2 < 1, which and Then, we depict the dependences of temperature on the real and imaginary parts of QNMs in Fig. 10 and Fig. 11 in the units of M and q, respectively. As we demonstrated in the previous section, the Davies points are still located at the maxima in the planes Ωq − T q and λq − T q, see For the complex frequency plane, the QNMs are exhibited in Fig. 12, where the Davies points are displayed as well. The spiral-like shape of QNMs is shown in the unit of q, but it is not so apparent in the unit of M .

Noncommutative Schwarzschild black holes
Now we turn to consider our second example, the noncommutative Schwarzschild BH [10] whose action is 1 with Then from Einstein's equation, R µν − 1 2 g µν R = 8π(T θ ) µν , one can obtain the shape function, where θ denotes the noncommutative parameter and γ 3 2 , r 2 4θ the lower incomplete Gamma function, The indices of scale factors for the relevant variables are shown in Tab. 3.
Therefore, we have two methods to rescale the variables. Firstly, θ is regarded as unit, so r and M as initial variables can be rescaled as follows, where u(x) and α(Θ) are two arbitrary functions which will be fixed below. Secondly, M is regarded as unit, then r and θ as initial variables can correspondingly be rescaled by where g(x) and ζ(m) are arbitrary, and x, Θ and m are dimensionless.
Considering the specific property of Gamma functions and requiring the linearity of rescaling, we prefer to adopt the first rescaling method and obtain the concise relations: u(x) ∝ x and α(Θ) ∝ Θ −1 .
Based on the relations, we find that it is convenient to make the following rescaling for r and θ, where x ≥ 0 and Θ ≥ 0. Note that the relation of r and x is linear and so is the relation of √ θ and Θ.
As a result, the horizon radius can be obtained from f (r H ) = 0, Here numerically computed by the first-derivative test, The root is located at x ext ≈ 1.51 which corresponds to the maximum value Θ max ≈ 0.53. Then we get the event horizon for the extremal noncommutative Schwarzschild BH by using Eq. (80), when M = M ext ≈ 1.90 √ θ. Moreover, the area of the extremal case is A ext ≈ 114.61θ, which coincides with the results in Refs. [10,36].

Geometric quantities and regularity
The regularity of black holes is related to geometric quantities of spacetime. The most immediate quantity is the Ricci scalar, which approaches to the following forms in the two limits, x → 0 and x → ∞, In addition, R has a critical value at x 0 = √ 2, i.e. R( √ 2) = 0, and changes its sign as r increasing from the inside to the outside range of the noncommutative Schwarzschild BH, which implies that the noncommutative Schwarzschild BH has a dS core and an AdS outer horizon. For other geometric quantities, such as the "square" of the Ricci tensor, one has R µν R µν = 4/(πM 4 Θ 6 ) on the one side x → 0, and R µν R µν → 0 on the other side x → ∞; for the Kretschmann scalar, We notice that all the above geometric quantities are regular, so the noncommutative black hole is regular everywhere. The above demonstrations can alternatively be seen clearly in Fig. 14.

Deformation of the first law of black bole mechanics
Using the same method as in the BV BH, we try to rebuild the relationship among entropy, temperature and mass in the noncommutative Schwarzschild BH. The surface gravity without the backreaction Using Eqs. (88) and (89), we calculate the variation of mass, Then considering the variation of area, dA = 8πr H dr H , we obtain the deformation of the first law of BH mechanics, Note that rH 2M → 1 as θ → 0, i.e., the deformation disappears. To reconstruct the first law of BH thermodynamics, let us apply T = κ/2π and rearrange Eq. (91) to be Since 0 ≤ r H ≤ 2M , the second part in dS is positive. This implies that dS ≥ dA/4. The entropy can be obtained by the integral, where z = r 2 H /(4θ) = A/(16πθ) and z ext = r 2 ext /(4θ) ≈ 2.28, see Fig. 15. As θ approaches to zero, the upper limit in the integral of entropy deviation tends to infinity, i.e.
z → ∞, and the integral is finite, which implies that the relation between S and A reduces to the standard form, S = A/4, due to the vanishing coefficient of δS. That is to say, the noncommutative Schwarzschild BH turns back to the normal Schwarzschild BH. Alternatively, the entropy can be obtained by the integral [37,38], which coincides with Eq. (93).
The temperature of the noncommutative Schwarzschild BH in the unit of θ reads and the heat capacity takes the form in terms of the rescaled horizon x H , The Davies point can be found numerically, i.e. x * H ≈ 2.38 or r * H ≈ 4.76 Fig. 16 for the graph of the heat capacity versus the horizon radius rescaled by θ. In addition, we give the dimensionless temperature in the unit of M , Moreover, the real and imaginary parts of QNMs in the unit of θ can be computed, respectively, The shape of QNMs in this unit does not change too much when compared with that in the unit of M , see the right graph of Fig. 18.
As to the spiral-like shape in the noncommutative Schwarzschild BH, we have a quite interesting discovery. The spiral-like shape does not exist when the QNMs are rescaled in the unit of M , but the obvious spiral-like shape appears when the QNMs are rescaled in the unit of θ. This implies that the spiral-like shape depends on the way of rescaling. See Fig. 19 for the details. The dependence of temperature on the real and imaginary parts of QNMs in the unit of M is shown in Fig. 20. As we have demonstrated, the Davies point is not located at the maximum of the curves. The dependence of temperature on the real and imaginary parts of QNMs in the unit of θ is shown in Fig. 21, where the Davies points are located at the maximum of the curves as expected. 6. 4D Einstein-Gauss-Bonnet black holes

Horizons and regularity
The 4D EGB BH was given in Ref. [21] where a conformal gravity with a trace anomaly was considered.
After a reparametrization, we can write the shape function, 2 where α is Gauss-Bonnet coupling constant, and we can verify that the corresponding metric is finite at the center, r = 0.
The branch with "+" sign is unstable as discussed in Ref. [21]. Here we just consider the branch with "−" sign, and write the outer and inner horizons as follows, Note that the 4D EGB BH turns back to the Schwarzschild BH when α → 0. Moreover, one has the horizon radius for the extreme case, r ext = √ α, and the corresponding area, A ext = 4πα.
Now we calculate the scale indices for the 4D EGB BH and list them in Tab. 4. Based on this table, M α r A S C κ T 1 2 1 2 2 2 −1 −1 we can choose the following rescaling for r and α, respectively, where x and a are dimensionless, and obtain the rescaled outer horizon, from which we can fix the range of a, 0 < a ≤ 1. The above equation can be rewritten as the horizon equation, from which we get the horizon for the extreme 4D EGB BH, x ext = 1.
Around the center, x = 0, the leading order of the Ricci scalar takes the form, while in the limit of x → ∞, it reads Namely, the 4D EGB BH is of dS core near the center and has asymptotic AdS behavior far out of the horizon. Although the Ricci scalar is divergent when x → 0, the gravitational force is repulsive at a short distance and thus an infalling particle never reaches the center as mentioned in Ref. [15]. Therefore, the regularity of 4D EGB BHs can be saved.

Deformation of the first law of black hole mechanics
As we have known, the regularity of 4D EGB BHs will deform the first law of mechanics. The surface gravity of this BH equals Using Eq. (108), we calculate the variation of mass, Then considering the variation of area, dA = 8πr H dr H , we obtain the deformation of the first law of BH mechanics, where the deformation disappears when α → 0. Correspondingly, the first law of BH thermodynamics We thus obtain the entropy, where A is bounded by A Sch = 16πM 2 . δS is regarded as the correction to the entropy, and it vanishes as α → 0, see Fig. 22.

Heat capacity and Davies points
The temperature can be represented in the unit of M as and in the unit of α as . (120) We plot the heat capacity with respect to the horizon in Fig. 23, which shows the existence of a second order phase transition in the 4D EGB BH.

Quasinormal modes in the eikonal limit
According to the equation of photon spheres [22], the relationship between the dimensionless radius x c and parameter a is Replacing a by the horizon x H with the help of Eq. (111), we obtain the cone equation, from which we solve the minimum of the upper cone, x min For the 4D EGB BH, the relation of the photon sphere radius versus the horizon radius is plotted in Fig. 24.
Since there is a gap between the upper and lower cones, we know that the imaginary part of QNMs must vanish in some range of the gap. We compute the real and imaginary parts of QNMs, respectively, in the unit of α, and The corresponding graph of Ω and λ versus α is shown in Fig. 25. As mentioned in the above two sections, the imaginary part of QNMs vanishes in the range of photon sphere radii: see the horizontal gap between the two blue curves in Fig. 25. Because the minimal photon sphere radius x min c allowed by the physical region II is larger than x 0 , such a phenomenon of a vanishing imaginary part will never happen.
In addition, we compute the real and imaginary parts of QNMs, respectively, in the unit of M , and We plot the relation of the QNMs with respect to the photon sphere radius in the unit of M for the 4D EGB BH in Fig. 26, where we can see the similar behaviors to those in the unit of α when we compare  We also plot the relations of the real part versus the imaginary part of QNMs in the unit of M and α for the 4D EGB BH in Fig. 27. Note that we find the similar spiral-like shape to that in the two models we have considered in the above two sections. That is, there exists a clear spiral-like behavior in the unit of α, but no such a behavior in the unit of M . Finally, we plot the dependence of the temperature on the real and imaginary parts of QNMs, respectively, in the unit of α for the 4D EGB BH in Fig. 28. We can see that the Davies points are located at the saddle points.

Discussions and conclusions
In this paper, starting with the 5D Myers-Perry BH as a sample, we investigate the three models of regular BHs in terms of a dimensionless scheme. They are the BV BH, the noncommutative Schwarzschild BH, and the EGB BH in the 4-dimensional spacetime. The regularity of the first two BHs is different from that of the last BH, that is, the former has no singular points in background spacetimes, while the latter represents the fact that a particle can never reach the divergent region because the gravitational force becomes repulsive and tends to infinity. What we focus on are the differences between regular BHs and singular (traditional) BHs, in particular, in the aspects of QNMs and phase transitions, and the relevant phenomena induced by QNMs and phase transitions.
• The entropy bounds are not universal in the models of regular BHs.
We have verified that the regularity of BHs is not compatible with the first law of BH mechanics.
By abandoning the linear correspondence between the mechanic and thermodynamic variables, we find that the regularity of spacetime will lead to two results, where one is a broken first law of BH mechanics and the other a deformation of entropy.
For the BV BH generated by nonlinear electrodynamics, the entropy deviation equals where A Sch = 16πM 2 is the area of the Schwarzschild BH. Since A < A Sch , δS is nonnegative.
For the noncommutative Schwarzschild BH, the entropy deviation is also nonnegative, For the 4D EGB BH, the nonnegative correction to entropy takes the form, where A ext ≤ A ≤ A Sch . Therefore, the entropy bounds of the three regular BHs must be larger than the Bekenstein bound of singular BHs, i.e. S = A/4 + δS > A/4. However, the origination of entropy corrections remains unclear at present. δS may be generated by some sort of entropy stored inside a BH, e.g. the Christodoulou-Rovelli (CR) entropy S CR [40], or by the so-called quantum entropy, where P is quantum pressure caused by the vacuum polarization, and V CR is the CR volume [41].
Here we just mention that S CR − P T dV CR should vanish for singular BHs, but may be nonzero for regular BHs. The detailed discussions will be reported elsewhere. The left graph of Fig. 29 depicts the case with ∂C α /∂M > 0, which implies that the Davies point is the maximum of temperature. Moreover, the Davies point separates the state of heat capacity into two phases. In the left phase, the heat capacity is positive before the mass reaches its critical value at which the second order phase transition occurs, namely, the temperature of BHs increases for a given "heat". After the mass increases and exceeds its critical value, i.e. in the right phase, the heat capacity becomes negative, namely, for a given "heat" the temperature of BHs decreases and such a process is accompanied with the Hawking radiation. The right graph of Fig. 29 describes the inverse procedure, where the Davies point is the minimum of temperature due to ∂C α /∂M < 0. In the left phase, the heat capacity is negative and such a process is accompanied with the Hawking radiation; then after the mass exceeds the critical point, i.e., in the right phase, the heat capacity becomes positive and the Hawking radiation terminates. We note that for the models of regular BHs we considered, ∂C α /∂M > 0, i.e., the Davies points are the maximum of temperature.
• QNMs of regular BHs in the eikonal limit have a spiral-like structure in a proper unit.
As we have observed, the spiral-like shapes discussed in Refs. [18,20] do not exist in the regular BHs if the QNMs are represented in the unit of M , but they appear if the other parameters are adopted as units. The reason is that the imaginary part of QNMs in the unit of M is a monotonic function of M , but a non-monotonic function of the other parameters.
We apply the light ring/QNM correspondence to calculate QNMs. According to the relationship between QNMs and photon sphere radii, we can see that the imaginary parts of QNMs disappear in the range of x ext < x c < x 0 . It seems that the regular BHs go into an oscillating stage without damping. We further study this phenomenon from cone equations and discover that such a phenomenon is in fact forbidden by the physical region of x H − x c graphs in which the minimal photon sphere radius x min c is greater than x 0 . In addition, when the regular BHs decay to the final state, where the horizon radius equals x ext and the photon sphere radius x min c , neither the real nor the imaginary parts of QNMs vanish. This is a universal property of the regular BHs considered in our current work, which does not appear in singular BHs, such as the Reissner-Nordström and Kerr BHs, etc. From the classical point of view, it may be difficult to understand why the BHs being in their final state still have damping contributions. This puzzle gives the motivation for us to future investigate from the quantum point of view, e.g. to study the canonical quantization of regular BHs in a minisuperspace [42].

Appendix A. The Lagrangian for the anisotropic fluid
The Lagrangian for the perfect fluid in general relativity has a long history [43,44,45,46,47,48].
For the noncommutative matter introduced in Ref. [10], the Lagrangian formalism considered in Ref. [49] explicitly depends on the spacetime coordinate. In this appendix, we give an alternative one.