# Thermodynamics for regular black holes as intermediate thermodynamic states
and quasinormal frequencies

Bai-Hao Huang<EMAIL_ADDRESS>School of Physics, Nankai
University, Tianjin 300071, China Han-Wen Hu<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071, China Liu Zhao
Corresponding author<EMAIL_ADDRESS>School of Physics, Nankai
University, Tianjin 300071, China

###### Abstract

The thermodynamics for regular black holes (RBHs) is considered under the
restricted phase space (RPS) formalism. It is shown that the RPS formalism
seems to hold for RBHs, however, in order for the extensive thermodynamic
parameters to be independent from each other, the RBHs need to be viewed as
intermediate thermodynamic states in a larger class of black holes (BHs) which
admit both regular and singular states. This idea is checked for several
classes of BHs. In particular, for the electrically charged Hayward class BHs,
it is shown that the regular states can either be thermodynamically stable or
unstable, depending on the amount of charges carried by the BHs. The
quasinormal frequencies for the Hayward class BHs are also analyzed, and it
turns out that, even for the thermodynamically unstable regular states, the
dynamic stability still holds, at least under massless scalar perturbations.

Keywords: regular black holes, thermodynamics, quasinormal frequencies,
stability

##  1 Introduction

Black holes (BHs) are important objects predicted by general relativity and
their existence is verified by gravitational wave observations. Theoretically,
under the strong energy condition and the weak cosmic censorship conjecture,
singularities are inevitable inside the event horizons of the BHs. However,
the existence of singularities signify the incompleteness of spacetime and the
failure of all physics laws [1, 2], there have been long-lasting efforts in
studying regular black holes (RBHs) since the 1970s [3]. Besides baldly
replacing the singular core of the black hole metric with a regular one, there
are also some theoretical reasons for envisaging the replacement of singular
BHs by RBHs. To name a few of the underlying reasonings, the ultraviolet
incompleteness of general relativity calls for a smallest spacetime resolution
of the Planck size, and also the strong energy condition may be weakened in
certain circumstances. Nowadays, a significant amount of interest has been
paid toward a particular class of RBHs which arise as solutions to the field
equations of general relativity coupled with nonlinear electrodynamics (NED)
[4, 5, 6, 7].

Besides pure theoretical motivations, the recent development in observational
techniques, such as the Event Horizon Telescope [8, 9], the LISA project[10]
and the LIGO-VIRGO-KAGRA coillaborations [11, 12], also provides the
possibility to verify the validity of various modified theories of gravity.
Therefore, it is meaningful to calculate the shadow and the quasinormal
frequencies of RBHs to assist in the analysis of experimental data. The study
of RBHs may also provide a potential solution for the information loss
paradox, which seems to be related to the singularity in the core of BHs [13].

In spite of various progresses in the study of RBHs, the thermodynamic
description for RBHs remains confusing. Under the guidance of Bardeen and
Hawking’s work [14], Rasheed redefined the electric charge, magnetic charge,
electric potential, and magnetic potential, and gave the first law of
RBHs[15]. However, it fails to hold in the Bardeen and Hayward RBHs [16, 17].
Ref.[16] suggested that $\mathrm{d}M$ in the first law needs a coefficient
that is not always equal to 1, which is followed up by many authors [16, 17,
18, 19, 20] and leads to a very different form of the first law and Smarr
formula. Such modifications suffer the problems of parameter non-independence
[21, 22, 23, 24] and non-extensiveness.

In this work, we employ the restricted phase space (RPS) formalism [25, 26] to
analyze the thermodynamic behaviors of RBHs. We find that the first law and
the Smarr formula of RBHs are very similar to those of singular BHs, however
the regularity condition calls for a constraint between the mass and the
charge, which also results in a parameter non-independence problem. This
situation leads us to suggest that RBHs may not be thermodynamically self-
consistent. In order to understand the thermodynamic behavior of RBHs, one
might need to view them as some intermediate states in the thermodynamic
processes of a larger class of BHs involving both singular and regular states.
One of the major motivation for this work is to verify this idea. Meanwhile,
we are also interested in the stability of RBHs. We will study the stability
of RBHs from both thermodynamic and dynamic perspectives. The thermodynamic
stability is analyzed by considering the behaviors under the RPS formalism,
while the dynamical stability will be analyzed by studying the quasinormal
frequencies.

This paper is organized as follows. In Sec. 2, some issues in existing
attempts for thermodynamics of RBHs are outlined. In Sec. 3, we derive the
first law and Smarr formula under the RPS formalism for four different classes
of black hole solutions which can become regular at some specific choices of
parameters, i.e. the Bardeen class, the Hayward class, the Bardeen-AdS class
[19], and the new class of RBHs presented recently in ref. [23]. The isocharge
$T-S$ processes for the electrically charged Hayward class BHs are analyzed in
Sec. 4 under the RPS formalism, which indicates that there can be at most a
single thermodynamically stable branch which exists only in a limited range of
temperature and only for BHs carrying some supercritical values of electric
charge. We also study the thermodynamic stability of the regular states in the
wider class of charged Hayward BHs in Sec. 5, using the idea of considering
RBHs as intermediate states. Sec. 6 is devoted to the study of quasinormal
frequencies (QNFs) for the whole class of charged Hayward BHs, with emphasis
on the dynamic stability of the regular states. Finally, we summarize the
results in Sec. 7. Throughout this paper, we use the subscript “reg” to denote
the thermodynamic quantities at the regular states.

##  2 Outline of issues in existing attempts for thermodynamics of RBHs

We begin by presenting the action of Einstein gravity coupled with NED [27,
28],

$S=\frac{1}{16\pi
G}\int_{\mathcal{M}}{\mathrm{d}^{4}x\sqrt{-g}\left[R-\mathcal{L}\left(F\right)\right]},$
(2.1)

where $F\equiv F_{\mu\nu}F^{\mu\nu}$ is electromagnetic invariant and
$\mathcal{L}\left(F\right)$ represents the Lagrangian density of the NED.
There can be different choices for $\mathcal{L}\left(F\right)$ which lead to
different classes of RBHs. We will specify the concrete form of
$\mathcal{L}\left(F\right)$ when it becomes necessary.

It is worth mentioning that sometimes the NED model is specified by the
Hamiltonian-like density [19, 29]

$\displaystyle\mathcal{H}(P)=2F\mathcal{L}_{F}-\mathcal{L}(F)$ (2.2)

instead of the Lagrangian density $\mathcal{L}(F)$, where

$\displaystyle
P_{\mu\nu}=\mathcal{L}_{F}F_{\mu\nu},\quad\mathcal{L}_{F}=\partial\mathcal{L}/\partial
F,\quad P=P_{\mu\nu}P^{\mu\nu}.$ (2.3)

It can be shown that

$\mathcal{H}_{P}=\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}P}=\frac{1}{\mathcal{L}_{F}}.$
(2.4)

The equations of motion which arise as the stationary conditions for the
action (2.1) read

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi
GT_{\mu\nu},\quad\nabla_{\mu}P^{\mu\nu}=0,$ (2.5)

where

$T_{\mu\nu}=\frac{1}{4\pi
G}\left(\mathcal{L}_{F}F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{1}{4}g_{\mu\nu}\mathcal{L}\right).$
(2.6)

We will be interested in the static, spherically symmetric metrics of the form

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+r^{2}(\mathrm{d}\theta^{2}+\sin^{2}\theta\
\mathrm{d}\varphi^{2}).$ (2.7)

Black hole solutions of the above form can carry either electric or magnetic
charges. The field strengths obeying the Maxwell-like equation
$\nabla_{\mu}P^{\mu\nu}=0$ and the Bianchi identity
${\nabla_{\mu}}^{*}F^{\mu\nu}=0$ in these two cases take the form

$P_{tr}=-\frac{Q_{e}}{r^{2}},\quad F_{\theta\varphi}=Q_{m}\sin\theta,$ (2.8)

where $Q_{e}$ and $Q_{m}$ are respectively the electric and magnetic charges
defined via [15]

$Q_{m}=\frac{1}{4\pi}\int_{\partial\Sigma}{F_{ab}},\quad
Q_{e}=\frac{1}{4\pi}\int_{\partial\Sigma}{{}^{*}P_{ab}},$ (2.9)

in which $\partial\Sigma$ is the boundary of the spacelike hypersurface
$\Sigma$ with the timelike Killing vector field $\xi^{a}$ acting as its normal
vector. As usual, one can introduce the electric and magnetic fields

$E_{a}=\xi^{b}F_{ab},\quad H_{a}=-\xi^{b}\,{}^{\ast}P_{ab},$

together with the corresponding potentials,

$\displaystyle E_{a}$ $\displaystyle=-\nabla_{a}\Phi,\quad
H_{a}=-\nabla_{a}\Psi.$ (2.10)

Considering that Rasheed’s first law and Smarr formula [15] do not work for
the RHBs, Zhang and Gao suggested that each parameter involved in the
Lagrangian of the matter field may introduce an extra term in the first law of
RBHs [17]. For instance, the first law and Smarr formula for the magnetically
supported Bardeen BHs are modified as

$\displaystyle\mathrm{d}M$
$\displaystyle=\frac{\kappa}{8\pi}\mathrm{d}A+\Psi_{\rm
H}\mathrm{d}Q_{m}+K_{q}\mathrm{d}Q_{m}+K_{M}\mathrm{d}M,$ (2.11)
$\displaystyle M$ $\displaystyle=\frac{\kappa}{4\pi}A+\Psi_{\rm
H}Q_{m}+K_{q}Q_{m}+K_{M}M,$ (2.12)

where $\Psi_{\rm H}$ is the magnetic potential on the event horizon, and
$K_{q},~{}K_{M}$ are defined as

$K_{q}=\frac{1}{4}\int_{r_{\rm
H}}^{\infty}{\mathrm{d}r\sqrt{-g}\frac{\partial\mathcal{L}}{\partial
Q_{m}}},\quad K_{M}=\frac{1}{4}\int_{r_{\rm
H}}^{\infty}{\mathrm{d}r\sqrt{-g}\frac{\partial\mathcal{L}}{\partial M}},$
(2.13)

and $\kappa$ and $A$ are respectively the surface gravity and the area of the
event horizon. Since the model under investigation is general relativity
coupled to NED, the Bekenstein-Hawking entropy formula is valid and the term
$\displaystyle\frac{\kappa}{8\pi}\mathrm{d}A$ in eq.(2.11) can be replaced by
$T\mathrm{d}S$ as usual. The problem of the above proposal lies in that the
meaning of $K_{q},~{}K_{M}$ are unclear and that the term $\mathrm{d}M$
acquires an extra coefficient $(1-K_{M})$ in the first law.

Alternatively, Wang and Fan [19, 20] suggested a different proposal which
brings the model parameter $\alpha$ that appears in the Lagrangian density of
the NED into the first law and the Smarr formula. For a class of RBHs carrying
both electric and magnetic charges, the resulting first law and the Smarr
formula read

$\displaystyle\mathrm{d}M_{\rm ADM}$
$\displaystyle=\frac{\kappa}{8\pi}\mathrm{d}A+\Phi_{\rm
H}\mathrm{d}Q_{e}+\Psi_{\rm H}\mathrm{d}Q_{m}+\Pi\mathrm{d}\alpha$ (2.14)
$\displaystyle M_{\rm ADM}$
$\displaystyle=\frac{\kappa}{4\pi}\mathrm{d}A+\Phi_{\rm H}Q_{e}+\Psi_{\rm
H}Q_{m}+2\Pi\alpha,$ (2.15)

where $\Psi_{\rm H}$ is as before, $\Phi_{\rm H}$ is the electric potential on
the event horizon, and $\Pi$ is defined as

$\Pi=\frac{1}{4}\int_{r_{\rm
H}}^{\infty}{\mathrm{d}r\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\alpha}}.$
(2.16)

The inclusion of a variable negative cosmological constant may lead to an
extra $V\mathrm{d}P$ term [19, 20]. Similar works appear in Ref.[30, 31],
where generalizations of the first law and Smarr formula for BHs with NED are
respectively deduced from the thought that each parameter $\alpha_{i}$ in
Lagrangian density of the NED introduces an extra term in the first law and
Smarr formula of RBHs and keep these parameters constant within fixed
spacetime (i.e.$\nabla_{a}\alpha_{i}=0$). The problem lies in that model
parameter $\alpha$ is related to the magnetic charge $Q_{m}$ and the ADM mass
$M_{\rm ADM}$ via [21]

$\alpha=\frac{8G^{2}Q^{6}_{m}}{M_{\rm ADM}^{4}},$ (2.17)

therefore, there is a problem of parameter non-independence. Moreover,
changing $\alpha$ implies changing the underlying model, and hence there is
also an ensemble of theories problem.

In spite of the numerous attempts mentioned above, it appears that a
consistent thermodynamic description for RBHs is still missing. In the next
section, we will try to establish a self consistent thermodynamic description
for RBHs using the recently proposed RPS formalism. It will be seen that, in
order to have a self consistent thermodynamic description, it is better to
view the RBHs as some intermediate states for a larger class of BHs which are
singular at wider range of values of the parameters, whilst become regular at
some specific values of parameters. In this way, all issues suffered by the
previous proposals are avoided.

##  3 RPS formalism for RBHs

In this section, we will employ the RPS formalism for describing the
thermodynamics of RBHs. The RPS formalism is inspired by Visser’s holographic
thermodynamics [32], but with an important difference, i.e. the cosmological
constant must be kept invariant. This allows for the RPS formalism to be
applicable to much wider classes of black hole solutions without urging an AdS
asymptotics [33, 34]. Moreover, the effective number of microscopic degrees of
freedom, $N=L^{n-2}/G$, and the corresponding conjugate chemical potential,
$\mu=GTI_{\rm E}/L^{n-2}$, are both defined independently without using the
Euler relation. All these features suggest that the RPS formalism may also be
applicable to RBHs, and it will be clear that it is indeed the case.

###  3.1 RPS formalism for the electrically charged Bardeen class BHs

First let us consider the RPS formalism for the electrically charged Bardeen
class BHs.

The Bardeen class BHs are supported by an NED with the Hamiltonian-like
density

$\displaystyle\mathcal{H}(P)=-\frac{4s\left(-\alpha
P\right)^{5/4}}{\alpha\left(1+\sqrt{-\alpha P}\right)^{1+s/2}},$ (3.1)

where $s$ is a dimensionless constant. Under the assumption that the black
hole solution is supported purely by an electric field, Ref.[35] suggested
that the shape function $f(r)$ in the metric should take the form

$f(r)=1-\frac{2Gm(r)}{r},\quad
m(r)=M_{g}-\frac{q^{3}}{G\alpha}\left[1-\frac{r^{s}}{\left(r^{2}+q^{2}\right)^{s/2}}\right],$
(3.2)

where the integration constant $q$ is related to the electric charge $Q_{e}$
via

$\displaystyle Q_{e}$ $\displaystyle=\frac{q^{2}}{G\sqrt{2\alpha}}.$ (3.3)

The form (3.2) of the shape function ensures that the ADM mass is always equal
to the integration constant $M_{g}$, i.e. $M_{\rm ADM}=M_{g}$.

The Kretschmann invariant for the above solution reads

$\displaystyle R^{\mu\nu\rho\kappa}R_{\mu\nu\rho\kappa}$
$\displaystyle=\frac{2}{r^{6}}\left[2\left(GM_{g}-\frac{q^{3}}{\alpha}\right)+2\frac{q^{3}r^{s}}{\alpha}\frac{\left[r^{2}-\left(s-1\right)q^{2}\right]}{\left(r^{2}+q^{2}\right)^{s/2+1}}\right]^{2}$
$\displaystyle+\frac{4}{r^{6}}\left[2\left(GM_{g}-\frac{q^{3}}{\alpha}\right)+\frac{2r^{s}}{\alpha}\frac{q^{3}}{\left(r^{2}+q^{2}\right)^{s/2}}\right]^{2}$
$\displaystyle+\frac{1}{r^{6}}\left[4\left(GM_{g}-\frac{q^{3}}{\alpha}\right)+\frac{2q^{3}r^{s}}{\alpha}\frac{\left[2r^{4}-\left(5s-4\right)r^{2}q^{2}+\left(s-1\right)\left(s-2\right)q^{4}\right]}{\left(r^{2}+q^{2}\right)^{s/2+2}}\right]^{2}.$
(3.4)

It is obvious that the solution has an intrinsic singularity at $r=0$ when
$GM_{g}\neq q^{3}/\alpha$, and it is regular at $r=0$ if
$GM_{g}=q^{3}/\alpha,~{}s\geqslant 3$. When $s=3$, this regular solution
corresponds to the original Bardeen RBH [3]. For this reason, the whole class
of black hole solution specified by the shape function (3.2) will be referred
to as Bardeen class BHs.

For generic choice of parameters (i.e. without imposing the regularity
condition $GM_{g}=q^{3}/\alpha,~{}s\geqslant 3$), the temperature $T$ and
entropy $S$ are given by

$T=\frac{1}{4\pi}\left(\frac{\partial f}{\partial r}\right)_{r=r_{\rm
H}}=\frac{1}{4\pi r_{\rm H}}-\frac{sq^{5}r_{\rm
H}^{s-2}}{2\pi\alpha\left(r_{\rm H}^{2}+q^{2}\right)^{s/2+1}},\quad
S=\frac{\pi r^{2}_{\rm H}}{G},$ (3.5)

where $r_{\rm H}$ is the radius of the event horizon which arises as the
largest root of $f(r)$. The value of the electric potential $\Phi_{\rm H}$ on
the event horizon reads

$\displaystyle\Phi_{\rm H}$
$\displaystyle=\frac{q}{\sqrt{2\alpha}}\left[3-\frac{3r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{s/2}}+\frac{sq^{2}r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{1+s/2}}\right].$ (3.6)

The on-shell Euclidean action $I_{\rm E}$ corresponding to the above solution
is [36, 37, 21]

$\displaystyle I_{\rm E}$ $\displaystyle=-\frac{1}{16\pi
G}\int_{\mathcal{M}}{\sqrt{g}\left[R-\mathcal{L}(F)\right]\mathrm{d}^{4}x}-\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}{\sqrt{h}\left(K\right)\mathrm{d}^{3}x}+\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}{\sqrt{h}\left(K_{0}\right)\mathrm{d}^{3}x}$
$\displaystyle=\frac{M_{g}\beta}{2}-\frac{\beta q^{3}}{G\alpha}+\frac{\beta
q^{3}}{G\alpha}\frac{r_{\rm H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{s/2}}.$
(3.7)

Therefore, according to the general formula from the RPS formalism, we can
introduce the chemical potential $\mu$ and the effective number of microscopic
degrees of freedom $N$ as follows,

$\mu=\frac{TI_{\rm E}}{N},\quad N=\frac{L^{2}}{G},$ (3.8)

where $L$ is a constant length scale.

By straightforward calculation, the following first law and Euler relation can
be verified to hold,

$\displaystyle\mathrm{d}M_{\rm ADM}$ $\displaystyle=T\mathrm{d}S+\Phi_{\rm
H}\mathrm{d}Q_{e}+\mu\mathrm{d}N,$ (3.9) $\displaystyle M_{\rm ADM}$
$\displaystyle=TS+\Phi_{\rm H}Q_{e}+\mu N.$ (3.10)

Now let us restrict ourselves to the regular case with $M_{g}=q^{3}/G\alpha$.
We can describe the regular value $M_{\rm reg}$ of $M_{\rm ADM}$ by use of the
equation $f(r_{\rm H})=0$,

$M_{\rm reg}=\frac{q^{3}}{G\alpha}=\frac{\left(r_{\rm
H}^{2}+q^{2}\right)^{s/2}}{2Gr_{\rm H}^{s-1}}.$ (3.11)

Here, $\alpha$ is required to remain as a constant,

$\alpha=\frac{2q^{3}r_{\rm H}^{s-1}}{\left(r_{\rm
H}^{2}+q^{2}\right)^{s/2}}=\mathrm{const}.$ (3.12)

Consequently we have

$\mathrm{d}r_{\rm H}=\frac{r_{\rm H}}{q}\frac{3r_{\rm
H}^{2}-\left(s-3\right)q^{2}}{r_{\rm
H}^{2}-\left(s-1\right)q^{2}}\mathrm{d}q.$ (3.13)

Substituting the condition $M_{g}=q^{3}/G\alpha$ into eqs.(3.5) and (3.8), the
temperature and chemical potential in the regular case can be rewritten as

$\displaystyle T_{\rm reg}=\frac{q^{3}r_{\rm H}^{s-2}\left[r_{\rm
H}^{2}-\left(s-1\right)q^{2}\right]}{2\pi\alpha\left(r_{\rm
H}^{2}+q^{2}\right)^{s/2+1}},\quad\mu_{\rm
reg}=\frac{G}{L^{2}}\left[-\frac{q^{3}}{2G\alpha}+\frac{q^{3}}{G\alpha}\frac{r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{s/2}}\right].$ (3.14)

By straightforward calculation, it can be verified that following first law
and Euler relation hold in the regular case,

$\displaystyle\mathrm{d}M_{\rm reg}$ $\displaystyle=T_{\rm
reg}\mathrm{d}S+\Phi_{\rm H}\mathrm{d}Q_{e}+\mu_{\rm reg}\mathrm{d}N,$ (3.15)
$\displaystyle M_{\rm reg}$ $\displaystyle=T_{\rm reg}S+\Phi_{\rm
H}Q_{e}+\mu_{\rm reg}N,$ (3.16)

where the differential relation (3.13) needs to be employed while verifying
the above relations.

Although eqs. (3.15) and (3.16) are very similar in form to eqs. (3.9) and
(3.10), their meanings are completely different. Eqs. (3.9) and (3.10)
describe the extensive thermodynamics for the whole class of electrically
charged Bardeen class BHs, while eqs. (3.15) and (3.16) describe only the
thermodynamics of the regular Bardeen class BHs. Inserting eq.(3.3) into the
regularity condition (3.11), we get

$\alpha=\frac{8G^{2}Q^{6}_{e}}{M_{\rm reg}^{4}},$ (3.17)

which indicates that $Q_{e}$ and $M_{\rm reg}$ are not independent of each
other. Therefore, eqs. (3.15) and (3.16) contain exactly the same problem of
parameter non-independence as appeared in earlier proposals using different
formalisms. We thus suggest that, instead of eqs. (3.15) and (3.16), eqs.
(3.9) and (3.10) should be adopted while describing the thermodynamic
behaviors of RBHs. This in turn suggests that RBHs should be viewed as
intermediate states for the larger Bardeen class BHs specified by the shape
function (3.2) which admit both regular and singular states.

###  3.2 RPS formalism for other BHs involving regular states

In order to support the proposal presented near the end of the last section,
let us proceed by briefly outlining RPS formalism for other classes of BHs
involving regular states.

  1. (i)

Hayward class BHs [7]. The Hamiltonian-like density of the NED is now given as

$\displaystyle\mathcal{H}(P)=-\frac{4s}{\alpha}\frac{\left(-\alpha
P\right)^{\left(s+3\right)/4}}{\left(1+\left(-\alpha
P\right)^{s/4}\right)^{2}},$ (3.18)

and the corresponding shape function reads

$\displaystyle f(r)=1-\frac{2GM_{g}}{r}-\frac{2q^{3}}{\alpha
r}\left[\frac{r^{s}}{r^{s}+q^{s}}-1\right],$ (3.19)

which is supported by an electric field of charge $Q_{e}$ and electric
potential $\Phi_{\rm H}$ on the event horizon,

$\displaystyle Q_{e}$ $\displaystyle=\frac{q^{2}}{G\sqrt{2\alpha}},$ (3.20)
$\displaystyle\Phi_{\rm H}$
$\displaystyle=\frac{q}{\sqrt{2\alpha}}\left[3-\frac{3r_{\rm H}^{s}}{r_{\rm
H}^{s}+q^{s}}+\frac{sr_{\rm H}^{s}q^{s}}{\left(r_{\rm
H}^{s}+q^{s}\right)^{2}}\right].$ (3.21)

The regularity conditions are still given by $M_{g}=q^{3}/G\alpha$ and
$s\geqslant 3$, among which the $s=3$ case is the original Hayward BH.
Therefore, the whole class of BHs described above will be referred to as
Hayward class BHs.

Besides $Q_{e}$ and $\Phi_{\rm H}$, the other macroscopic parameters of the
Hayward class BHs are given as follows,

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi r_{\rm H}}-\frac{sq^{s+3}r_{\rm
H}^{s-2}}{2\pi\alpha\left(r_{\rm H}^{s}+q^{s}\right)^{2}},\quad S=\frac{\pi
r^{2}_{\rm H}}{G},$ (3.22) $\displaystyle\mu$
$\displaystyle=\frac{G}{L^{2}}\left[\frac{M_{g}}{2}-\frac{q^{3}}{G\alpha}+\frac{q^{3}}{G\alpha}\frac{r_{\rm
H}^{s}}{r_{\rm H}^{s}+q^{s}}\right],\quad N=\frac{L^{2}}{G}.$ (3.23)

It can be checked that, for the Hayward class BHs, we still have $M_{\rm
ADM}=M_{g}$, and the first law (3.15) and Euler relation (3.16) still hold,
and the regular states should be regarded as intermediate states of the larger
Hayward class BHs.

  2. (ii)

Bardeen-AdS class BHs [38, 19]. One can introduce a negative cosmological
constant $\Lambda$ into the action for the Bardeen class BHs, yielding

$S=\frac{1}{16\pi
G}\int_{\mathcal{M}}{\mathrm{d}^{4}x\sqrt{-g}\left[R-2\Lambda-\mathcal{L}\left(F\right)\right]}.$
(3.24)

The corresponding electrically supported black hole solutions will be referred
to as the Bardeen-AdS class BHs, which is specified by the shape function

$f(r)=1-\frac{2GM_{g}}{r}-\frac{2q^{3}}{\alpha
r}\left[\frac{r^{s}}{\left(r^{2}+q^{2}\right)^{s/2}}-1\right]+\frac{r^{2}}{l^{2}},$
(3.25)

where $l^{2}=-3/\Lambda$.

It follows that the AD mass for this class of BHs is equal to $M_{g}$.
Moreover, the regularity conditions remain unchanged,
$M_{g}=q^{3}/G\alpha,~{}s\geqslant 3$. The macroscopic parameters for this
class of BHs are given as follows,

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi r_{\rm H}}+\frac{3r_{\rm
H}}{4\pi l^{2}}-\frac{q^{5}sr_{\rm H}^{s-2}}{2\pi\alpha\left(r_{\rm
H}^{2}+q^{2}\right)^{s/2+1}},\qquad S=\frac{\pi r^{2}_{\rm H}}{G},$ (3.26)
$\displaystyle\Phi_{\rm H}$
$\displaystyle=\frac{q}{\sqrt{2\alpha}}\left[3-\frac{3r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{s/2}}+\frac{sq^{2}r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{1+s/2}}\right],\quad
Q_{e}=\frac{q^{2}}{G\sqrt{2\alpha}},$ (3.27) $\displaystyle\mu$
$\displaystyle=\frac{G}{l^{2}}\left[\frac{M_{g}}{2}-\frac{r_{\rm
H}^{3}}{2Gl^{2}}-\frac{q^{3}}{G\alpha}+\frac{q^{3}}{G\alpha}\frac{r_{\rm
H}^{s}}{\left(r_{\rm H}^{2}+q^{2}\right)^{s/2}}\right],\quad
N=\frac{l^{2}}{G}.$ (3.28)

It is easy to verify the first law

$\mathrm{d}M_{\rm AD}=T\mathrm{d}S+\Phi_{\rm H}\mathrm{d}Q_{e}+\mu\mathrm{d}N$

and Euler relation

$M_{\rm AD}=TS+\Phi_{\rm H}Q_{e}+\mu N$

hold irrespective of whether the regularity conditions are satisfied. However,
when the regularity conditions hold, the problem of parameter non-independence
arises once again. Therefore, the regular cases should still be regarded as
intermediate states of the larger Bardeen-AdS class BHs.

  3. (iii)

Models presented in Ref. [23]. The Hamitonian-like density of the NED field
and the simplest shape function presented in Ref. [23] read

$\displaystyle\mathcal{H}(P)=\frac{P}{1-4\alpha P},\quad\left(\alpha\geq
0\right),\quad
f(r)=1-\frac{2GM_{g}}{r}+\frac{p^{2}}{r^{2}}{{}_{2}F_{1}}\left[\frac{1}{4},1;\frac{5}{4};-\frac{2\alpha
p^{2}}{r^{4}}\right],$ (3.29)

where ${}_{2}F_{1}\left[\frac{1}{4},1;\frac{5}{4};-\frac{2\alpha
p^{2}}{r^{4}}\right]$ denotes a hypergeometric function. The curvature
singularity is absent when $M_{g}=\pi p^{3/2}/4G\left(8\alpha\right)^{1/4}$.
Besides, the ADM mass is equal to the integration constant $M_{g}$. The
macroscopic parameters for this class of BHs are given as follows,

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi r_{\rm H}}-\frac{p^{2}r_{\rm
H}}{4\pi\left(r_{\rm H}^{4}+2\alpha p^{2}\right)},\quad S=\frac{\pi r_{\rm
H}^{2}}{G},$ (3.30) $\displaystyle\Phi_{\rm H}$
$\displaystyle=\frac{3p}{4r_{\rm
H}}{{}_{2}F_{1}}\left[\frac{1}{4},1;\frac{5}{4};-\frac{2\alpha p^{2}}{r_{\rm
H}^{4}}\right]+\frac{pr_{\rm H}^{3}}{4\left(r_{\rm H}^{4}+2\alpha
p^{2}\right)},\quad Q_{e}=\frac{p}{G},$ (3.31) $\displaystyle\mu$
$\displaystyle=\frac{G}{L^{2}}\left[\frac{M_{g}}{2}-\frac{p^{2}}{2Gr_{\rm
H}}{{}_{2}F_{1}}\left[\frac{1}{4},1;\frac{5}{4};-\frac{2\alpha p^{2}}{r_{\rm
H}^{4}}\right]\right],\quad N=\frac{L^{2}}{G}.$ (3.32)

It can be checked by direct calculations that the first law (3.15) and the
Euler relation (3.16) still hold irrespective of whether the regularity
condition $M_{g}=\pi p^{3/2}/4G\left(8\alpha\right)^{1/4}$ is obeyed. When the
regularity condition is fulfilled, parameter non-independence problem arises
once again, which supports our suggestion that the regular BHs should be
viewed as intermediate states of the larger class of BHs involving both
singular and regular states.

##  4 Thermodynamic behaviors of Hayward class BHs

Now that the RPS formalism is established for the above mentioned four classes
of BHs involving regular states, we would like to proceed a little step by
analyzing the thermodynamic behaviors of one out of the four classes.

We take the Hayward class BHs as example and set $s=3$. In order to describe
the thermodynamic behaviors, it is necessary to rewrite the mass $M(=M_{\rm
ADM})$, temperature $T$, electric potential $\Phi_{\rm H}$ and chemical
potential $\mu$ as functions in the additive variables $S,~{}Q_{e},~{}N$. For
simplicity of notations, we rescale the electric charge $Q_{e}$ into
$Q=Q_{e}\pi\sqrt{2\alpha}$. Then the parameters $G,~{}r_{\rm H},~{}q$ can be
expressed in terms of $S,Q,N$ by by solving eqs. (3.22),(3.20) and (3.23),

$G=\frac{L^{2}}{N},\quad r_{\rm H}=\sqrt{\frac{SL^{2}}{\pi N}},\quad
q=\sqrt{\frac{QL^{2}}{\pi N}}.$ (4.1)

It is then easy to get

$\displaystyle M(S,Q,N)$
$\displaystyle=\frac{1}{2L}\sqrt{\frac{NS}{\pi}}+\frac{N}{L^{2}\alpha}\left(\frac{QL^{2}}{\pi
N}\right)^{3/2}-\frac{N}{L^{2}\alpha}\left(\frac{QL^{2}}{\pi
N}\right)^{3/2}\frac{S^{3/2}}{S^{3/2}+Q^{3/2}},$ (4.2) $\displaystyle
T(S,Q,N)$
$\displaystyle=\frac{1}{4\pi^{1/2}}\sqrt{\frac{N}{SL^{2}}}-\frac{3Q^{3}S^{1/2}}{2\pi^{3/2}\alpha\left(S^{3/2}+Q^{3/2}\right)^{2}}\sqrt{\frac{L^{2}}{N}},$
(4.3) $\displaystyle\Phi_{\rm H}(S,Q,N)$
$\displaystyle=\sqrt{\frac{QL^{2}}{2\alpha\pi
N}}\left[\frac{3Q^{3/2}}{S^{3/2}+Q^{3/2}}+\frac{3S^{3/2}Q^{3/2}}{\left(S^{3/2}+Q^{3/2}\right)^{2}}\right],$
(4.4) $\displaystyle\mu(S,Q,N)$ $\displaystyle=\frac{1}{4L}\sqrt{\frac{S}{\pi
N}}-\frac{L}{2\alpha}\left(\frac{Q}{\pi
N}\right)^{3/2}+\frac{L}{2\alpha}\left(\frac{Q}{\pi
N}\right)^{3/2}\frac{S^{3/2}}{S^{3/2}+Q^{3/2}}.$ (4.5)

It is obvious that, under the rescaling $S\rightarrow\lambda
S,~{}Q\rightarrow\lambda Q,~{}N\rightarrow\lambda N$ of the additive variables
$S,~{}Q,~{}N$, the mass $M$ scales as $M\rightarrow\lambda M$, whilst
$T,~{}\Phi_{\rm H},~{}N$ remain unchanged. These homogeneity behaviors are
expected for a standard extensive thermodynamic system. Therefore, we can
proceed using standard techniques for analyzing thermodynamic systems to study
the behavior of the Hayward class BHs.

Let us look at the isocharge $T-S$ processes at fixed $N$. It is natural to
seek for inflection points to see whether there could be any critical points.
It turns out that the inflection point equations

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{Q,N}=\left(\frac{\partial^{2}T}{\partial S^{2}}\right)_{Q,N}=0$
(4.6)

indeed have a solution with $N>0$ and $T>0$. We label the parameters
$S,~{}Q,~{}T$ at the inflection point as $S_{c},~{}Q_{c},~{}T_{c}$, and their
values are given as follows,

$\displaystyle Q_{c}$
$\displaystyle=\frac{\pi\alpha}{6L^{2}}\frac{\left(A^{3/2}+1\right)^{3}}{A\left(5A^{3/2}-1\right)}N,$
(4.7) $\displaystyle S_{c}$
$\displaystyle=AQ_{c}=\frac{\pi\alpha}{6L^{2}}\frac{\left(A^{3/2}+1\right)^{3}}{\left(5A^{3/2}-1\right)}N,$
(4.8) $\displaystyle T_{c}$
$\displaystyle=\frac{\sqrt{6}}{4\pi\alpha^{1/2}}\frac{4A^{3/2}-2}{\left(A^{3/2}+1\right)^{3/2}\left(5A^{3/2}-1\right)^{1/2}},$
(4.9)

where

$\displaystyle A$ $\displaystyle=\left(\frac{8+3\sqrt{6}}{10}\right)^{2/3}.$
(4.10)

Notice that $T_{c}$ is independent of $N$, which is a characteristic feature
required in extensive thermodynamics.

Using these “critical” values, we can introduce the relative electric charge,
entropy and temperature as follows,

$\displaystyle\mathcal{Q}=\frac{Q}{Q_{c}},\quad\mathcal{S}=\frac{S}{S_{c}},\quad\mathcal{T}=\frac{T}{T_{c}}=\frac{\left(5A^{3/2}-1\right)}{4A^{3/2}-2}\frac{1}{\mathcal{S}^{1/2}}-\frac{\left(A^{3/2}+1\right)^{3}}{4A^{3/2}-2}\frac{\mathcal{S}^{1/2}\mathcal{Q}^{3}}{\left(A^{3/2}\mathcal{S}^{3/2}+\mathcal{Q}^{3/2}\right)^{2}}.$
(4.11)

Moreover, we also introduce the Helmholtz free energy $F=M-TS$ and its
relative value $\displaystyle\mathcal{F}=\frac{F}{F_{c}}$. It follows that

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{B}\left[\frac{3A\left(5A^{3/2}-1\right)^{1/2}\mathcal{S}^{1/2}}{\left(A^{3/2}+1\right)^{3/2}}+\frac{\mathcal{Q}^{3}\left(5\mathcal{S}^{3/2}A^{3/2}+2\mathcal{Q}^{3/2}\right)\left(A^{3/2}+1\right)^{3/2}}{\left(\mathcal{S}^{3/2}A^{3/2}+\mathcal{Q}^{3/2}\right)^{2}A^{1/2}\left(5A^{3/2}-1\right)^{1/2}}\right],$
(4.12) $\displaystyle B$
$\displaystyle=\frac{20A^{3}+4A^{3/2}+2}{\left(A^{3/2}+1\right)^{3/2}A^{1/2}\left(5A^{3/2}-1\right)^{1/2}}.$
(4.13)

(a) $\mathcal{T}-\mathcal{S}$ curve of Hayward BH

(b) $\mathcal{F}-\mathcal{T}$ curve of Hayward BH

Figure 1: $\mathcal{T}-\mathcal{S}$ (1(a)) and $\mathcal{F}-\mathcal{T}$
(1(b)) curves of Hayward BH at fixed $Q,~{}N$.

Fig. 1(a) and Fig. 1(b) depict the $\mathcal{T}-\mathcal{S}$ and
$\mathcal{F}-\mathcal{T}$ curves respectively at fixed $Q_{e},~{}N$. Contrary
to the cases of RN-AdS and Kerr-AdS BHs which exhibit two stable phases on the
small and the large entropy ends, the $\mathcal{T}-\mathcal{S}$ and the
$\mathcal{F}-\mathcal{T}$ curves for the Hayward class BHs seem to have been
turned upside-down. Consequently, there can be at most one stable phase for
the Hayward class BHs which exists only for “supercritical” values of the
electric charge in a limited range of the entropy (i.e. ranging from the local
minimum to the local maximum). The stability in this range of parameters can
be justified by the positivity of the heat capacity

$C_{Q,N}=\left(\frac{\partial M}{\partial
T}\right)_{Q,N}={T}\left[\left(\frac{\partial T}{\partial
S}\right)_{Q,N}\right]^{-1}.$

Therefore, strictly speaking, the inflection point described above is actually
not a critical point for phase transitions but rather a critical point for the
stable phase to appear.

##  5 RBHs as intermediate thermodynamic states

The analysis for the $\mathcal{T}-\mathcal{S}$ processes made in Section 4
works for the complete Hayward class BHs involving both singular and regular
states. However, where exactly the regular state appears in the
$\mathcal{T}-\mathcal{S}$ processes is not touched upon. The purpose of this
section is to identify the position of the regular states in concrete
$\mathcal{T}-\mathcal{S}$ processes.

Before dwelling into details, let us first make some discussions about the
regularity conditions and the number of roots for the shape function $f(r)$.
Let us recall that one of the regularity condition,

$\displaystyle M_{g}=\frac{q^{3}}{G\alpha},$ (5.1)

is a constraint between the mass and the charge. Such a constraint resembles
the saturated Bogomol’nyi bound

$Q^{2}=4\pi\varepsilon_{0}GM^{2}$

for the extremal RN spacetime. Such a similarity suggests a comparison between
the Hayward class BHs and the RN BHs, which can be best illustrated by
questing the number of roots for the shape function $f(r)$.

Let us fix $s=3$ and rewrite the shape function (3.19) in terms of the
rescaled parameters
$R=r/GM_{g},~{}J=q/GM_{g},~{}\tilde{\alpha}=\alpha/G^{2}M_{g}^{2}$ as

$f(R,J)=1-\frac{2\left(1-\tilde{\alpha}^{-1}J^{3}\right)}{R}-\frac{2\tilde{\alpha}^{-1}J^{3}R^{2}}{R^{3}+J^{3}}.$
(5.2)

Meanwhile, the regularity condition (5.1) can be re-expressed using the
rescaled parameter as

$\displaystyle\tilde{\alpha}^{-1}J^{3}_{\rm reg}=1.$ (5.3)

Therefore, inserting $J=J_{\rm reg}$ into eq. (5.2), we get the shape function
for the regular Hayward BH,

$f(R,J_{\rm reg})=1-\frac{2R^{2}}{R^{3}+J^{3}_{\rm reg}}.$ (5.4)

Figure 2: the curve of $f(R,J_{\rm reg})$ with different value of $J_{\rm
reg}$.

It follows that the number of roots for $f(R,J_{\rm reg})$ varies according to
different choices of the parameter $J_{\rm reg}$. Fig. 2 depicts the curves of
$f(R,J_{\rm reg})$ versus $R$ for different choices of $J_{\rm reg}$. It turns
out that for $J_{\rm reg}>\dfrac{2}{3}\sqrt[3]{4}$, $f(R,J_{\rm reg})$ has no
real root, implying that the spacetime contains no event horizon. When $J_{\rm
reg}=\dfrac{2}{3}\sqrt[3]{4}$, $f(R,J_{\rm reg})$ has a single root. For
$0<J_{\rm reg}<\dfrac{2}{3}\sqrt[3]{4}$, $f(R,J_{\rm reg})$ has two distinct
real roots. Thus we see that in order that the shape function (5.4) describes
a regular black hole, the parameter $J_{\rm reg}$ needs to have an upper bound
which equals $\dfrac{2}{3}\sqrt[3]{4}$.

It is meaningful to rewrite $J_{\rm reg}$ in terms of the relative electric
charge $\mathcal{Q}$ defined in eq. (4.11),

$J_{\rm reg}=\frac{q}{GM_{\rm
reg}}=\frac{\alpha}{q^{2}}=\frac{6A\left(5A^{3/2}-1\right)}{\left(A^{3/2}+1\right)^{3}\mathcal{Q}}.$
(5.5)

Using this equation, the upper bound for $J_{\rm reg}$ can be turned into the
lower bound for $\mathcal{Q}$,

$\displaystyle\mathcal{Q}=\frac{6A\left(5A^{3/2}-1\right)}{J_{\rm
reg}\left(A^{3/2}+1\right)^{3}}\geq\frac{6A\left(5A^{3/2}-1\right)}{\frac{2}{3}\sqrt[3]{4}\left(A^{3/2}+1\right)^{3}}.$
(5.6)

Therefore, in order to have a regular black hole state, the spacetime needs to
be sufficiently charged.

Now let us proceed to study the thermodynamic stability near the regular
state. First of all, let us substitute eq.(4.1) together with the regularity
condition (5.1) into the equation $f(r_{H})=0$ and solve the result for
$\alpha$. This gives

$\alpha=\frac{2q^{3}r_{\rm H}^{2}}{r_{\rm H}^{3}+q^{3}}=\frac{L^{2}}{\pi
N}\frac{2Q^{3/2}S}{S^{3/2}+Q^{3/2}}=\mathrm{const}.$ (5.7)

Inserting this result into eq.(4.3) we get

$\displaystyle T_{\rm reg}$
$\displaystyle=\frac{N^{1/2}}{4\pi^{1/2}L\,S^{1/2}}\frac{S^{3/2}-2Q^{3/2}}{S^{3/2}+Q^{3/2}}=\frac{N^{1/2}}{4\pi^{1/2}L\,S^{1/2}}\frac{X-2}{X+1},$
(5.8)

where the dimensionless parameter $X$ is defined as

$X\equiv\frac{S^{3/2}}{Q^{3/2}}=\left(\frac{\mathcal{S}A}{\mathcal{Q}}\right)^{3/2}.$

The positivity of the temperature requires $X>2$.

It is important to notice that the heat capacity of the RBHs should not be
calculated using eq. (5.8), because, due to the constraint (5.7), $S$ is not
independent of $Q$ in the regular states. To calculate the heat capacity of
the regular states, one should employ eq. (4.3) to calculate the partial
derivative $\displaystyle\left(\frac{\partial T}{\partial S}\right)_{Q,N}$,
and then evaluate the result at the regular states. The final result reads

$\displaystyle\left.\left(\frac{\partial T}{\partial
S}\right)_{Q,N}\right|_{\rm reg}$
$\displaystyle=-\frac{N^{1/2}}{8\pi^{1/2}L\,S^{3/2}}\frac{S^{3}-13S^{3/2}Q^{3/2}+4Q^{3}}{\left(S^{3/2}+Q^{3/2}\right)^{2}}$
$\displaystyle=-\frac{N^{1/2}}{8\pi^{1/2}L\,S^{3/2}}\frac{X^{2}-13X+4}{(X+1)^{2}}.$
(5.9)

It is evident that the thermodynamic stability of the RBHs requires
$X^{2}-13X+4<0$. Combined with the requirement $X>2$, we find that the RBHs
are thermodynamically stable only when

$\displaystyle 2<X<X_{0}\equiv\dfrac{13+\sqrt{153}}{2}.$ (5.10)

Combining eqs. (4.7) with (5.7), we get

$\displaystyle\frac{\mathcal{S}\mathcal{Q}^{3/2}}{\mathcal{S}^{3/2}A^{3/2}+\mathcal{Q}^{3/2}}=\frac{3\left(5A^{3/2}-1\right)}{\left(A^{3/2}+1\right)^{3}}.$
(5.11)

This gives an implicit relationship between the relative entropy and the
relative charge at the regular states. It can be seen that, as long as the
bound (5.6) is not saturated, eq.(5.7) regarded as an algebraic equation for
$\mathcal{S}$ possesses two positive roots, among which only the bigger one
corresponds to the black hole event horizon. Consequently, there is precisely
one regular state on each isocharge $\mathcal{T}-\mathcal{S}$ curve.

For the bigger root of $\mathcal{S}$, it can be checked that $X$ is single-
valued in $\mathcal{Q}$ and is monotonically decreasing as $\mathcal{Q}$
decreases. After some algebra, the thermodynamic stability condition (5.10)
can be turned into the following constraint condition over $\mathcal{Q}$,

$\displaystyle\dfrac{6A\left(5A^{3/2}-1\right)}{\sqrt[3]{32/27}\left(A^{3/2}+1\right)^{3}}\equiv\mathcal{Q}_{L}<\mathcal{Q}<\mathcal{Q}_{U}\equiv\dfrac{X_{0}+1}{X_{0}^{2/3}}\dfrac{3A\left(5A^{3/2}-1\right)}{\left(A^{3/2}+1\right)^{3}}.$
(5.12)

Both the upper and lower bounds for $\mathcal{Q}$ can be approximated by float
point numbers, i.e.

$\displaystyle\mathcal{Q}_{L}\simeq 3.09133,\quad\mathcal{Q}_{U}\simeq
4.11553.$ (5.13)

(a) $\mathcal{T}-\mathcal{S}$ curves with regular states marked

(b) $\mathcal{F}-\mathcal{T}$ curves with regular states marked

Figure 3: $\mathcal{T}-\mathcal{S}$ (3(a)) and $\mathcal{F}-\mathcal{T}$
(3(b)) curves of Hayward BHs with regular states marked. The curves are
intentionally zoomed-in in order to have a better resolution to show the exact
position of the regular states on the curves. Figure 4:
$\mathcal{T}-\mathcal{S}$ process between regular states cannot be isocharge
process.

Figs. 3(a) and 3(b) depict the $\mathcal{T}-\mathcal{S}$ and
$\mathcal{F}-\mathcal{T}$ curves for certain choices of $\mathcal{Q}$ with the
regular states marked by a cross ($\times$). The selected values of
$\mathcal{Q}$ are such that some of the values lie in between
$\mathcal{Q}_{L}$ and $\mathcal{Q}_{U}$ and some exceed $\mathcal{Q}_{U}$. It
can be seen that for those $\mathcal{Q}$ that lie in between the bounds, the
corresponding regular states appear on the increasing segment of the
$\mathcal{T}-\mathcal{S}$ curve, signifying thermodynamic stability. On the
other hand, for those $\mathcal{Q}$ that exceed the upper bound
$\mathcal{Q}_{U}$, the corresponding regular states appear on the decreasing
segment of the $\mathcal{T}-\mathcal{S}$ curve, signifying thermodynamic
instability. Similar conclusion can also be drown from the
$\mathcal{F}-\mathcal{T}$ curves, on which stable regular states appear on the
lowest branch of $\mathcal{F}$, while the unstable ones do not.

Due to the non-independence between $\mathcal{S}$ and $\mathcal{Q}$ for the
regular states, any $\mathcal{T}-\mathcal{S}$ process leading from one regular
state to another cannot be an isocharge process. This is illustrated in Fig.
4.

Before closing this section, let us remark that, according to eqs. (5.3) and
(5.5), we have, at the regular states,

$\displaystyle\tilde{\alpha}=J^{3}_{\rm
reg}=\left(\frac{6A\left(5A^{3/2}-1\right)}{\left(A^{3/2}+1\right)^{3}\mathcal{Q}}\right)^{3}.$
(5.14)

The thermodynamic stability condition for the regular states can be
reformulated by use of the parameter $\tilde{\alpha}$, i.e.

$\displaystyle\tilde{\alpha}_{L}<\tilde{\alpha}<\tilde{\alpha}_{U},$ (5.15)

wherein

$\displaystyle\tilde{\alpha}_{L}=\dfrac{16(13+\sqrt{153})^{2}}{(15+\sqrt{153})^{3}}\approx
0.50228,\quad\alpha_{U}=\frac{32}{27}\approx 1.18519.$ (5.16)

The bound (5.15) follows from the substituting eq.(5.14) into (5.12). For
$\tilde{\alpha}$ in between the above bounds, the corresponding regular state
will be thermodynamically stable.

##  6 Quasinormal frequencies and dynamic stability

Besides thermodynamic stability, the dynamic stability of BHs is also
important in order to have a full understanding about the behaviors of BHs
under perturbation. The standard way for analyzing dynamic stability of BHs is
to calculate the quasinormal frequencies (QNFs) of the BHs under small
perturbation. Such analysis has been done for some of the known RBHs in Refs.
[39, 40, 41, 42].

In this section, we will use the 13th-order WKB method under the Padé
approximation [43, 44] to calculate the quasinormal frequencies (QNFs) of the
electrically charged Hayward class BHs under the perturbation of a massless
scalar field. We assume that the only parameter which changes as the BH
evolves is $J$, which, for fixed black hole mass, corresponds to the change in
charge. What we actually do is to calculate the QNFs for the whole class of
BHs with the shape function (5.2) at fixed values of $\tilde{\alpha}$ without
imposing the regularity condition (5.3), and then look at the results at the
particular choices of $J$ which obey the regularity condition. This makes a
difference from previous works [39, 40, 41, 42], because those works
considered exclusively the QNFs within the regular BH configurations.

The massless Klein-Gordon equation reads

$\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi=0.$
(6.1)

Since the spacetime is spherically symmetric, it is natural to make a
separation of variables which takes the spherical harmonic functions
$Y_{lm}(\theta,\varphi)$ as a factor,

$\Psi(t,r,\theta,\varphi)=\mathrm{e}^{-\mathrm{i}\omega_{l}t}Y_{lm}(\theta,\varphi)\frac{\phi_{l}(r)}{r}.$
(6.2)

Then the Klein-Gordon equation can be reduced into the Schödinger-like
equation for the radial factor $\phi_{l}(r)$, which, for brevity, will be
written simply as $\phi(r)$:

$\frac{\mathrm{d}^{2}\phi}{\mathrm{d}r_{*}^{2}}+\omega^{2}\phi=V_{\text{eff}}(r)\phi,$
(6.3)

where the Regge-Wheeler coordinate $r_{*}$ is defined via
$\dfrac{\mathrm{d}r_{*}}{\mathrm{d}r}=\dfrac{1}{f(r)}$, and the effective
potential takes the form

$V_{\mathrm{eff}}(r)=f(r)\left[\frac{l(l+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}\right].$
(6.4)

The quasinormal modes are defined by the solution of eq. (6.3) obeying the
boundary conditions

$\phi(r)\sim\mathrm{e}^{\pm\mathrm{i}\omega r_{*}},\quad
r_{*}\rightarrow\pm\infty.$ (6.5)

In previous models studies on RBHs, there is also an inherent correlation
between the thermodynamics and dynamics [45, 46]. Some of studies indicates
that the thermodynamic stability and dynamic stability can be different for
the same BH solution with the same parameter set. We would like to check
whether such phenomena also occurs for the regular states. For this purpose,
we take two different values for the parameter $\tilde{\alpha}$, i.e.
$\tilde{\alpha}=1$ in the thermodynamically stable range and
$\tilde{\alpha}=0.1$ in the thermodynamically unstable range, and study the
corresponding dynamic stability.

(a) $\real(\omega)$ versus $J$

(b) $-\imaginary(\omega)$ versus $J$

(c) QNFs in the complex plane

Figure 5: QNFs at different values of $l$ with $\tilde{\alpha}=1$. The arrows
in Panel 5(a) and 5(b) point to the increasing direction of $l$, which ranges
from $1$ to $10$.

It should be reminded that for fixed $\tilde{\alpha}$, the event horizon
equation $f(R,J)=0$ implies that there is an upper bound for $J$. Therefore,
any $\omega-J$ curve should have an end point at $J=J_{\rm max}$.

Fig. 5(a) and Fig. 5(b) respectively depict the real and imaginary parts of
the QNFs as functions in $J$ at different values of $l$, with the regular
states marked by round dots. It can be seen that the imaginary parts of the
QNFs at the regular states are always negative, signifying dynamic stability.
Meanwhile, away from the regular states, the real part of the QNFs increases
with $J$ at fixed $l$, and also with $l$ at fixed $J$, while the absolute
value of the negative imaginary part decreases with $J$ at fixed $l$, and also
with $l$ at fixed $J$.

Fig. 5(c) depicts the QNFs on the complex plane with the imaginary axis turned
upside down. One can see that for sufficiently large $l$, the upper and lower
bounds for the imaginary parts gradually become constant.

(a) $\real(\omega)$ versus $J$

(b) $-\imaginary(\omega)$ versus $J$

(c) QNFs in the complex plane

Figure 6: QNFs at different values of $l$ with $\tilde{\alpha}=0.1$. The
arrows in Panel 6(a) and 6(b) point to the increasing direction of $l$, which
ranges from $1$ to $10$.

One might expect that something different might happen when $\tilde{\alpha}$
takes values beyond the scope (5.15). However, the fact is not the case, as is
shown in Fig. 6, which are parallel to Fig. 5, but with $\tilde{\alpha}=0.1$.
The negativity of the imaginary parts of the QNFs as depicted in Fig. 6(b)
indicates that, the regular states in this case, although being
thermodynamically unstable, are actually dynamically stable under massless
scalar perturbations. Besides, all qualitative behaviors of the QNFs with
$\tilde{\alpha}=0.1$ are basically identical to the case with
$\tilde{\alpha}=1$. Therefore, our work indicates once again that, for the
same BH solution with the same parameter sets, the thermodynamic stability and
the dynamic stability need not to be identical.

##  7 Conclusion and outlook

The RPS formalism in which the Euler relation is put at the center position
appears to hold for RBHs. However, the requirement of parameter independence
in BH thermodynamics calls for the view of treating RBHs as intermediate
thermodynamic states in a larger class of BHS involving both regular and
singular states. This idea is verified in several classes of BHs incolving
regular states, including the Bardeen(-AdS) class, Hayward class and a class
of BH solutions of the model introduced in [23]. The idea that RBHs should be
viewed as intermediate states has not been considered seriously before, and
thus our approach is totally different from the previous researches such as
[17, 19, 20] on the thermodynamics of RBHs.

Following our proposal, it is shown that, for the electrically charged Hayward
class BHs, the regular states may be thermodynamically either stable or
unstable, depending on the amount of charges carried by the BHs. However, by
studying the QNFs of the whole Hayward class BHs, it is shown that, even for
the thermodynamically unstable regular states, the dynamic stability can still
hold, at least under massless scalar perturbations. This gives further
evidence to the view that the thermodynamic and dynamical stabilities for the
same BH solution with the same parameter set need not to be identical.

Last but not the least, the successful application of the RPS formalism to the
several classes of BHs involving regular states indicates the strength and
wide applicability of this formalism itself, which seems to be worth of
further explorations.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under Grant No. 12275138.

## Data Availability Statement

This research makes no use of new data. The QNFs analyzed in Section 6 is
completely based on numerical solutions to eq.(6.3), for which the procedure
is standard.

## Declaration of competing interest

The authors declare no competing interest.

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