# Regular black holes: A short topic review

Chen Lan<EMAIL_ADDRESS>Department of Physics, Yantai University, 30
Qingquan Road, Yantai 264005, China Hao Yang<EMAIL_ADDRESS>School
of Physics, Nankai University, 94 Weijin Road, Tianjin 300071, China Yang Guo
<EMAIL_ADDRESS>School of Physics, Nankai University, 94 Weijin Road,
Tianjin 300071, China Yan-Gang Miao Corresponding author:
<EMAIL_ADDRESS>School of Physics, Nankai University, 94 Weijin Road,
Tianjin 300071, China

###### Abstract

The essential singularity in Einstein’s gravity can be avoidable if the
preconditions of Penrose’s theorem can be bypassed, i.e., if the strong energy
condition is broken in the vicinity of a black hole center. The singularity
mentioned here includes two aspects: (i) the divergence of curvature
invariants, and (ii) the incompleteness of geodesics. The both aspects are now
taken into account in order to determine whether a black hole contains
essential singularities. In this sense, black holes without essential
singularities are dubbed regular (non-singular) black holes. The regular black
holes have some intriguing phenomena that are different from those of singular
black holes, and such phenomena have inspired numerous studies. In this
review, we summarize the current topics that are associated with regular black
holes.

###### Contents

  1. 1 Introduction
  2. 2 Construction of regular black holes
    1. 2.1 How to construct non-rotating regular black holes?
      1. 2.1.1 The case with one shape function
      2. 2.1.2 The case with two shape functions
    2. 2.2 How many curvature invariants do we need to define a regular black hole?
    3. 2.3 How to construct rotating regular black holes?
      1. 2.3.1 What are the problems faced by the Newman-Janis algorithm?
      2. 2.3.2 How to modify the Newman-Janis algorithm?
      3. 2.3.3 What are the regularity conditions of rotating regular black holes?
  3. 3 Interpretation of regular black holes
    1. 3.1 How to understand regular black holes correctly?
    2. 3.2 How to find the sources of non-rotating regular black holes?
    3. 3.3 What are the difficulties for us to find the sources of rotating regular black holes?
    4. 3.4 Can regular black holes have scalar hairs?
  4. 4 Energy conditions of regular black holes
    1. 4.1 Is the strong energy condition a key to lead to a regular black hole?
    2. 4.2 What are the energy conditions of regular black holes?
  5. 5 Thermodynamics of regular black holes
    1. 5.1 What is the entropy of regular black holes?
    2. 5.2 What is the correct first law of thermodynamics for regular black holes?
  6. 6 Rugular black hole chemistry and thermodynamic geometry
    1. 6.1 What is the regular black hole chemistry?
      1. 6.1.1 Thermodynamic phase transition and shift of critical points
      2. 6.1.2 Regular black hole as a heat engine
    2. 6.2 How to eliminate the singularity of the thermodynamic geometry for regular black holes?
      1. 6.2.1 Construction of thermodynamic geometry
      2. 6.2.2 Singularity of thermodynamic geometry and elimination
  7. 7 Conclusion and outlook

##  1 Introduction

Regular black holes (RBHs) are a collection of black holes (BHs) that has
coordinate singularities (horizons) but lacks essential singularities in the
entire spacetime. In most cases, the strategy to determine a RBH refers [1, 2,
3] to the spacetime with finite curvature invariantsaaaFor instance, the Ricci
curvature $R=g^{\mu\nu}R_{\mu\nu}$, the contraction of two Ricci tensors
$R_{\mu\nu}R^{\mu\nu}$, and the Kretschmann scalar
$R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$. everywhere, particularly at the
BH center. This is related to Markov’s limiting curvature conjecture [4, 5, 6,
7], which states that the curvature invariants must be uniformly restricted by
a certain universal value. However, such a strategy fails [8, 9] in the well-
known Taub-NUT BH because the null and timelike geodesics are incomplete at
the horizon, which contradicts [10, 11] to the alternative strategy to
determine a regular spacetime based on the geodesic completeness.bbbAccording
to this strategy, a spacetime is regular if its null and timelike geodesics
are complete, i.e., the affine parameter of a test particle will not terminate
at any finite value. The strategy of complete geodesics also encounters [12,
13] counterexamples, see e.g. Refs. [14, 15], where the geodesics are complete
but the curvature invariants are divergent, which consequently contradicts to
Markov’s limiting curvature conjecture. In this sense, the two strategies
should be complementary to each other in order to judge RBHs.

The studies of RBHs can date back to Sakharov and Gliner’s works [16, 17],
where they stated that the essential singularities can be avoided if the
vacuum is replaced by a vacuum-like medium endowed with a de Sitter metric.
This idea has been developed further by Dynmnikova, Gurevich, and Starobinsky
[18, 19]. The first model of RBHs was implemented by Bardeen [20], now called
the Bardeen BH, which was constructed via simply replacing the mass of
Schwarzschild BHs with a $r$-dependent function. As a result, the essential
singularity of the Kretschmann scalar is removed in the Bardeen BH, meanwhile
the core of this BH is of de Sitter, i.e., the Ricci curvature is positive in
the vicinity of this BH center.

After three decades of Bardeen’s proposal, Ayón-Beato and García provided [21]
the first interpretation of the Bardeen BH in field theory, i.e., they
speculated a source, a magnetic monopole in the context of nonlinear
electrodynamics, which can lead to the Bardeen BH solution from Einstein’s
field equations. Recently, a large number of RBH models have been given in
this way. In particular, such an approach has been extended [3, 22, 23, 24,
25] to explain all RBH models with the spherical symmetry. It is different
from the usual way in finding BH solutions by solving Einstein’s field
equations. According to this approach, one writes the desired RBH and magnetic
monopole solutions at first, and then determines the corresponding action of
nonlinear electrodynamics. Because of such a special logic, i.e., from the
solutions to the action of matters, the RBHs have nontrivial (phantom) scalar
hairs [23, 26, 27]. Moreover, they are regarded to be classical objects as
they are the solutions of Einstein’s field equations.

Correspondingly, there are two different ways to construct RBH models: one is
to solve Einstein’s field equations that are associated with a kind of special
sources, e.g., the matters with spatial distributions [1, 28, 29, 30, 31, 32,
33, 34]; and the other is to derive RBHs as quantum corrections to the
classical BHs with singularity, e.g., the loop quantum gravity and asymptotic
safety method [35, 36, 37, 38, 39, 40, 41, 42, 43]. Based on the former way,
the RBHs behave semiclassically; whereas based on the latter, those RBH models
exhibit quantum behaviors.

Besides the structures of RBHs [44, 45, 46, 47, 48, 49, 50, 51], the study
also extends to the other areas of BH physics, including thermodynamics [22,
52, 53, 54, 55], dynamics [56, 57, 58, 59, 60, 61], shadows [62, 63, 64, 65,
66, 67, 68, 69, 70], quasinormal modes [56, 71, 72, 73], superradiance [74],
and synchrotron radiations [75], etc. Currently, the study of RBHs has made a
great progress in depth and breadth, which leads to the necessity for us to
summarize the new results in a systematic way. Comparing with the previous
reviews [19, 29, 76], we would like to gather the new progress developed
recently. This is the main motivation for drafting this review.

Our review is organized as follows. In Sec. 2, we address the issue on the
construction for both the non-rotating and rotating RBHs, where we also
discuss the number of curvature invariants among the Zakhary-Mcintosh
invariants for determining whether a BH is regular or not. Sec. 3 includes the
clarification for understanding RBHs and the establishment of the sources of
RBHs in terms of Petrov’s approach. We end this section by showing the
peculiarity of RBHs on scalar hairs that does not break the non-hair theorem.
In Sec. 4 we demonstrate the role played by the strong energy condition in
RBHs, and provide an illustration or a resolution of the issue on the
violation of the other energy conditions in RBHs. Sec. 5 is dedicated to the
thermodynamics of RBHs, where we give a discussion on the entropy-area law,
based on which the self-consistent first law of thermodynamics is given in
Sec. 5.2. In Sec. 6, we discuss the chemistry and thermodynamic geometry for
RBHs because they are the nontrivial extensions of interesting issues
associated with singular BHs (SBHs). Finally, we conclude in Sec. 7 with some
outlook.

##  2 Construction of regular black holes

In this section, we summarize the approaches for both the non-rotating and
rotating RBHs, meanwhile we analyze the minimum set of curvature invariants
that are needed to judge a RBH. Here the curvature invariants refer to the
Zakhary-Mcintosh (ZM) invariants [77, 78] that form a complete set of Riemann
invariants and contain seventeen elements. The reason that we adopt the ZM
invariants rather than the usual ones, such as the Ricci scalar and
Kretschmann scalar [79, 22, 6], will be explained in Sec. 2.2.

###  2.1 How to construct non-rotating regular black holes?

Despite the complexity of ZM invariants, the calculation of ZM invariants
becomes simple for those BHs with the spherical symmetry. The RBHs with the
spherical symmetry have two types of metrics: A metric in the first type
involves one shape function, see the specific square of line elements,

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f^{-1}(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},$
(1)

and a metric in the second type involves two shape functions, see the specific
square of line elements,

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f^{-1}(r)A^{2}(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},$
(2)

which is equivalent to

$\mathrm{d}s^{2}=-f(\xi)\mathrm{d}t^{2}+f^{-1}(\xi)\mathrm{d}\rho^{2}+r^{2}(\xi)\mathrm{d}\Omega^{2},$
(3)

where $\xi$ is a newly defined variable,

$\xi\equiv\int\mathrm{d}r\,A(r).$ (4)

In the following of this subsection, we shall give some restrictions to the
two types of metrics, which will reveal that the RBHs depicted by the two
types of metrics have finite curvature invariants.

####  2.1.1 The case with one shape function

It is quite general for us to write the shape function as follows,

$f(r)=1-\frac{2M\sigma(r)}{r},$ (5)

where $\sigma(r)$ is a function of the radial variable $r$ and $M$ is BH mass.
In order to observe the regularity, we expand $\sigma(r)$ by the power series
around $r=0$,

$\sigma(r)=\sigma_{1}r+\sigma_{2}r^{2}+\sigma_{3}r^{3}+O(r^{4}),$ (6)

where $\sigma_{i}$ are constant coefficients. Then, by substituting Eqs. (1),
(5) and (6) into the ZM invariants, we can find the conditions of finite
curvatures, that is, the coefficients $\sigma_{1}$ and $\sigma_{2}$ must
vanish,

$\sigma_{1}=0=\sigma_{2}.$ (7)

As examples, we write down the behaviors of three usual candidates among the
seventeen ZM invariants around $r=0$: The Ricci scalar $R$, the Weyl scalar
$W$, and the Kretschmann scalar $K$ have the asymptotic behaviors for RBHs,

$R=24M\sigma_{3}+O(r),\qquad W=O(r^{2}),\qquad K=96M^{2}\sigma_{3}^{2}+O(r).$
(8)

Alternatively, we can select three curvatures from the seventeen ZM invariants
and write $\sigma(r)$, $\sigma^{\prime}(r)$, and $\sigma^{\prime\prime}(r)$ as
functions of these three curvatures because the ZM invariants contain
$\sigma(r)$ and only its first and second order derivatives,
$\sigma^{\prime}(r)$ and $\sigma^{\prime\prime}(r)$. Then, requiring the
finiteness of the three curvatures, we can find the behavior of $\sigma(r)$
around the center $r=0$, i.e., $\sigma(r)$ should not decrease slower than
$r^{3}$ as $r$ approaches to zero [80], otherwise some of the ZM invariants
will diverge at $r=0$.

####  2.1.2 The case with two shape functions

For the RBHs with two shape functions [6, 23, 81, 82], we apply the similar
procedure to the above, that is, we expand the both of shape functions (see
Eq. (2)) by the power series,

$A(r)=A_{0}+A_{1}r+A_{2}r^{2}+O(r^{3}),$ (9a)
$f(r)=B_{0}+B_{1}r+B_{2}r^{2}+O(r^{3}).$ (9b)

After substituting Eqs. (2), (9a) and (9b) into the ZM invariants, we can find
the conditions for finite curvatures,

$A_{0}=B_{0},\qquad A_{1}=B_{1}=0,$ (10)

i.e., the first order of $r$ must be absent in the power expansions. We still
give three curvature invariants for RBHs as examples when $r$ goes to zero,

$R=\frac{6(A_{2}-2B_{2})}{A_{0}}+O(r),\qquad
W=O(r^{2}),\qquad\mathcal{S}=\frac{3A_{2}^{2}}{A_{0}^{2}}+O(r),$ (11)

where $\mathcal{S}$ and $\mathcal{S}_{\mu\nu}$ are defined [6] by
$\mathcal{S}\equiv\mathcal{S}^{\mu\nu}\mathcal{S}_{\mu\nu}$ and
$\mathcal{S}_{\mu\nu}\equiv R_{\mu\nu}-g_{\mu\nu}R/4$, respectively.

###  2.2 How many curvature invariants do we need to define a regular black
hole?

Generally, the finite curvature invariants and geodesic completeness are not
equivalent to each other,cccFor certain cases, these two conditions are
equivalent, e.g., for spherically symmetric BHs with one shape function, i.e.,
$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f^{-1}(r)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}$.
but they can be regarded as two independent necessary conditions for checking
whether a BH is regular. Moreover, the former is coordinate independent, while
the latter depends on the choice of coordinates. For instance, in the Rindler
spacetime [11, 83],
$\mathrm{d}s^{2}=-z^{2}\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}$,
the geodesics cannot be extended along the $z$-direction because the
corresponding affine parameter is finite at $z=0$. In other words, the point
$z=0$ acts as a singularity in this spacetime. However, after an appropriate
transformation,

$t\to\tanh^{-1}\frac{T}{Z},\qquad x\to X,\qquad y\to Y,\qquad
z\to\sqrt{Z^{2}-T^{2}},$ (12)

the original metric converts to that of the Minkowski spacetime,
$\mathrm{d}s^{2}=-\mathrm{d}T^{2}+\mathrm{d}X^{2}+\mathrm{d}Y^{2}+\mathrm{d}Z^{2}$,
i.e., there are no singularities anywhere. From this point of view, we can see
that the condition of finite curvature invariants has its advantage, i.e., it
does not need to be concerned about selecting appropriate coordinates.

Nevertheless, it is unclear how many curvatures have to be used in order to
determine a RBH. Usually, there are three candidatesdddAlternatively, $K$ and
$R_{2}$ are replaced by $W\equiv C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$,
the contraction of two Weyl tensors, and
$\mathcal{S}\equiv\mathcal{S}_{\mu\nu}\mathcal{S}^{\mu\nu}$, where
$\mathcal{S}_{\mu\nu}\equiv R_{\mu\nu}-g_{\mu\nu}R/4$ [6]. connected by the
Ricci decomposition, i.e., the Ricci scalar $R$, the Kretschmann scalar $K$
and the contraction of two Ricci tensors $R_{2}$. The question is whether $R$,
$K$ and $R_{2}$ are enough to reveal the singularities in all the seventeen
curvature invariants. The answer is of course negative. Let us see the well-
known Taub-NUT BH [84, 85] as a sample,

$\mathrm{d}s^{2}=-f(r)\left[\mathrm{d}t+2n\cos(\theta)\mathrm{d}\phi\right]^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+\zeta^{2}\left[\mathrm{d}\theta^{2}+\sin^{2}(\theta)\mathrm{d}\phi^{2}\right]$
(13)

with

$f(r)=\frac{\Delta}{\zeta^{2}},\qquad\Delta=r^{2}-2Mr-n^{2},\qquad\zeta=\sqrt{r^{2}+n^{2}},$
(14)

where $M$ is mass, $n$ called the NUT parameter is positive, and the horizon
is located at $r_{\rm H}=M+\sqrt{M^{2}+n^{2}}$. It is shown [9, 86] that the
geodesics are incomplete at the horizon.

We turn to the investigation of the curvature invariants of the Taub-NUT BH.
The Ricci tensor vanishes, $R_{\mu\nu}=0$. The Kretschmann scalar reads

$K\sim\frac{48\left(n^{2}-M^{2}\right)}{n^{6}}+O\left(r\right),$ (15)

which is finite as $r$ approaches to zero. Moreover, $R_{2}$ is also finite
when $r$ goes to zero. As a result, the three curvature invariants, $R$, $K$
and $R_{2}$ are finite in the Taub-NUT BH spacetime. But we cannot conclude
that the other ZM invariants are regular everywhere. In fact, there are two ZM
invariants that are divergent at the horizon. They are constructed [78] by the
Weyl tensor, $C_{\mu\nu\alpha\beta}$, and its dual,
$C^{*}_{\mu\nu\alpha\beta}=\epsilon_{\mu\nu\rho\sigma}C^{\rho\sigma}_{\;\;\;\alpha\beta}/2$,
as follows:

$I_{2}=-C_{\mu\nu}^{\;\;\alpha\beta}C_{\alpha\beta}^{*\;\;\mu\nu}=\frac{\Phi(r,\theta)}{\Delta^{2}\sin^{2}\theta},\qquad
I_{4}=-C_{\mu\nu}^{\;\;\alpha\beta}C_{\alpha\beta}^{*\;\;\gamma\zeta}C_{\gamma\zeta}^{\;\;\mu\nu}=\frac{\Psi(r,\theta)}{\Delta^{2}\sin^{2}\theta},$
(16)

where $\Phi(r,\theta)$ and $\Psi(r,\theta)$ are holomorphic functions of $r$
and $\theta$ and have no zeros at the horizon, $r_{\rm H}$, and at
$\theta=0,\uppi$. Since $I_{2}$ and $I_{4}$ are proportional to
$1/\Delta^{2}$, the essential singularity reappears at $r_{\rm H}$, which
coincides with the result deduced from the incomplete geodesics [86, 9].
Moreover, $I_{2}$ and $I_{4}$ reveal additional singularities at
$\theta=0,\;\uppi$, where $\theta=\uppi$ is known as the “Misner string” [86].

What we learn from this example is that the finiteness of the usual candidates
of curvature invariants is not sufficient in general, which leads to the
conclusion that a complete set of Riemann invariants is necessary from the
perspective of finite curvatures in order to determine whether there exist
essential singularities.

###  2.3 How to construct rotating regular black holes?

It is considerably difficult to obtain rotating RBH’s solutions from the
Einstein field equations because the complexity of Einstein’s field equations
in the case of rotation is much greater than that of the static case.
Therefore, the widely-used method for constructing rotating BHs is the Newman-
Janis algorithm (NJA) [87].

The NJA originated from the connection between a static BH and a rotating one
in general relativity. It is well known that the Schwarzschild, Reissner-
Nordström (RN), Kerr, and Kerr-Newman (KN) BHs were obtained by solving
Einstein’s field equations in vacuum. These solutions have clear physical
explanations. By comparing the metrics of these BHs, Newman and Janis proposed
the NJA to mathematically describe the transformation from spherically
symmetric Schwarzschild BHs to axially symmetric Kerr BHs. The algorithm can
also describe the transformation from RN BHs to KN BHs.

Subsequently, Gurses and Gursey extended [88] this algorithm for the Kerr-
Schild type of BHs, namely, the metric of one shape function mentioned in Sec.
2.1. Further, Drake and Szekeres generalized [89] the NJA for general
spherically symmetric BHs. Based on the above algorithms, many spherically
symmetric RBHs have been extended to their axially symmetric counterparts,
such as the noncommutative BHs [29, 90, 91], loop quantum corrected BHs [92,
93], Bardeen BHs [20, 94], and Hayward BHs [95, 94], etc.

####  2.3.1 What are the problems faced by the Newman-Janis algorithm?

The NJA faces the uncertainty of complex transformation of metrics. This
problem arises from the complex transformation of coordinates [96, 97, 98]:

$r\rightarrow r+\mathrm{i}a\cos\theta,\qquad u\rightarrow
u-\mathrm{i}a\cos\theta,$ (17)

where $(u,r,\theta,\varphi)$ are the advanced null coordinates and $a$ is the
rotation parameter. According to the transformation, we need to convert a
static and spherically symmetric metric function into a rotational and axially
symmetric function, and ensure that the latter is real but not complex. As a
result, this conversion of metric functions must follow certain rules.
However, such rules are ambiguous in the NJA.

The commonly-used rule is obtained by comparing the RN metric with the KN
metric. The $tt$-component of the RN metric reads

$g_{(\rm{RN})tt}=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}},$ (18)

where $M$ is mass and $q$ is charge. The $tt$-component of the KN metric takes
the form,

$g_{(\rm{KN})tt}=1-\frac{2Mr}{r^{2}+a^{2}\cos^{2}\theta}+\frac{q^{2}}{r^{2}+a^{2}\cos^{2}\theta}.$
(19)

The conversion rule is as follows:

$r^{2}=rr^{*}\rightarrow(r+ia\cos\theta)(r-ia\cos\theta)=r^{2}+a\cos^{2}\theta,$
(20)
$\frac{1}{r}=\frac{1}{2}\left(\frac{1}{r}+\frac{1}{r^{*}}\right)\rightarrow\frac{1}{2}\left(\frac{1}{r+ia\cos\theta}+\frac{1}{r-ia\cos\theta}\right)=\frac{r}{r^{2}+a^{2}\cos^{2}\theta}.$
(21)

However, the above conversion rule may not be applicable due to the complexity
of RBH metrics. Here we take the black-bounce spacetime [81] as an example,
where its metric’s $tt$-component reads

$g_{(\rm{BB})tt}=1-\frac{2M}{\sqrt{r^{2}+l^{2}}},$ (22)

where $l$ is the regularization parameter. When $l$ vanishes, the metric
becomes the Schwarzschild’s. Therefore, the rotation formulation of this
metric should reduce to the Kerr metric when $l=0$. But this is not the case.
The rotating black-bounce metric under the above conversion rule takes the
form,

$g_{(\rm{rBB})tt}=1-\frac{2M}{\sqrt{r^{2}+a^{2}\cos^{2}\theta+l^{2}}},$ (23)

whereas the $tt$-component of the Kerr metric is

$g_{(\rm{K})tt}=1-\frac{2Mr}{r^{2}+a^{2}\cos^{2}\theta}.$ (24)

Obviously, $g_{(\rm{rBB})tt}$ does not reduce to $g_{(\rm{K})tt}$ when $l=0$.
As a result, the above conversion rule does not apply to the black-bounce
spacetime. Due to the failure of this rule, we need to find such a rule that
is applicable to more models. Furthermore, the ambiguity related to coordinate
transformations leads to difficulties in the generalization of the NJA.

####  2.3.2 How to modify the Newman-Janis algorithm?

In order to avoid the ambiguity caused by the complex transformation, Azreg-
Ainou modified [97, 96] the NJA as follows.

For a general static metric,

$\mathrm{d}s^{2}=-G(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{F(r)}+H(r)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$
(25)

one introduces the advanced null coordinates $(u,r,\theta,\varphi)$ defined by

$\mathrm{d}u=\mathrm{d}t-\frac{\mathrm{d}r}{\sqrt{FG}},$ (26)

and expresses the contravariant form of the metric in terms of a null tetrad,

$g^{\mu\nu}=-l^{\mu}n^{\nu}-l^{\nu}n^{\mu}+m^{\mu}m^{*\nu}+m^{\nu}m^{*\mu},$
(27)

where

$l^{\mu}=\delta^{\mu}_{r},$ (28a)
$n^{\mu}=\sqrt{\frac{F}{G}}\delta^{\mu}_{u}-\frac{F}{2}\delta^{\mu}_{r},$
(28b)
$m^{\mu}=\frac{1}{\sqrt{2H}}\left(\delta^{\mu}_{\theta}+\frac{i}{\sin\theta}\delta^{\mu}_{\varphi}\right),$
(28c)
$l_{\mu}l^{\mu}=m_{\mu}m^{\mu}=n^{\nu}n_{\nu}=l_{\mu}m^{\mu}=n_{\mu}m^{\mu}=0,$
(28d) $l_{\mu}n^{\mu}=-m_{\mu}m^{*\mu}=1.$ (28e)

Then, one introduces the rotation via the complex transformation, Eq. (17),
under which $\delta^{\mu}_{\nu}$ transform as vectors:

$\delta^{\mu}_{r}\rightarrow\delta^{\mu}_{r},\qquad\delta^{\mu}_{u}\rightarrow\delta^{\mu}_{u},\qquad\delta^{\mu}_{\theta}\rightarrow\delta^{\mu}_{\theta}+ia\sin\theta(\delta^{\mu}_{u}-\delta^{\mu}_{r}),\qquad\delta^{\mu}_{\varphi}\rightarrow\delta^{\mu}_{\varphi}.$
(29)

For a SBH, the metric function of its rotating counterpart can be determined
under the above complex transformation. However, such a transformation does
not work well for a RBH as we discussed in Sec. 2.3.1. Thus, one assumes that
$\\{G,F,H\\}$ transform to $\\{A,B,\Psi\\}$:

$\\{G(r),F(r),H(r)\\}\rightarrow\\{A(r,\theta,a),B(r,\theta,a),\Psi(r,\theta,a)\\}$
(30)

where $\\{A,B,\Psi\\}$ are real functions to be determined, and they should
recover their static counterparts in the limit $a\rightarrow 0$, namely,

$\lim_{a\to 0}A(r,\theta,a)=G(r),\qquad\lim_{a\to
0}B(r,\theta,a)=F(r),\qquad\lim_{a\to 0}\Psi(r,\theta,a)=H(r).$ (31)

According to Eq. (29) and Eq. (30), the null tetrad becomes

$l^{\mu}=\delta^{\mu}_{r},$ (32a)
$n^{\mu}=\sqrt{\frac{B}{A}}\delta^{\mu}_{u}-\frac{B}{2}\delta^{\mu}_{r},$
(32b)
$m^{\mu}=\frac{1}{\sqrt{2\Psi}}\left[\delta^{\mu}_{\theta}+ia\sin\theta(\delta^{\mu}_{u}-\delta^{\mu}_{r})+\frac{i}{\sin\theta}\delta^{\mu}_{\varphi}\right],$
(32c)

and the corresponding metric with rotation takes the form,

$\begin{split}\mathrm{d}s^{2}=&-A\mathrm{d}u^{2}-2\sqrt{\frac{A}{B}}\mathrm{d}u\mathrm{d}r-2a\sin^{2}\theta\left(\sqrt{\frac{A}{B}}-A\right)\mathrm{d}u\mathrm{d}\varphi+2a\sin^{2}\theta\sqrt{\frac{A}{B}}\mathrm{d}r\mathrm{d}\varphi\\\
&+\Psi\mathrm{d}\theta^{2}+\sin^{2}\theta\left[\Psi+a^{2}\sin^{2}\theta\left(2\sqrt{\frac{A}{B}}-A\right)\right]\mathrm{d}\varphi^{2}.\end{split}$
(33)

Next, one rewrites the above metric with the Boyer-Lindquist coordinates, and
let the metric have only one off-diagonal term $g_{t\phi}$. To reach the aim,
one needs the following coordinate transformation,

$\mathrm{d}u=\mathrm{d}t+\lambda(r)\mathrm{d}r,\quad\mathrm{d}\varphi=\mathrm{d}\phi+\chi(r)\mathrm{d}r,$
(34)

where $\\{\lambda(r),\chi(r)\\}$ must depend only on $r$ to ensure
integrability. If the transformation Eq. (30) is a priori determined,
$\\{\lambda(r),\chi(r)\\}$ may not exist. Considering these constraints, one
has the formulations of
$\\{A(r,\theta,a),B(r,\theta,a),\lambda(r),\chi(r)\\}$,

$A(r,\theta)=\frac{(FH+a^{2}\cos^{2}\theta)\Psi}{(K+a^{2}\cos^{2}\theta)^{2}},$
(35a) $B(r,\theta)=\frac{FH+a^{2}\cos^{2}\theta}{\Psi},$ (35b)
$\lambda(r)=-\frac{K+a^{2}}{FH+a^{2}},$ (35c) $\chi(r)=-\frac{a}{FH+a^{2}},$
(35d)

where $K(r)$ is defined by

$K(r)\equiv\sqrt{\frac{F(r)}{G(r)}}H(r).$ (36)

As a result, one obtains the metric for rotating RBHs with the Kerr-like form,

$\mathrm{d}s^{2}=\frac{\Psi}{\rho^{2}}\left[-\left(1-\frac{2f}{\rho^{2}}\right)\mathrm{d}t^{2}+\frac{\rho^{2}}{\Delta}\mathrm{d}r^{2}-\frac{4af\sin^{2}\theta}{\rho^{2}}\mathrm{d}t\mathrm{d}\phi+\rho^{2}\mathrm{d}\theta^{2}+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}\mathrm{d}\phi^{2}\right],$
(37)

where

$\begin{split}&\rho^{2}\equiv K+a^{2}\cos^{2}\theta,\qquad 2f(r)\equiv K-FH\\\
&\Delta(r)\equiv
FH+a^{2},\qquad\Sigma\equiv(K+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta.\end{split}$
(38)

In the above metric, $\Psi(r,\theta,a)$ remains unknown and may be determined
by some specific physical interpretations. For example, if the source is
interpreted as an imperfect fluid rotating about the $z$ axis, $\Psi$ obeys
[96] the Einstein field equations,

$\left(K+a^{2}y^{2}\right)^{2}\left(3\Psi_{,r}\Psi_{,y^{2}}-2\Psi\Psi_{,ry^{2}}\right)=3a^{2}K_{,r}\Psi^{2},$
(39)
$\left[K_{,r}^{2}+K(2-K_{,rr})-a^{2}y^{2}(2+K_{,rr})\right]\Psi+(K+a^{2}y^{2})(4y^{2}\Psi_{,y^{2}}-K_{,r}\Psi_{,r})=0$
(40)

where “,” is the indexical notation for derivatives and $y\equiv\cos\theta$.
However, it is almost impossible to determine $\Psi(r,\theta,a)$ in this way
because of the high complexity. For the RBH metrics mentioned in Sec. 2.1, one
usually chooses

$\Psi(r,\theta,a)=H(r)+a^{2}\cos^{2}\theta.$ (41)

While doing so may lose a reasonable physical explanation, it needs to be
tested case by case whether such a choice really loses a physical explanation.
It is worth mentioning that Eq. (41) is compatible with the NJA and is
available to construct a rotating RBH, but it is still unclear whether Eq.
(41) is the only choice.

####  2.3.3 What are the regularity conditions of rotating regular black
holes?

For the seed metric with one shape function, the metric of rotating RBHs takes
the form via the NJA,

$\mathrm{d}s^{2}=-\frac{\Delta}{\rho^{2}}(\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\phi)^{2}+\frac{\rho^{2}}{\Delta}\mathrm{d}r^{2}+\rho^{2}\mathrm{d}\theta^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[a\mathrm{d}t-(r^{2}+a^{2})\mathrm{d}\phi\right]^{2},$
(42)

where

$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,\qquad\Delta=r^{2}-2M\sigma(r)r+a^{2}.$
(43)

This metric reduces to the Kerr metric when $\sigma(r)=1$, and to the KN
metric when $\sigma(r)=1-q^{2}/(2Mr)$.

Further, the metric Eq. (42) belongs [99, 76] to the Petrov type D because
$\Psi_{2}$ can be the only non-vanishing scalar when
$\Psi_{0},\Psi_{1},\Psi_{3}$ and $\Psi_{4}$ vanish simultaneously, where the
five complex scalar functions can be expressed by the Weyl tensor
$C_{\kappa\lambda\mu\nu}$ as follows:

$\Psi_{0}=C_{\kappa\lambda\mu\nu}l^{\kappa}m^{\lambda}l^{\mu}m^{\nu},$ (44a)
$\Psi_{1}=C_{\kappa\lambda\mu\nu}l^{\kappa}k^{\lambda}l^{\mu}m^{\nu},$ (44b)
$\Psi_{2}=C_{\kappa\lambda\mu\nu}l^{\kappa}m^{\lambda}m^{*\mu}k^{\nu},$ (44c)
$\Psi_{3}=C_{\kappa\lambda\mu\nu}k^{\kappa}l^{\lambda}k^{\mu}m^{*\nu},$ (44d)
$\Psi_{4}=C_{\kappa\lambda\mu\nu}k^{\kappa}m^{*\lambda}k^{\mu}m^{*\nu}.$ (44e)

Therefore, the algebraically complete set of second order invariants is
$\\{R,I,I_{6},K\\}$ [77, 99, 76], which means that Eq. (42) is regular if the
set of invariants does not diverge anywhere. The definitions of $R$ and $K$
have been introduced in Sec. 2.2, and the definitions of $I$ and $I_{6}$ are

$I\equiv\frac{1}{24}C^{*}_{\alpha\beta\gamma\delta}C^{*\alpha\beta\gamma\delta},$
(45)
$I_{6}\equiv\frac{1}{12}{\mathcal{S}_{\alpha}}^{\beta}{\mathcal{S}_{\beta}}^{\alpha},$
(46)

which also belong to the seventeen ZM invairants. According to the set of
invariants, one can deduce the necessary and sufficient condition for the
regularity of Eq. (42): If $\sigma(r)$ is a $C^{3}$ function, then it demands

$\sigma(0)=0,\qquad\sigma^{\prime}(0)=0,\qquad\sigma^{\prime\prime}(0)=0.$
(47)

For the seed metric with two shape functions, a general analytical method is
still lacking for the regularity conditions of rotating RBHs, and what one can
do is to verify the regularity only by calculating $R$ and $K$ in most cases
[100, 101, 102].

##  3 Interpretation of regular black holes

A complete RBH theory refers to physical interpretations from either quantum
theory of gravity, such as loop quantum gravity and asymptotic safety, or the
construction of gravitational sources in the context of classical field
theory. In this section, we explain RBHs from the perspective of coordinate
transformations at first, and then we summarize the techniques for
constructing gravitational sources for both non-rotating and rotating RBHs. At
the end of this section, we give a short discussion on the scalar hair of RBHs
because it relates to the classical field interpretations.

###  3.1 How to understand regular black holes correctly?

It is a confusing issue whether RBHs exist in nature or they are just
mathematical tricks, which was emphasized in Refs. [81, 103], where a RBH is
constructed by a seeming “coordinate transformation”. To make the following
discussions clear, we ask the question in another way:

> Is a RBH a full spacetime, or is it just a “good” coordinate system that
> does not cover the entire spacetime in the radial direction?

Let us examine the Schwarzschild BH as an example to see the essence of the
above question, where the Penrose diagram is shown in Fig. 1(a).

(a) $r\in[0,\infty)$.

(b) $\xi\in[0,\infty)$

Figure 1: Penrose diagrams of a Schwarzschild BH in two different radial
coordinates.

In order to construct a RBH, we drag the coordinate system downward by a
transformation $r\to r(\xi)$, in such a way that the coordinate system cannot
cover the singularity after the operation, see Fig. 1(b), where
$r\to\sqrt{\xi^{2}+l^{2}}$ with $l>0$. In other words, we put the singularity
$r=0$ down on the “non-physical” domain in the new coordinate system if the
new radial coordinate $\xi$ is defined in $\xi\in[0,\infty)$, i.e., the
singularity is dragged to the imaginary axis after analytically continuing
$\xi$ into a complex plane. This operation is equivalent to restricting the
old radial coordinate in $r\in[l,\infty)$ by hands. The singularity is
subtracted from the “old” spacetime, such that the “new” spacetime is regular.
In other words, the “new” coordinates cover a smaller portion of manifold than
the old coordinates in the Schwarzschild spacetime, but the topology of
manifold never changes. We call the Schwarzschild BH in $\xi$ as a fake RBH.

Now let us see a real RBH which does not redisplay the singularity by a
transformation. Taking the Bardeen BH as an example [20],

$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+f^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},\qquad
f=1-\frac{2Mr^{2}}{(r^{2}+g^{2})^{3/2}},$ (48)

where $M$ is mass and $g$ magnetic charge of monopoles, we know that the
Kretschmann scalar is regular in $r\in[0,\infty)$. After a replacement,
$r^{2}\to\xi^{2}-g^{2}$, the metric Eq. (48) becomes

$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+\frac{f^{-1}\xi^{2}}{\xi^{2}-g^{2}}\mathrm{d}\xi^{2}+(\xi^{2}-g^{2})\mathrm{d}\Omega^{2},\qquad
f=1-\frac{2M(\xi^{2}-g^{2})}{\xi^{3}}.$ (49)

If we take Eq. (49) as an independent metric describing a “new” spacetime, the
corresponding Kretschmann scalar diverges around $\xi=0$,

$K\sim\frac{900g^{8}M^{2}}{\xi^{14}}+O\left(\frac{1}{\xi^{13}}\right).$ (50)

It seems that the Bardeen BH redisplays the singularity at $\xi=0$, and even
that almost all the RBHs regain singularities by a replacement $r\to r(\xi)$.
However, this is not the case because $\xi$ could never be smaller than $g$,
otherwise the signature in Eq. (49) gets mess and the integral measure
$\sqrt{-g}$ becomes complex.

The above two examples reflect the fact that the singularity cannot be
resolved by a coordinate transformation, which is consistent with the essence
of singularities in BHs.

Further, if one analytically continues the radial coordinate to a complex
plane, the complex singularities may emerge again. Considering the Kretschmann
scalar of Bardeen BHs,

$K=\frac{12M^{2}\left(-4g^{6}r^{2}+47g^{4}r^{4}-12g^{2}r^{6}+8g^{8}+4r^{8}\right)}{\left(g^{2}+r^{2}\right)^{7}},$
(51)

we observe that the singularities are moved to the non-physical domain, e.g.,
$r=\pm\mathrm{i}g\in\mathbb{C}$. Generally, one can classify RBHs into three
types by the characteristics of singularities [73]:

  * •

The first type corresponds to those RBHs whose geodesics are complete in the
domain of $r\in[0,\infty)$ although their curvature invariants have essential
singularities at $r=0$ from the perspective of complex analysis, e.g.,
$\sigma(r)=\mathrm{e}^{-1/r}$, see Ref. [79];

  * •

The second type corresponds to those RBHs whose singularities of curvature
invariants are moved to the non-physical domain, e.g., the Bardeen and Hayward
BHs [104];

  * •

The third type corresponds to those RBHs whose curvature invariants have no
singularities on the entire complex plane, e.g., the noncommutative geometry
inspired BH [28].

This classification directly affects the calculation on the asymptotic
frequencies of quasinormal modes (QNMs) by the monodromy method [73].

###  3.2 How to find the sources of non-rotating regular black holes?

Gliner discussed [17] an algebraic property of a four-dimensional energy-
momentum tensor (EMT) denoted by $[(1111)]$, where symbol $1$ corresponds one
diagonal component of the EMT and the parentheses imply equal components,eeeAs
in Ref. [17], we do not distinguish between time and space components by a
comma. see Refs. [84, 105] for the Segré notations. The matter with the
algebraic property $[(1111)]$, called a $\mu$-vacuum, has a de Sitter-like
metric and thus avoids singularities. Later, Gliner’s work was extended [1,
106], where there are four types of algebraic properties in general for
spherically symmetric BHs,

$[(1111)],\qquad[(11)(11)],\qquad[11(11)],\qquad[(111)1].$ (52)

The matter with these algebraic properties can generate RBHs.

For instance, one RBH given in Ref. [1] has the property $[(11)(11)]$.
Generally, all RBHs with metric Eq. (1) can have this algebraic property
because the Einstein tensor is of the following form,

$G^{0}_{\;0}=G^{1}_{\;1}=\frac{f^{\prime}(r)}{r}+\frac{f(r)}{r^{2}}-\frac{1}{r^{2}},$
(53a)
$G^{2}_{\;2}=G^{3}_{\;3}=\frac{f^{\prime}(r)}{r}+\frac{f^{\prime\prime}(r)}{2},$
(53b)

where we have supposed that Einstein’s theory of gravity still holds,
$G^{\mu}_{\;\;\nu}=8\uppi T^{\mu}_{\;\;\nu}$, thus we can discuss the
algebraic properties of EMTs in terms of Einstein’s tensor.

One example given in Ref. [81] has the property $[11(11)]$. For the metric
with the form of Eq. (3), one has the components of the Einstein tensor,

$G^{0}_{\;\;0}=\frac{f^{\prime}\rho^{\prime}}{\rho}+\frac{f\rho^{\prime
2}}{\rho^{2}}+\frac{2f\rho^{\prime\prime}}{\rho}-\frac{1}{\rho^{2}},$ (54a)
$G^{1}_{\;\;1}=\frac{f^{\prime}\rho^{\prime}}{\rho}+\frac{f\rho^{\prime
2}}{\rho^{2}}-\frac{1}{\rho^{2}},$ (54b)
$G^{2}_{\;\;2}=G^{3}_{\;\;3}=\frac{f^{\prime}\rho^{\prime}}{\rho}+\frac{f^{\prime\prime}}{2}+\frac{f\rho^{\prime\prime}}{\rho},$
(54c)

where no equality, $G^{0}_{\;\;0}=G^{1}_{\;\;1}$, exists generally. However,
when $\rho\propto r$, $G^{0}_{\;\;0}=G^{1}_{\;\;1}$ appears, and then
$[11(11)]$ reduces to $[(11)(11)]$.

For the EMT with the algebraic property $[(111)1]$, there is an example in
Refs. [23, 45]. Here $[(111)1]$ implies
$G^{0}_{\;\;0}=G^{2}_{\;\;2}=G^{3}_{\;\;3}$.

It seems that the algebraic properties do not give us any aid in the
construction of RBHs. However, these properties are extremely important. The
main reason is that they are associated closely with the construction of RBHs
that is different from the construction of SBHs. Let us see the details as
follows.

Generally, a complete theory of RBHs can be established with one of two
distinct logics. The first is the so-called bottom-up approach, in which the
metric with finite curvature invariants is derived based on the First
Principle, such as the loop quantum gravity or the theory of asymptotic
safety, etc. The second logic is the so-called top-down approach, in which the
metric with finite curvature invariants or complete geodesics is postulated at
first, and then the classical field that yields such a metric is found out.
Therefore, it is necessary to clarify the algebraic properties of the
gravitational field for searching matter sources in the second approach.

For instance, the RBH with metric Eq. (1) cannot be interpreted by a scalar
phantom field depending only on the radial coordinate, but the RBH with metric
Eq. (3) can. The reason is that the algebraic property of a scalar phantom
field EMT is consistent with the Einstein tensor based on Eq. (3), i.e., the
components of Einstein’s tensor match the algebra, $[(111)1]$.

Furthermore, the algebraic properties depend on specific gravitational
theories. For the metric Eq. (1), the algebra is $[(11)(11)]$ in the Einstein
gravity, but it changes in the $F(R)$ theory. For instance, if we choose a
special case of Starobinsky’s action [107, 108, 109],

$F(R)=R+\alpha R^{2},$ (55)

the gravitational equations read

$F^{\mu}_{\;\;\nu}\equiv
F^{\prime}(R)R^{\mu}_{\;\;\nu}-\frac{1}{2}F(R)g^{\mu}_{\;\;\nu}-[\nabla^{\mu}\nabla_{\nu}-g^{\mu}_{\;\;\nu}\Box]F^{\prime}(R)=8\uppi
T^{\mu}_{\;\;\nu}.$ (56)

For the metric Eq. (1), the components of tensor $F^{\mu}_{\;\;\nu}$ take the
forms,

$\begin{split}2r^{4}F^{0}_{\;\;0}=&-4\alpha-2r^{3}f^{\prime}\left(2\alpha
f^{\prime\prime}+\alpha rf^{(3)}-1\right)\\\ &+2f\left[12\alpha-2\alpha
r^{2}\left(r^{2}f^{(4)}+2f^{\prime\prime}+6rf^{(3)}\right)+8\alpha
rf^{\prime}+r^{2}\right]\\\ &+4\alpha r^{2}f^{\prime 2}+\alpha
r^{4}f^{\prime\prime 2}-20\alpha f^{2}-2r^{2},\end{split}$ (57a)
$\begin{split}2r^{4}F^{1}_{\;\;1}=&-4\alpha-2r^{3}f^{\prime}\left(2\alpha
f^{\prime\prime}+\alpha rf^{(3)}-1\right)\\\ &+2f\left[-12\alpha-4\alpha
r^{2}\left(4f^{\prime\prime}+rf^{(3)}\right)+8\alpha
rf^{\prime}+r^{2}\right]\\\ &+4\alpha r^{2}f^{\prime 2}+\alpha
r^{4}f^{\prime\prime 2}+28\alpha f^{2}-2r^{2},\end{split}$ (57b)
$\begin{split}2r^{4}F^{2}_{\;\;2}=2r^{4}F^{3}_{\;\;3}=&4\alpha+2rf^{\prime}\left[-12\alpha-2\alpha
r^{2}\left(5f^{\prime\prime}+rf^{(3)}\right)+r^{2}\right]\\\ &+4\alpha
f\left[2r^{2}f^{\prime\prime}-r^{3}\left(5f^{(3)}+rf^{(4)}\right)+8rf^{\prime}+6\right]\\\
&+8\alpha r^{2}f^{\prime 2}+r^{4}f^{\prime\prime}\left(1-\alpha
f^{\prime\prime}\right)-28\alpha f^{2}.\end{split}$ (57c)

Generally, we have the algebraic structure, $[11(11)]$. In other words, the
RBH with the metric Eq. (1) can be generated by the matter with the algebra,
$[11(11)]$.

Similarly, the change of algebraic structures appears in the conformal gravity
with the following action [110],

$S=\int\mathrm{d}^{4}x\sqrt{-g}\;W,$ (58)

where $W$ is the Weyl scalar defined by contracting two Weyl tensors. The
variation to the action gives the gravitation-dependent field,
$B^{\mu}_{\;\;\nu}$, which is called the Bach tensor. For the metric Eq. (1),
we obtain

$\begin{split}24r^{4}B^{0}_{\;\;0}=&-4rf\left[r\left(r^{2}f^{(4)}-f^{\prime\prime}+3rf^{(3)}\right)+2f^{\prime}\right]\\\
&+r^{2}\left(rf^{\prime\prime}-2f^{\prime}\right)^{2}-2r^{4}f^{(3)}f^{\prime}+4f^{2}-4,\end{split}$
(59a)
$24r^{4}B^{1}_{\;\;1}=-2r^{3}f^{(3)}\left(rf^{\prime}-2f\right)+\left[r\left(rf^{\prime\prime}-2f^{\prime}\right)+2f\right]^{2}-4,$
(59b)
$\begin{split}24r^{4}B^{2}_{\;\;2}=24r^{4}B^{3}_{\;\;3}=&-r^{2}\left(rf^{\prime\prime}-2f^{\prime}\right)^{2}+2r^{4}f^{(3)}f^{\prime}-4f^{2}+4\\\
&+2rf\left[r\left(r^{2}f^{(4)}-2f^{\prime\prime}+2rf^{(3)}\right)+4f^{\prime}\right],\end{split}$
(59c)

whose algebraic structure is also $[11(11)]$.

###  3.3 What are the difficulties for us to find the sources of rotating
regular black holes?

The physical interpretation of a rotating RBH should coincide with its non-
rotating counterpart called a seed metric. For the seed metric with one shape
function, the physical interpretation contains two aspects: An imperfect fluid
and a gravitational field coupled to nonlinear electrodynamics. Although the
interpretation from an imperfect fluid is trivial, it is often adopted for the
spacetime without electromagnetic fields. Meanwhile, the resulting rotating
RBH matches [99, 76, 111] the Segré type, $[(11)(11)]$. So the interpretation
works well in the aspect of imperfect fluids for the models in Refs. [90, 96].

For the spacetime with electromagnetic fields, it is widely used [2, 21, 79]
for the physical interpretation that a gravitational field is coupled to
nonlinear electrodynamics. However, it is difficult to extend this
interpretation to a rotating spacetime. The main reason is that the number of
non-zero components of $F_{\mu\nu}$ is changed from one to four by the
introduction of rotation, that is, $F_{01},F_{02},F_{13}$ and $F_{23}$ are
non-trivial, where the field strength is defined by
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. In the metric
Eq. (42), these four components satisfy [112, 113] the relations,

$F_{31}=a\sin^{2}\theta F_{10},\qquad aF_{23}=(r^{2}+a^{2})F_{02}.$ (60)

The gravitational field coupled to nonlinear electrodynamics is described by
the action,

$S=\frac{1}{16\uppi}\int{\mathrm{d}^{4}x\sqrt{-g}[R-\mathscr{L}(F)]},$ (61)
$F=F_{\mu\nu}F^{\mu\nu}.$ (62)

Using the Einstein field equations,

$G_{\mu\nu}=2\mathscr{L}_{F}F_{\mu\alpha}{F_{\nu}}^{\alpha}-\frac{1}{2}\delta_{\mu\nu}\mathscr{L},$
(63)

one can determine $\mathscr{L}$ and $\mathscr{L}_{F}$, where
$\mathscr{L}_{F}\equiv\mathrm{d}\mathscr{L}/\mathrm{d}F$. To further determine
$F_{\mu\nu}$ one needs to utilize the dynamic equations. The variation of the
action with respect to $A^{\mu}$ yields the dynamic equations,

$\nabla_{\mu}\left(\mathscr{L}_{F}F^{\mu\nu}\right)=0,\qquad{\nabla_{\mu}}^{*}F^{\mu\nu}=0,$
(64)

where
${}^{*}F^{\mu\nu}\equiv\frac{1}{2}\eta^{\mu\nu\alpha\beta}F_{\alpha\beta}$,
and $\eta^{0123}=-1/\sqrt{-g}$. Then one obtains that the non-zero components
of $F_{\mu\nu}$ satisfy the following equations,

$\frac{\partial}{\partial
r}\left[(r^{2}+a^{2})\sin\theta\mathscr{L}_{F}F_{10}\right]+\frac{\partial}{\partial\theta}\left[\sin\theta\mathscr{L}_{F}F_{20}\right]=0,$
(65a) $\frac{\partial}{\partial
r}\left[a\sin\theta\mathscr{L}_{F}F_{10}\right]+\frac{\partial}{\partial\theta}\left[\frac{1}{a\sin\theta}\mathscr{L}_{F}F_{20}\right]=0,$
(65b) $\frac{\partial}{\partial
r}F_{20}-\frac{\partial}{\partial\theta}F_{10}=0,$ (65c)
$\frac{\partial}{\partial\theta}\left[a^{2}\sin^{2}\theta
F_{10}\right]-\frac{\partial}{\partial r}\left[(r^{2}+a^{2})F_{20}\right]=0.$
(65d)

Because $\mathscr{L}_{F}$ is quite complicated and these equations are highly
nonlinear, it is almost impossible to solve these equations directly. Instead
of solving the above equations, one then turns to the nonlinear
electromagnetic field by studying [114, 115, 116] the change of gauge fields
under the NJA. When the RN metric is transformed into the KN metric, the gauge
potential $A_{\mu}$ changes [117] in the following way.

In the RN metric, $A_{\mu}$ can be written as

$A_{\mu}=\frac{q}{r}\delta^{u}_{\mu},$ (66)

and its contravariant counterpart takes the form,

$A^{\mu}=-\frac{q}{r}\delta^{\mu}_{r}=-\frac{q}{r}l^{\mu},$ (67)

where $l^{\mu}$ is the tetrad in Eq. (28). Under the transformation governed
by Eqs. (20) and (29), the gauge potential becomes

$\tilde{A}^{\mu}=-\frac{qr}{\rho^{2}}\delta^{\mu}_{r},$ (68)

and its 1-form reads

$\tilde{A}=\frac{qr}{\rho^{2}}(\mathrm{d}u-a\sin^{2}\theta\mathrm{d}\theta),$
(69)

which can be written as

$\tilde{A}=\frac{qr}{\rho^{2}}\left(\mathrm{d}t-\frac{\rho^{2}}{\Delta}\mathrm{d}r-a\sin^{2}\theta\mathrm{d}\theta\right),$
(70)

due to $\mathrm{d}u=\mathrm{d}t-\frac{\rho^{2}}{\Delta}\mathrm{d}r$. Because
the factor $\frac{qr}{\Delta}$ depends only on $r$, the term of $\mathrm{d}r$
can be removed by a gauge transformation and the final formulation of the
gauge potential can be simplified to be

$\tilde{A}=\frac{qr}{\rho^{2}}\left(\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\phi\right),$
(71)

which is just the gauge potential of the KN metric.

However, this method encounters a problem in the NJA, that is, the conversion
rule Eq. (20) may not be applicable to the gauge potentials in RBHs. For
example, the gauge field of spherically symmetric RBHs with the magnetic
charge $Q_{m}$ is $A_{\mu}=Q_{m}\cos\theta\delta^{\phi}_{\mu}$, then the gauge
field of rotating RBHs will become [116]

$A_{\mu}=-\frac{Q_{m}a\cos\theta}{\rho^{2}}\delta^{t}_{\mu}+\frac{Q_{m}(r^{2}+a^{2})\cos\theta}{\rho^{2}}\delta^{\phi}_{\mu},$
(72)

if one uses the above method. But $\mathscr{L}_{F}$ calculated by Eq. (72) is
different [118] from that by Eq. (63), which means that this method should be
modified in the case of RBHs.

For the seed metric with two shape functions, there are mainly two types of
RBHs: One is associated with the loop quantum gravity and the other type is
associated with the black-bounce spacetimes. For the first type, the physical
interpretation of rotating metrics is just from the loop quantum gravity [43,
102]. And for the second type, the physical interpretation involves two
aspects, where one is the stress-energy tensor of a scalar field with nonzero
self-interaction potentials and the other is a magnetic field in the framework
of nonlinear electrodynamics [119]. In addition, there are two different
physical interpretations for rotating black-bounce spacetimes, where one is
the gravitational field coupled with a nonlinear electrodynamic field together
with a contribution of charged dusts and the other is an anisotropic fluid
[100]. However, it needs further studies to judge which interpretation is more
reasonable.

###  3.4 Can regular black holes have scalar hairs?

SBHs are governed by the non-scalar-hair theorem [120, 121]. In RBHs, the
situation is improved. For instance, the metric of conformal RBHs in Ref.
[110] reads

$\mathrm{d}s^{2}=\left(1+\frac{L^{2}}{r^{2}}\right)^{2n}\left(-f\mathrm{d}t^{2}+f^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}\right),$
(73)

where $L$ is regularization parameter with the length dimension and
$f=1-2M/r$. This RBH model can be produced by the following action,

$I_{\rm
conf}=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\phi\left(\frac{1}{6}R\phi-\Box\phi\right),$
(74)

where $\phi$ is scalar field. The equation of motion for the scalar field
$\phi$ is

$\Box\phi=\frac{1}{6}R\phi,$ (75)

and its solution takes [122] the form,

$\phi=\left(1+\frac{L^{2}}{r^{2}}\right)^{-n}\left[\frac{c_{1}}{2M}\ln\left(1-\frac{2M}{r}\right)+c_{2}\right],$
(76)

where $c_{1}$ and $c_{2}$ are integration constants. This solution is
divergent at the horizon, $r_{\rm H}=2M$, which implies $c_{1}=0$, i.e., the
solution can be simplified to be

$\phi=c_{2}\left(1+\frac{L^{2}}{r^{2}}\right)^{-n}.$ (77)

Since $n\geq 1$, $\phi$ is bounded by $0\leq\phi\leq c_{2}$. In other words,
Eq. (74) has a non-trivial scalar hair because of a non-minimal coupling
[120].

Another example is shown in Ref. [23, 45], where the action is cast in the
Einstein gravity with a minimally coupled (phantom) scalar field $\phi$,

$I_{\rm
phatom}=\int\mathrm{d}^{4}x\sqrt{-g}\left[R-\partial_{\mu}\phi\partial^{\mu}\phi-2V(\phi)\right],$
(78)

where $V(\phi)$ is potential, see Ref. [23, 45] for its formulation. This
model gives a RBH solution that has the form of Eq. (3) with two shape
functions, where one of the functions reads

$f(\rho)=1-\frac{\rho_{0}\left(\uppi
b^{2}-2b\rho+\uppi\rho^{2}\right)}{2b^{3}}+\frac{\rho_{0}\left(b^{2}+\rho^{2}\right)}{b^{3}}\tan^{-1}\left(\frac{\rho}{b}\right),$
(79)

where we have used the condition $2bc=-\uppi\rho_{0}$ with $\rho_{0}>0$ and
$b>0$ to replace $c$ in the original formula in Ref. [23, 45]. In addition,
the potential takes the form,

$V(\phi)=-\frac{\rho_{0}}{2b^{3}}\left[2\sqrt{2}\phi+3\sin(\sqrt{2}\phi)+(\uppi-\sqrt{2}\phi)\cos(\sqrt{2}\phi)-2\uppi\right],$
(80)

thus we obtain

$\phi
V^{\prime}(\phi)=\frac{\rho_{0}\phi}{2b^{3}}\left[(\sqrt{2}\uppi-2\phi)\sin(\sqrt{2}\phi)-2\sqrt{2}\cos(1+\sqrt{2}\phi)\right].$
(81)

Because $\phi V^{\prime}(\phi)$ is not always positive, it is not restricted
[120] by the no-hair theorem. Thus the non-trivial scalar-hair solution
exists,

$\phi=\pm\sqrt{2}\tan^{-1}\left(\frac{\rho}{b}\right)+\phi_{0},$ (82)

where $\phi_{0}$ is integration constant, and it is bounded by
$\phi_{0}-\uppi/\sqrt{2}<\phi<\phi_{0}+\uppi/\sqrt{2}$.

##  4 Energy conditions of regular black holes

The energy conditions are important to the study of RBHs. On the one hand,
they are related to the formation of RBHs, and on the other hand they are
regarded as a criteria if a RBH is realistic or not. In this section, we
explain these two aspects.

###  4.1 Is the strong energy condition a key to lead to a regular black hole?

It was originally thought [1, 123] that RBHs can be constructed when the
singularity at their centers is replaced by the de Sitter core, which implies
the violation of the strong energy condition (SEC).

Because of this, RBHs are not governed by the Penrose singularity theorem,
which can be understood from the Raychaudhuri equation [83, 124],

$\frac{\mathrm{d}\Theta}{\mathrm{d}\tau}=-R_{\mu\nu}u^{\mu}u^{\nu},$ (83)

where $\tau$ is proper time, $u^{\mu}$ is four-velocity and $\Theta$ depicts
the expansion of geodesic congruence. For simplicity, we have already ignored
higher order terms associated with expansion, rotation and shear in the right
hand of Eq. (83). Then, choosing $u^{\mu}=(1,0,0,0)$, we arrive at

$\frac{\mathrm{d}\Theta}{\mathrm{d}\tau}=-R_{00}=-4\uppi
G\left(\rho+\sum_{i=1}^{3}p_{i}\right),$ (84)

where $\rho$ is energy density and $p_{i}$ are three components of pressure.
The violation of SEC, $\rho+\sum_{i=1}^{3}p_{i}<0$, implies an increasing
$\Theta$ along the proper time, i.e., the interaction is repulsive.

Nevertheless, it was also discovered [125, 79, 126, 49] that RBHs can have an
anti-de Sitter or a flat core. For instance, the RBH constructed in Ref. [125]
is described by

$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+f^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},\qquad
f=1+\frac{r^{4}}{r^{4}+2qQ_{m}^{2}}\left(-\frac{2M}{r}+\frac{Q_{m}^{2}}{r^{2}}\right),$
(85)

where $q$ is a positive parameter describing the non-minimal coupling of Yang-
Mills fields, $Q_{m}$ is magnetic charge, and the cosmological constant is set
to be zero in order to highlight the essence. The anti-de Sitter core can be
seen clearly from the following asymptotic relations,

$f\sim 1+\frac{r^{2}}{2q}+O(r^{3}),\qquad R\sim-\frac{6}{q}+O\left(r\right)$
(86)

as $r$ approaches to zero. Moreover, a spherically symmetric RBH model with a
flat core is highlighted in Ref. [126, 127], where the shape function reads
[128]

$f=1-\frac{2M}{r}\mathrm{e}^{-a/r}.$ (87)

It is obvious that $f\to 1$ and $R\to 0$ as $r$ approaches to zero because the
parameter $a$ is positive. These two examples meet the SEC in the cores
because the AdS and Minkowski spacetimes satisfy the SEC.

The problem immediately arises, if the SEC does not break, i.e., the gravity
is attractive in the core, how the collapse can be avoided. One interesting
resolution is based [46] on the introduction of the Tolman mass which can be
regarded as a kind of integral SEC [129, 130],

$m_{\rm T}=\frac{1}{4\uppi}\int\sqrt{-g}R_{00}\mathrm{d}^{3}x=\int
r^{2}R_{00}\mathrm{d}r.$ (88)

The integral SEC breaks in the core ($r\in[0,r_{-}]$, $r_{-}$ is the innermost
horizon) if the Tolman mass is negative. Due to the negative Tolman mass in
these two models described by Eq. (85) and Eq. (87), we can conclude that the
two models violate the integral SEC in their cores.

In summary, it is the impressionist SEC that plays the key role in
constructing RBHs, not the pointillist one [131]. Because of this, it is not
important whether the cores of RBHs are de Sitter, anti-de Sitter or flat
although the violation of SEC is a necessary condition for the formation of
RBHs from gravitational collapse [132].

###  4.2 What are the energy conditions of regular black holes?

If the SEC answers how RBHs are formed, then the other three energy conditions
actually focus on whether RBHs are realistic [133, 44, 134, 135, 136, 112]
through the characteristics of classical matters.

Besides the SEC, the other three energy conditions are the weak energy
condition (WEC), the null energy condition (NEC), and the dominant energy
condition (DEC) [131]. The violation of WEC implies a negative local energy
density, and the violation of DEC means the superluminal speed of energy
density flow. Moreover, the NEC is a relaxation of WEC, i.e., the energy
density can be negative as long as the sum of energy density and pressure is
positive [83].

For the RBHs with one shape function, see Eq. (5), the other three energy
conditions except the SEC can be reduced to the following differential
inequalities,

$\begin{split}\text{WEC}&:\quad\sigma^{\prime}\geq
0\;\cup\;r\sigma^{\prime\prime}\leq 2\sigma^{\prime},\\\ \text{NEC}&:\quad
r\sigma^{\prime\prime}\leq 2\sigma^{\prime},\\\
\text{DEC}&:\quad\sigma^{\prime}\geq 0\;\cup\;-2\sigma^{\prime}\leq
r\sigma^{\prime\prime}\leq 2\sigma^{\prime},\end{split}$ (89)

where the prime denotes the derivatives w.r.t. $r$. The relation between these
three energy conditions can be represented by

$\text{NEC}\subset\text{WEC}\subset\text{DEC}.$ (90)

Among these three independent differential inequalities in Eq. (89),
$\sigma^{\prime}\geq 0$ implies that $\sigma$ is a monotone increasing
function of $r$; while $r\sigma^{\prime\prime}\leq 2\sigma^{\prime}$ provides
$\sigma\leq\sigma_{0}r^{3}$, where $\sigma_{0}\equiv\lim_{r\to
0}\sigma/r^{3}$; the last one $r\sigma^{\prime\prime}\geq-2\sigma^{\prime}$
gives a solution, $r\sigma\geq 0$, under the boundary conditions,
$\sigma|_{r=0}=0=\sigma^{\prime}|_{r=0}$.

For the RBHs with two shape functions, see Eq. (3), the situation becomes
complicated. The differential inequalities now involve an additional unknown
function, $r(\xi)$, which causes the inequalities to be unsolvable without
extra specific constrains. Therefore, we cannot extract any valuable
information from these differential inequalities.

In addition, some RBH models violate [137, 138, 73] the three energy
conditions. For instance, the well-known Bardeen and Hayward BHs break the
DEC, and the RBH generated by the non-minimal coupled Wu-Yang monopole brakes
[32, 33, 139, 140, 68] the WEC, etc. A remedy is proposed in Ref. [141] by
deforming the shape function. For a generic $\sigma$ function, its deformed
formulation reads

$\sigma=\frac{M^{\mu\nu-3}r^{3}}{\left(r^{\mu}+q^{\mu}\right)^{\nu}},$ (91)

where $M$ is mass and $q$ regularization parameter, which contains the Bardeen
and Hayward BHs as special cases. Meanwhile, it will meet the three energy
conditions if the parameters $\mu$ and $\nu$ take the values in the following
regions,

$\displaystyle\frac{2}{\nu}<\mu\leq\frac{1}{2}\sqrt{\frac{49\nu+96}{\nu}}-\frac{7}{2}$
$\displaystyle\quad\text{when}\quad\frac{2}{5}<\nu\leq 3;$ (92a)
$\displaystyle\frac{2}{\nu}<\mu\leq\frac{3}{\nu}$
$\displaystyle\quad\text{when}\quad\nu>3,$ (92b)

where $M^{\mu\nu-3}$ is introduced for balancing the dimension, and this
parameterization is not unique.

##  5 Thermodynamics of regular black holes

The thermodynamics of RBHs is a rather confusing battleground in the study of
RBHs. The confusion comes from the existence of extra terms in the first law
of mechanics such that the correspondence between mechanical and thermodynamic
quantities is problematic. In this section, we give our clarifications on
these problems.

###  5.1 What is the entropy of regular black holes?

It was reported [142, 143, 30, 144, 145, 146, 147] that RBHs have a deviation
term in entropy, i.e., the entropy of RBHs breaks the area law, $S\neq A/4$,
but the opposite opinion was also reported [148, 22, 52, 149, 150, 55]. This
puzzle gives rise to an influence on the first law of thermodynamics and the
interpretation of RBHs, where ambiguous deviation terms appear for the former,
while the verification of Hawking’s quantum theory is hard to be executed for
the latter.

Taking the Hayward BH [95] as an example, whose shape function is

$f=1-\frac{2M}{r}\frac{r^{3}}{r^{3}+2Ml^{2}},$ (93)

where $l$ is length scale introduced for regularization, we can obtain the
entropy from the first law of thermodynamics, $\mathrm{d}M=T\mathrm{d}S$,

$S=\int^{r_{+}}_{r_{-}}\frac{\mathrm{d}M}{T}=S_{\rm BH}+\Delta S,$ (94)

where $r_{+}$ and $r_{-}$ are outer and inner horizons, respectively, $S_{\rm
BH}$ is the Bekenstein-Hawking entropy,

$S_{\rm BH}=\uppi\left(r_{+}^{2}-r_{-}^{2}\right),$ (95a) and $\Delta S$ is a
deviation term, $\Delta S=\frac{\uppi
l^{4}\left(r_{+}^{2}-r_{-}^{2}\right)}{\left(r_{-}^{2}-l^{2}\right)\left(r_{+}^{2}-l^{2}\right)}+2\uppi
l^{2}\ln\left(\frac{r_{+}^{2}-l^{2}}{r_{-}^{2}-l^{2}}\right).$ (95b)

We note $\Delta S>0$ because of $r_{+}\geq r_{-}>l$, otherwise there will be
no horizons. As a result, if Eq. (94) is still used to calculate the entropy,
the area law is violated, $S\neq A/4$, even though a pressure term,

$\mathcal{P}=-\frac{3}{8\uppi l^{2}},$ (96)

is added [151].

As a matter of fact, if we interpret the metric, Eq. (1), as a spacetime
produced by a field of Dirac magnetic monopoles [2, 152], the entropy can be
derived by Hawking’s path integral method for a given RBH depicted by Eq. (1).

To start with, we write down the full action,

$I=\frac{1}{16\uppi}\int\mathrm{d}^{4}x\sqrt{-g}\,\left[R-\mathcal{L}(F)\right],$
(97)

where $F$ is the contraction of electromagnetic tensors, $F\equiv
F_{\mu\nu}F^{\mu\nu}=2q/r^{4}$, and $q$ is the magnetic charge of monopole. By
solving one of Einstein’s equations [22],

$\frac{\mathcal{L}}{2}-\frac{2M\sigma^{\prime}(r)}{r^{2}}=0,$ (98)

and replacing $r$ by $r=[F/(2q)]^{1/4}$, we can determine the Lagrangian
$\mathcal{L}(F)$.

To calculate the entropy, we apply the path-integral method in the zero-loop
approximation [153],

${Z}=\int{D}g\,{D}A\;\mathrm{e}^{-I}\approx\mathrm{e}^{-I_{\rm cl}},$ (99)

where the full Euclidean action $I_{\rm cl}$ consists of four parts,

$I_{\rm cl}=I_{\rm EH}+I_{\rm GHY}-I_{0}+I_{M}.$ (100)

$I_{\rm EH}$ is the Einstein-Hilbert action,

$I_{\rm EH}=-\frac{1}{16\uppi}\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{-g}\,R,$
(101)

$I_{\rm GHY}$ is the Gibbons-Hawking-York boundary term,

$I_{\rm
GHY}=-\frac{1}{8\uppi}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{-h}\,K,$
(102)

$I_{0}$ is the subtraction term,

$I_{0}=-\frac{1}{8\uppi}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{-h}\,K_{0},$
(103)

and $I_{M}$ is the matter action of nonlinear electrodynamic source, where $R$
is the Ricci curvature of bulk space, $K$ and $K_{0}$ are extrinsic curvatures
of surface and background reference, respectively.

For the metric Eq. (1), we obtain

$\displaystyle I_{\rm EH}=\frac{\beta}{2}\left(r_{\rm H}-M\right)-\uppi r_{\rm
H}^{2},$ (104) $\displaystyle I_{\rm
GHY}=\frac{M}{2}\beta\left(r\sigma^{\prime}+3\sigma\right)-r\beta\sim\frac{3M}{2}\beta-r\beta,$
(105) $\displaystyle
I_{0}=-r\beta\sqrt{1-\frac{2M\sigma}{r}}\sim-r\beta+M\beta+O\left(r^{-1}\right),$
(106)

where $\beta$ is the inverse of BH temperature, $1/T$, and the last two
asymptotic relations at the boundary $r\to\infty$ are derived based on the
assumption that the RHB is asymptotic to the Schwarzschild BH at infinity
[79], i.e., $\lim_{r\to\infty}\sigma=1$ and
$\lim_{r\to\infty}r\sigma^{\prime}=0$. Meanwhile, the action of matter can be
determined with the help of one component of Einstein’s equations Eq. (98),

$I_{M}=\beta M-\frac{\beta r_{\rm H}}{2}.$ (107)

Substituting Eqs. (104)-(107) into Eq. (100), we arrive at the total Euclidean
action,

$I_{\rm cl}=\beta M-\uppi r_{\rm H}^{2}.$ (108)

On the other hand, the thermodynamic law for the canonical ensemble is
$\mathcal{F}=M-TS$, where $\mathcal{F}=TI_{\rm cl}$ is Helmholtz free energy.
Thus, we can read $S=\uppi r_{\rm H}^{2}$ from Eq. (108), which exhibits the
well-known entropy-area law of BHs. The same result can be obtained by Wald’s
Noether-charge approach. Since the Lagrangian of the model Eq. (1) is just
$R$, the entropy density can be read off directly from Table I in Ref. [154].

However, the problem is far more complicated than it appears — the above
calculation relies on the interpretation of the metric. That is, the
calculation depends on the nonlinear magnetic interpretation of metrics, see
Eq. (97). If we reinterpret the source as dyons, the path-integral method is
not applicable because no actions for dyons can be constructed [22].
Furthermore, if we interpret a metric by using an alternative gravity, such as
$f(R)$, the entropy-area law will be changed.

###  5.2 What is the correct first law of thermodynamics for regular black
holes?

One feature of RBHs is that their mechanical theorems differ from those of
SBHs [155]. More precisely, unexpected extra terms will appear in the first
mechanical laws of RBHs, and the number of extra terms depends on the number
of parameters involved in Lagrangian of matters.

For instance, the Lagrangian of Bardeen BHs contains two parameters, mass $M$
and magnetic charge $q$ [21], thus the first mechanical law reads [155]

$\mathrm{d}M=\frac{\kappa}{8\uppi}\mathrm{d}A+\Psi_{H}\mathrm{d}q+K_{M}\mathrm{d}M+K_{q}\mathrm{d}q,$
(109)

where $\kappa$ is the surface gravity, $\Psi_{H}$ is the magnetic potential
and the last two terms are extra. As a result, one encounters several problems
when attempting to construct the first thermodynamic law, e.g., what is the
correspondence between mechanical and thermodynamic variables and what is the
dimension of the thermodynamic phase space?

To construct the first thermodynamic law, Fan and Wang [22] introduced a
parameter $\alpha$ in the action of nonlinear electrodynamics. According to
their method, the first thermodynamic law for the Bardeen BH, Eq. (48), can be
cast in the following form,

$\mathrm{d}E=T\mathrm{d}S+\Psi_{H}\mathrm{d}Q_{m}+\Pi\mathrm{d}\alpha,$ (110)

where the three thermodynamic variables

$E=M,\qquad Q_{m}=\sqrt{Mq/2},\qquad\alpha=q^{3}/M=8Q^{6}_{m}/M^{4}$ (111)

are not independent from each other in the phase space. In other words, the
dimension of thermodynamic phase space is two because $\alpha$ is a redundant
dimension. The other correspondences between thermodynamic and mechanical
variables are

$T\longleftrightarrow\frac{\kappa}{2\uppi},\qquad
S\longleftrightarrow\frac{A}{4}.$ (112)

Nevertheless, Eq. (110) is still problematic because

$\displaystyle\frac{A}{4}=S\neq\int\frac{\mathrm{d}E}{T}=\int\frac{\mathrm{d}M}{T},$
(113)

where the integral is calculated under the condition,
$\mathrm{d}Q_{m}=0=\mathrm{d}\alpha$. The reason of $S\neq\int\mathrm{d}M/T$
or $T\neq(\partial E/\partial S)_{Q,\alpha}$ is that the relations,
$\mathrm{d}Q_{m}=0=\mathrm{d}\alpha$, imply that $M$ is also a constant, see
Eq. (111), which leads to a trivial integral, $\int\mathrm{d}M/T\equiv 0$.

If only one parameter is fixed, say $\mathrm{d}Q_{m}=0$, we have

$\frac{A}{4}=\int\frac{\mathrm{d}M}{T}\left(1+\frac{32Q_{m}^{6}\Pi}{M^{5}}\right)\neq\int\frac{\mathrm{d}M}{T},$
(114)

namely, the simple thermodynamic relation, $S=\int\mathrm{d}M/T$, is broken.
This result gives rise to a tendency to abandon [142, 143, 30, 144, 145, 146,
147, 156, 157, 158] the area-entropy law,

$\displaystyle S=\int\frac{\mathrm{d}E}{T}\neq\frac{A}{4}.$ (115)

In the above treatment, the entropy $S$ is not an independent available. In
other words, it cannot be determined without the first thermodynamic law. The
worst point is that the broken area-entropy relation contradicts the results
obtained from either Hawking’s path-integral or Wald’s entropy formula.

Therefore, the question naturally arises — what a correct first thermodynamic
law is. We list its most important features below:

  1. 1.

The area law should be maintained, i.e., $S=A/4$, if one explains RBHs in the
context of Einstein’s gravity. Generally, the entropy in the first
thermodynamic law should be consistent with that calculated from either
Hawking’s path-integral or Wald’s entropy formula.

  2. 2.

Every thermodynamic variable should be independent from the first
thermodynamic law, i.e., it can be determined without the first thermodynamic
law, but the thermodynamic formula, $S=\int\mathrm{d}E/T$, must hold. Here are
some counterexamples: temperature is not independent in Ref. [159], deviation
of internal energy $X$ is not independent in Ref. [160], etc.

  3. 3.

Every thermodynamic variable should be independent from each other, e.g.,
$\mathrm{d}M=T\mathrm{d}S+K_{1}\mathrm{d}\alpha+K_{2}\mathrm{d}\beta+\ldots$,
is ill-defined if $\alpha=M$ and $\beta=TM$ that make $\alpha$ and $\beta$
dependent on $M$ in the thermodynamic phase space.

In order to establish a well-defined first thermodynamic law that meets the
above conditions for RBHs with two parameters, such as Bardeen BHs, we have
proposed [80] the following form,

$\mathrm{d}U=T\mathrm{d}S-P_{+}\mathrm{d}V,$ (116)

where the total internal energy reads

$U=\frac{r_{+}}{2},$ (117)

$P_{+}$ and $V$ are the thermodynamic pressure and volume, respectively,

$P_{+}=\left.\frac{G^{r}_{\;r}}{8\uppi}\right|_{r=r_{+}},\qquad
V=\frac{4}{3}\uppi r_{+}^{3}.$ (118)

##  6 Rugular black hole chemistry and thermodynamic geometry

Thermodynamic geometry is a very powerful tool for exploring the
microstructures of RBHs and more recently BH chemistry has been able to
understand BHs in a different way. The study of BH chemistry and Ruppeiner
geometry can help us to further understand the thermodynamic properties of
RBHs.

###  6.1 What is the regular black hole chemistry?

####  6.1.1 Thermodynamic phase transition and shift of critical points

In recent years, there have been a lot of studies on BH chemistry showing that
RBHs behave very similar chemical phenomena to those of SBHs, such as
gas/liquid phase transitions [156, 161, 162], van der Waals-like fluid
properties [163, 164, 165, 166], triple points [167, 168], and heat engines
[169, 170, 171, 172, 173]. The BH chemistry is based on equations of state.
For a given equation of state, the critical point ($T_{c}$, $P_{c}$, $V_{c}$)
of phase transitions can be determined by the following condition,

$\left(\frac{\partial P}{\partial
V}\right)_{T}=0,\qquad\left(\frac{\partial^{2}P}{\partial
V^{2}}\right)_{T}=0.$ (119)

The behaviors of critical points allow us to make a deep analogy between BHs
and gas/liquid systems. In particular, the phase transitions of van der Waals-
like fluid have been well tested [174, 175] in different theories of gravity.
However, due to the problematic first law of thermodynamics, the Maxwell equal
area law is no longer valid for RBHs. The exotic equal area law for Hayward
AdS BHs reads [52]

$\oint VdP=-\oint\Psi dQ_{m}+\oint\text{extra term},$ (120)

which makes Maxwell’s equal area law invalid in the $P-V$ plane. Thus, the
critical point ($T_{c}$ , $P_{c}$) of the first order small/large BH
transition does not coincide with the inflection point of isotherms.

####  6.1.2 Regular black hole as a heat engine

That BHs behave like the van der Waals fluid and have gas/liquid phase
transitions also makes it possible to treat BHs as heat engines. The first
holographic heat engine was proposed [176] by Johnson, where BHs act as a
working substance. The heat engine flows in a cycle are shown in Fig. 2, where
$Q_{H}$ is net input heat flow and $Q_{C}$ is net output heat flow.

Figure 2: The heat engine flows.

Using equations of state of AdS BHs, for instance, we can compute the total
mechanical work as follows,

$W=Q_{H}-Q_{C}.$ (121)

Therefore, the efficiency of heat engines is defined by

$\eta\equiv\frac{W}{Q_{H}}=1-\frac{Q_{C}}{Q_{H}}.$ (122)

Recently, some progress has been made [177, 169, 171, 178] in the studies of
heat engines for RBHs, where the relation between efficiency and entropy
(pressure) has been obtained [171] and the comparisons of efficiency among
RBHs have also been made. We note that it is subtle to construct heat engines
and especially the engine cycles in the $P-V$ plane for RBHs, and that some
other progress has provided [179, 180, 181] a new perspective on thermodynamic
properties of RBHs.

###  6.2 How to eliminate the singularity of the thermodynamic geometry for
regular black holes?

####  6.2.1 Construction of thermodynamic geometry

The similar situation has happened in the studies of thermodynamic geometry,
where such a geometry has been shown to be a powerful tool for understanding
the thermodynamic properties and microstructures of SBHs. The Ruppeiner metric
is defined [182] by the Hessian matrix of thermodynamic entropy,

$g^{\rm R}_{\mu\nu}\equiv-\frac{\partial^{2}S(X)}{\partial X^{\mu}\partial
X^{\nu}},$ (123)

and the Weinhold metric is defined [183] by the Hessian matrix of BH mass,

$g^{\rm W}_{\mu\nu}=\frac{\partial^{2}M(X)}{\partial X^{\mu}\partial
X^{\nu}},$ (124)

where $X^{\mu}$ denote thermodynamic variables. Further, we can see that there
is the following conformal relationship between the two kinds of thermodynamic
geometries if we write the metrics in terms of line elements,

$\mathrm{d}s^{2}_{\rm R}=\frac{1}{T}\mathrm{d}s^{2}_{\rm W}.$ (125)

Note that the above two kinds of thermodynamic geometries are valid only for
SBHs because they are based on the corresponding first law of thermodynamics.
For RBHs, however, the first law of thermodynamics contains additional terms.
Thus, we have to restrict the first law in a subphase space, and rewrite it,
as an example, for a RBH described by a four-dimensional subphase space,

$\mathrm{d}M=\frac{T}{1-\Pi}\mathrm{d}S-\frac{P}{1-\Pi}\mathrm{d}V,$ (126)

where $\Pi$ is related to additional terms. In this case, if we constructed
thermodynamic geometry from the problematic first law Eq. (126), the
thermodynamic metric would also contain additional terms associated with
$\Pi$. But $\Pi$ has no thermodynamic counterpart. Correspondingly, the
conformal relationship between Ruppeiner geometry and Weinhold geometry
becomes

$\displaystyle\mathrm{d}s^{2}_{\rm R}=\frac{1-\Pi}{T}\mathrm{d}s^{2}_{\rm W}.$
(127)

It is important to emphasize that the construction of thermodynamic geometry
for RBHs is a challenging topic, and it must be based on the correct first law
of thermodynamics. The microstructure of RBHs has also been studied [163, 184,
185] in different gravity theories with the help of thermodynamic geometry. As
is known, RBHs are very special BH models and have been widely concerned [167,
186, 187, 173] recently. Their first law of thermodynamics may need to be
modified, and accordingly their thermodynamic geometry may be modified as
well. It is hopeful to deeply understand the thermodynamics of RBHs from the
perspective of thermodynamic geometry.

####  6.2.2 Singularity of thermodynamic geometry and elimination

A very special feature of charged AdS BHs is that the heat capacity at
constant volume vanishes, i.e., $C_{V}=0$. This property makes the
thermodynamic line element, Eq. (127), singular and the corresponding
thermodynamic information is unavailable from thermodynamic geometry. A new
normalized scalar curvature is defined [188],

$\displaystyle R_{N}=R\,C_{V},$ (128)

under which the divergence of Ruppeiner scalar curvature can subtly be
eliminated. For RBHs, extra terms will appear in thermodynamic line elements
due to the modification in the first law of thermodynamics, which can be used
to eliminate the divergence of thermodynamic scalar curvature. Therefore, RBHs
can be viewed as a treatment option to address the divergence of thermodynamic
scalar curvature.

##  7 Conclusion and outlook

We have presented several topics of RBHs in the current review, where the
importance of topics plays a major role in our choice. There are still three
interesting issues we want to mention here. The first is whether RBHs are
trivial from the point of view of coordinate transformations, the second issue
is what sources that generate rotating RBHs are, and the last issue is what
differences between RBHs and singular ones are.

For the first issue, it is known that every RBH corresponds to a metric with
several coordinate singularities. Thus, one can always find a transformation
to remove these singularities. Consequently, the recast metric is completely
equivalent to the previous one from the spacetime perspective but it has no
longer horizons, i.e., no more so-called black holes. Let us phrase the
question more clearly: Whether RBH is a realistic object or just a
mathematical expression of spacetime?

The second issue arises due to the construction of rotating RBHs. Bambi and
Modesto utilized [94] the NJA to create the metrics of rotating RBHs. On the
one hand, the NJA depends on theories of gravity. Taking a RBH model in the
Chern-Simons gravity [189] as an example, the NJA can provide a rotating
metric, but the Pontryagin density, ${}^{*}RR$, does not vanish [190], which
means that the constructed metric will not satisfy equations of motion. On the
other hand, as far as we know, there is still a lack of investigation on the
classical field source of rotating RBHs.

At last, the most intriguing issue is possibly what the differences between
RBHs and singular ones are. To answer this question, one must distinguish
which differences are caused by the absence of singularities and which are
caused by the models’ characteristics, such as the differences between
Schwarzschild and RN BHs. This issue plays an elementary role in the future
study of RBHs.

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China under Grant No. 12175108.

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