|
According 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 [11, 12].
2
where the geodesics are complete but the curvature invariants are divergent, which consequently
contradicts 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 [17,18], 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 [19–21], see also the reviews [22,23]. The first model of RBHs was implemented
by Bardeen [24], now called the Bardeen BH, which was constructed by 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 the BH center.
After three decades of Bardeen’s proposal, Ay´on-Beato and Garc´ıa provided [25] 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, 26–31] to interpret all RBH models with
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 [27, 32–34]. Moreover, they are regarded to be classical objects
as they are the solutions of Einstein’s field equations.
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, 35–41]; 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 [42–50]. Based on the
former way, the RBHs behave semiclassically; whereas based on the latter, those RBH models
exhibit quantum behaviors. In other words, RBHs are now regarded as a tool to study the classical
limit of quantum black holes because we do not have a complete theory of quantum gravity at
the moment.
Besides the structures of RBHs [51–58], the study also extends to the other areas of BH
physics, including thermodynamics [26, 59–62], dynamics [63–68], shadows [69–78], quasinormal
modes [63,79–88], superradiance [89], and synchrotron radiations [90–92] , etc. All of the research
on RBHs aims to explore how RBHs differ from singular black holes (SBHs) and thus shed light
on further research for quantum gravity. Currently, the study of RBHs has made 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 [23,36,93], 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 of the construction for both
the non-rotating and rotating RBHs, where we also discuss the number of curvature invariants
among the Zakhary-Mcintosh invariants (a complete set of curvature 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
3
showing the peculiarity of RBHs on scalar hairs, which 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 of 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 entropyarea
law, based on which the self-consistent first law of thermodynamics is given in Sec. 5.2. In
Sec. 6, we discuss thermodynamic geometry for RBHs because they are the nontrivial extensions
of interesting issues associated with SBHs. Finally, we conclude in Sec. 7 with some outlooks.
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 an
RBH. Here the curvature invariants refer to the Zakhary-Mcintosh (ZM) invariants [94, 95] that
form a complete set of Riemann invariants and contain seventeen elements, see also App. B. The
reason that we adopt the ZM invariants rather than the usual ones, such as the Ricci scalar and
Kretschmann scalar [7, 26, 96], 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 spherical symmetry. The RBHs with spherical symmetry have two types of
metrics: A metric in the first type involves one shape function, see the specific square of line
elements,
ds2 = −f(r)dt2 + f−1(r)dr2 + r2dΩ2, (1)
and a metric in the second type involves two shape functions, see the specific square of line
elements,
ds2 = −f(r)dt2 + f−1(r)A2(r)dr2 + r2dΩ2, (2)
which is equivalent to
ds2 = −f(ξ)dt2 + f−1(ξ)dξ2 + r2(ξ)dΩ2, (3)
where ξ is a newly defined variable,
ξ ≡
Z
dr 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 −
2Mσ(r)
r
, (5)
4
where σ(r) is a function of the radial variable r and M is BH mass. In order to observe the
regularity, we expand σ(r) by the power series around r = 0,
σ(r) = σ1r + σ2r2 + σ3r3 + O(r4), (6)
where σi 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 σ1 and σ2 must
vanish,
σ1 = 0 = σ2. (7)
As an example, we write the behaviors of three usual candidates among the seventeen ZM invariants
around r = 0: The Ricci scalar R, the Weyl scalar W = WαβμνWαβμν, and the Kretschmann
scalar K have the asymptotic behaviors for RBHs,
R = 24Mσ3 + O(r), W = O(r2), K = 96M2σ2
3 + O(r). (8)
Alternatively, we can select three curvatures from the seventeen ZM invariants and write σ(r),
σ′(r), and σ′′(r) as functions of these three curvatures because the ZM invariants contain σ(r)
and only its first and second order derivatives, σ′(r) and σ′′(r). Then, requiring the finiteness of
the three curvatures, we can find the behavior of σ(r) around the center r = 0, i.e., σ(r) should
not decrease slower than r3 as r approaches to zero [97], 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 [7, 27, 98, 99], we apply a similar procedure to the
above, that is, we expand both of shape functions (see Eq. (2)) by the power series,
A(r) = A0 + A1r + A2r2 + O(r3), (9a)
f(r) = B0 + B1r + B2r2 + O(r3). (9b)
After substituting Eqs. (2), (9a) and (9b) into the ZM invariants, we can find the conditions for
finite curvatures,
A0 = B0, A1 = B1 = 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 an example when r goes to zero,
R =
6(A2 − 2B2)
A0
+ O(r), W = O(r2), S =
3A22
A20
+ O(r), (11)
where S and Sμν are defined [7] by S ≡ SμνSμν and Sμν ≡ Rμν − gμνR/4, respectively.
5
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,c but they can be regarded as two independent necessary conditions for checking whether
a BH is regular. Moreover, the former is coordinate-independent, i.e., a coordinate singularity
does not appear in curvature invariants, whereas in order to prove the regularity by the latter,
one has to eliminate the coordinate singularity by selecting an appropriate coordinate. Thus, the
latter often involves the choice of a coordinate system in practice. For instance, in the Rindler
spacetime [12, 100], ds2 = −z2dt2 + dx2 + dy2 + dz2, 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 → tanh−1 T
Z
, x → X, y → Y, z →
√
Z2 − T2, (12)
the original metric converts to that of the Minkowski spacetime, ds2 = −dT2+dX2+dY 2+dZ2,
i.e., there is no singularity 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, there are two questions associated with the criterion of finite curvature invariants.
The first is whether the curvature invariants can reveal the singularity of spacetime, and the second
is how many curvature invariants have to be used in order to determine a RBH if the first question
has a positive answer.
It is known that the components of Riemann tensors are not suitable to describe spacetime [101]
because they depend on the choices of coordinate systems. However, the scalars constructed by
Riemann tensors and metrics can improve the situation. These curvature scalars are important
for investigating singularities. In other words, the curvature scalars are believed to describe the
primary properties of spacetime, in particular, to determine the existence of spacetime singularities.
Considering the independent components of Riemann tensors and metrics together with the
constraints from coordinate transformation, one can construct 14 curvature scalarsd in the 4D
spacetime [4]. For simple cases, there exist only three curvature scalarse that are connected by
the Ricci decomposition, i.e., the Ricci scalar (R), the Kretschmann scalar (K) and the contraction
of two Ricci tensors (R2).
For certain cases in the presence of matter, the 14 scalars are not complete, i.e., more than 14
scalars are required [102]. Here the completeness implies the minimal number of invariants for all 6
Petrov types and 15 Segr´e types [103]. It has been proved [94] that the complete set of curvature
invariants should contain 17 elements that are known as ZM invariants. These invariants are
cFor certain cases, these two conditions are equivalent, e.g., for spherically symmetric BHs with one shape
function, i.e., ds2 = −f(r)dt2 + f−1(r)dr2 + r2dΩ2.
dThis number comes from 20 independent components of a Riemann tensor plus 10 independent components
of a metric but minus 16 constraints imposed by the general coordinate transformation.
eAlternatively, K and R2 are replaced by W ≡ CμναβCμναβ, the contraction of two Weyl tensors, and S ≡
SμνSμν, where Sμν ≡ Rμν − gμνR/4 [7].
6
originally defined by the spinorial quantities in the Penrose-Newman formalism, and are recast in
the explicit algebraic expressions in Ref. [95].
Thus, the two questions mentioned above become whether the ZM invariants can determine
the singularities of spacetime and how many elements are required in this set. The second question
has been analyzed for several cases, for instance, four scalars are necessary for rotating RBHs [104],
while two scalars are enough for non-rotating ones [105]. As to the first question, the situation is
relatively complicated and no definite answer is given at the moment.
Let us see the well-known Taub-NUT BH [103, 106] as a sample,
ds2 = −f(r) [dt + 2n cos(θ)dϕ]2 +
dr2
f(r)
+ ζ2
dθ2 + sin2(θ)dϕ2
(13)
with
f(r) =
Δ
ζ2 , Δ = r2 − 2Mr − n2, ζ =
√
r2 + n2, (14)
where M is mass, the NUT parameter n is positive and referred to as magnetic mass, and the
horizon is located at rH = M +
√
M2 + n2. Now the Taub-NUT BH is regarded [107, 108] as
the electric-magnetic duality of Schwarzschild BHs. It is shown [10, 109] 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, Rμν = 0. The Kretschmann scalar reads
K ∼
48 (n2 −M2)
n6 + O (r) , (15)
which is finite as r approaches zero. Moreover, R2 is also finite when r goes to zero. As a result,
the three curvature invariants, R, K, and R2 are finite in the Taub-NUT BH spacetime. As done
for the center r = 0, we can show that the horizon has no singularity when we check R, K, and
R2. Further, we confirm that all the ZM invariants are regular everywhere in the Taub-NUT BH
spacetime, i.e., the finite-curvature method cannot ensure the completeness of geodesics [10,109].
This implies that the ZM curvature invariants are not able to fully reflect the singularity of
spacetime, or more invariants than the ZM ones are needed. The validity of these statements
remains open at the moment.
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) [110].
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-Nordstr¨om (RN), Kerr, and Kerr-
Newman (KN) BHs were obtained by solving Einstein’s field equations in electro-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
7
symmetric Schwarzschild BHs to axially symmetric Kerr BHs. The algorithm can also describe
the transformation from RN BHs to KN BHs.
Subsequently, G¨urses and G¨ursey extended [111] 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 [112] 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 [36, 113, 114], loop quantum corrected BHs [115, 116], Bardeen
BHs [24, 117], and Hayward BHs [117, 118], phantom RBHs [119], 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 [120–122]:
r → r + ia cos θ, u → u − ia cos θ, (16)
where (u, r, θ, φ) 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(RN)tt = 1 −
2M
r
+
q2
r2 , (17)
where M is mass and q is charge. The tt-component of the KN metric takes the form,
g(KN)tt = 1 −
2Mr
r2 + a2 cos2 θ
+
q2
r2 + a2 cos2 θ
. (18)
The conversion rule is as follows:
r2 = rr∗ → (r + ia cos θ)(r − ia cos θ) = r2 + a cos2 θ, (19)
1
r
=
1
2
1
r
+
1
r∗
→
1
2
1
r + ia cos θ
+
1
r − ia cos θ
=
r
r2 + a2 cos2 θ
. (20)
However, the above conversion rule may not be applicable to RBH metrics due to their complexity.
Here we take the black-bounce spacetime [98] as an example, where its metric’s ttcomponent
reads
g(BB)tt = 1 −
2M
√
r2 + l2
, (21)
where l is the regularization parameter. When l vanishes, the metric becomes the form of
Schwarzschild’s spacetime. 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(rBB)tt = 1 −
2M
√
r2 + a2 cos2 θ + l2
, (22)
8
whereas the tt-component of the Kerr metric is
g(K)tt = 1 −
2Mr
r2 + a2 cos2 θ
. (23)
Obviously, g(rBB)tt does not reduce to g(K)tt when l = 0. As a result, the above conversion rule
does not apply to the black-bounce spacetime. Owing to the failure of this rule, we need to
find such a rule that applies 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
[120, 121] the NJA as follows.
For a general static metric,
ds2 = −G(r)dt2 +
dr2
F(r)
+ H(r)
dθ2 + sin2 θdφ2
, (24)
one introduces the advanced null coordinates (u, r, θ, φ) defined by
du = dt −
dr
√
FG
, (25)
and expresses the contravariant form of the metric in terms of a null tetrad,
gμν = −lμnν − lνnμ + mμm∗ν + mνm∗μ, (26)
where
lμ = δμ
r , (27a)
nμ =
r
F
G
δμ
u −
F
2
δμ
r , (27b)
mμ =
1
√
2H
δμ
θ +
i
sin θ
δμ
φ
, (27c)
lμlμ = mμmμ = nνnν = lμmμ = nμmμ = 0, (27d)
lμnμ = −mμm∗μ = 1. (27e)
Then, one introduces the rotation via the complex transformation, Eq. (16), under which δμ
ν
transform as a vector:
δμ
r → δμ
r , δμ
u → δμ
u, δμ
θ → δμ
θ + ia sin θ(δμ
u − δμ
r ), δμ
φ → δμ
φ. (28)
For an 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,Ψ}:
{G(r), F(r),H(r)} → {A(r, θ, a),B(r, θ, a), Ψ(r, θ, a)}, (29)
9
where {A,B,Ψ} are real functions to be determined, and they should recover their static counterparts
in the limit a → 0, namely,
lim
a→0
A(r, θ, a) = G(r), lim
a→0
B(r, θ, a) = F(r), lim
a→0
Ψ(r, θ, a) = H(r). (30)
According to Eq. (28) and Eq. (29), the null tetrad becomes
lμ = δμ
r , (31a)
nμ =
r
B
A
δμ
u −
B
2
δμ
r , (31b)
mμ =
1
√
2Ψ
δμ
θ + ia sin θ(δμ
u − δμ
r ) +
i
sin θ
δμ
φ
, (31c)
and the corresponding metric with rotation takes the form,
ds2 = − Adu2 − 2
r
A
B
dudr − 2a sin2 θ
r
A
B
− A
!
dudφ + 2a sin2 θ
r
A
B
drdφ
+ Ψdθ2 + sin2 θ
"
Ψ + a2 sin2 θ
2
r
A
B
− A
!#
dφ2.
(32)
Next, one rewrites the above metric with the Boyer-Lindquist coordinates, and lets the metric
have only one off-diagonal term gtϕ. To reach the goal, one needs the following coordinate
transformation,
du = dt + λ(r)dr, dφ = dϕ + χ(r)dr, (33)
where {λ(r), χ(r)} must depend only on r to ensure integrability. If the transformation Eq. (29)
is a priori determined, {λ(r), χ(r)} may not exist. Considering these constraints, one has the
formulations of {A(r, θ, a),B(r, θ, a), λ(r), χ(r)},
A(r, θ) =
(FH + a2 cos2 θ)Ψ
(K + a2 cos2 θ)2 , (34a)
B(r, θ) =
FH + a2 cos2 θ
Ψ
, (34b)
λ(r) = −
K + a2
FH + a2 , (34c)
χ(r) = −
a
FH + a2 , (34d)
where K(r) is defined by
K(r) ≡
s
F(r)
G(r)
H(r). (35)
As a result, one obtains the metric for rotating RBHs with the Kerr-like form,
ds2 =
Ψ
ρ2
−
1 −
2f
ρ2
dt2 +
ρ2
Δ
dr2 −
4af sin2 θ
ρ2 dtdϕ + ρ2dθ2 +
Σsin2 θ
ρ2 dϕ2
, (36)
10
where
ρ2 ≡ K + a2 cos2 θ, 2f(r) ≡ K − FH
Δ(r) ≡ FH + a2, Σ ≡ (K + a2)2 − a2Δsin2 θ.
(37)
In the above metric, Ψ(r, θ, 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, Ψ obeys [120] the Einstein field equations,
K + a2y2 2
(3Ψ,rΨ,y2 − 2ΨΨ,ry2) = 3a2K,rΨ2, (38)
K2
,r + K(2 − K,rr) − a2y2(2 + K,rr)
Ψ + (K + a2y2)(4y2Ψ,y2 − K,rΨ,r) = 0, (39)
where “,” in the subscript of variables means derivatives and y ≡ cos θ. However, it is almost
impossible to determine Ψ(r, θ, a) in this way because of the high complexity. For the RBH metrics
mentioned in Sec. 2.1, one usually chooses
Ψ(r, θ, a) = H(r) + a2 cos2 θ. (40)
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. (40)
is compatible with the NJA and is available to construct a rotating RBH, but it is still unclear
whether Eq. (40) is the only choice or not.
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,
ds2 = −
Δ
ρ2 (dt − a sin2 θdϕ)2 +
ρ2
Δ
dr2 + ρ2dθ2 +
sin2 θ
ρ2
adt − (r2 + a2)dϕ
2
, (41)
where
ρ2 = r2 + a2 cos2 θ, Δ = r2 − 2Mσ(r)r + a2. (42)
This metric reduces to the Kerr metric when σ(r) = 1, and to the KN metric when σ(r) =
1 − q2/(2Mr).
Further, the metric Eq. (41) belongs [93, 104] to the Petrov type D because Ψ2 can be the
only non-vanishing scalar when Ψ0,Ψ1,Ψ3 and Ψ4 vanish simultaneously, where the five complex
scalar functions can be expressed by the Weyl tensor Cκλμν as follows:
Ψ0 = Cκλμνlκmλlμmν, (43a)
Ψ1 = Cκλμνlκkλlμmν, (43b)
Ψ2 = Cκλμνlκmλm∗μkν, (43c)
Ψ3 = Cκλμνkκlλkμm∗ν, (43d)
Ψ4 = Cκλμνkκm∗λkμm∗ν. (43e)
11
Therefore, the algebraically complete set of second-order invariants is {R, I, I6,K} [93, 94, 104],
which means that Eq. (41) is regular if the set of invariants does not diverge anywhere. The
definitions of R and K have been given in Sec. 2.2, and the definitions of I and I6 are
I ≡
1
24
C∗
αβγδC∗αβγδ, (44)
I6 ≡
1
12
Sα
βSβ
α, (45)
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. (41): If σ(r) is a C3 function,
then it demands
σ(0) = 0, σ′(0) = 0, σ′′(0) = 0. (46)
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 [123–125].
3 Interpretation of regular black holes
A complete RBH theory refers to physical interpretations from either the quantum theory
of gravity, such as loop quantum gravity and asymptotic safety method, 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 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. [98, 126], 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. 1a.
In order to construct a RBH, we drag the coordinate system downward by a transformation
r → r(ξ), in such a way that the coordinate system cannot cover the singularity after the operation,
see Fig. 1b, where r →
p
ξ2 + l2 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 ξ is defined
in ξ ∈ [0,∞), i.e., the singularity is dragged to the imaginary axis after analytically continuing
ξ into a complex plane. This operation is equivalent to restricting the old radial coordinate in
r ∈ [l,∞) by hands. The singularity is subtracted from the “old” spacetime, such that the “new”
12
1 1 2 3
ˆX
1.5
1.0
0.5
0.5
1.0
1.5
ˆT
r=0
(a) r ∈ [0,∞).
1 1 2 3
ˆX
1.5
1.0
0.5
0.5
1.0
1.5
ˆT
»=0
(b) ξ ∈ [0,∞)
Figure 1: Penrose diagrams of a Schwarzschild BH in two different radial coordinates.
spacetime is regular. In other words, the “new” coordinates cover a smaller portion of the manifold
than the old coordinates in the Schwarzschild spacetime, but the topology of the manifold never
changes. We call the Schwarzschild BH in ξ 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 [24],
ds2 = −fdt2 + f−1dr2 + r2dΩ2, f = 1 −
2Mr2
(r2 + g2)3/2 , (47)
where M is mass and g magnetic charge of monopoles, we know that the Kretschmann scalar is
regular in r ∈ [0,∞). After a replacement, r2 → ξ2 − g2, the metric Eq. (47) becomes
ds2 = −fdt2 +
f−1ξ2
ξ2 − g2 dξ2 + (ξ2 − g2)dΩ2, f = 1 −
2M(ξ2 − g2)
ξ3 . (48)
If we take Eq. (48) as an independent metric describing a “new” spacetime, the corresponding
Kretschmann scalar diverges around ξ = 0,
K ∼
900g8M2
ξ14 + O
1
ξ13
. (49)
It seems that the Bardeen BH redisplays the singularity at ξ = 0, and even that almost all the
RBHs regain singularities by a replacement of r → r(ξ). However, this is not the case because ξ
could never be smaller than g, otherwise the signature in Eq. (48) changes in such a way that the
line-element represents a manifold with two time dimensions (breaking of the causality) and two
space dimensions and the integral measure
√
−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 =
12M2 (−4g6r2 + 47g4r4 − 12g2r6 + 8g8 + 4r8)
(g2 + r2)7 , (50)
13
we observe that the singularities are moved to the non-physical domain, e.g., r = ±ig ∈ C.
Generally, one can classify RBHs into three types by the characteristics of singularities [87]:
• The first type corresponds to those RBHs whose geodesics are complete in the domain of
r ∈ [0,∞) although their curvature invariants have essential singularities at r = 0 from the
perspective of complex analysis, e.g., σ(r) = e−1/r, see Ref. [96];
• 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 [127];
• 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 [35].
This classification directly affects the calculation of the asymptotic frequencies of quasinormal
modes (QNMs) by the monodromy method [87].
3.2 How to find the sources of non-rotating regular black holes?
Gliner discussed [18] an algebraic property of a four-dimensional energy-momentum tensor
(EMT) denoted by [(1111)], where symbol 1 corresponds to one diagonal component of the EMT
and the parentheses imply equal components,f see Refs. [103, 128] for the Segr´e notations. The
matter with the algebraic property [(1111)], called a μ-vacuum, has a de Sitter-like metric and
thus avoids singularities. Later, Gliner’s work was extended [1, 129], where there are four types
of algebraic properties in general for spherically symmetric BHs,
[(1111)], [(11)(11)], [11(11)], [(111)1]. (51)
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,
G0
0 = G1
1 =
f′(r)
r
+
f(r)
r2
−
1
r2 , (52a)
G2
2 = G3
3 =
f′(r)
r
+
f′′(r)
2
, (52b)
where we have supposed that Einstein’s theory of gravity still holds, Gμ
ν = 8πTμ
ν, thus we can
discuss the algebraic properties of EMTs in terms of Einstein’s tensor.
One example given in Ref. [98] has the property [11(11)]. For the metric with the form of Eq.
(3), one has the components of the Einstein tensor,
G0
0 =
f′ρ′
ρ
+
fρ′2
ρ2 +
2fρ′′
ρ
−
1
ρ2 , (53a)
G1
1 =
f′ρ′
ρ
+
fρ′2
ρ2
−
1
ρ2 , (53b)
fAs in Ref. [18], we do not distinguish between time and space components by a comma.
14
G2
2 = G3
3 =
f′ρ′
ρ
+
f′′
2
+
fρ′′
ρ
, (53c)
where G0
0 does not equal G1
1 generally. However, when ρ ∝ r, G0
0 = G1
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. [27, 52]. Here
[(111)1] implies G0
0 = G2
2 = G3
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 which 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. 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 [130–132],
F(R) = R + αR2, (54)
the gravitational equations read
Fμ
ν ≡ F′(R)Rμ
ν −
1
2
F(R)gμ
ν − [∇μ∇ν − gμ
ν□]F′(R) = 8πTμ
ν. (55)
For the metric Eq. (1), the components of tensor Fμ
ν take the forms,
2r4F0
0 = − 4α − 2r3f′
2αf′′ + αrf(3) − 1
+ 2f
12α − 2αr2
r2f(4) + 2f′′ + 6rf(3)
+ 8αrf′ + r2
+ 4αr2f′2 + αr4f′′2 − 20αf2 − 2r2,
(56a)
2r4F1
1 = − 4α − 2r3f′
2αf′′ + αrf(3) − 1
+ 2f
−12α − 4αr2
4f′′ + rf(3)
+ 8αrf′ + r2
+ 4αr2f′2 + αr4f′′2 + 28αf2 − 2r2,
(56b)
2r4F2
2 = 2r4F3
3 =4α + 2rf′
−12α − 2αr2
5f′′ + rf(3)
+ r2
+ 4αf
2r2f′′ − r3
5f(3) + rf(4)
+ 8rf′ + 6
+ 8αr2f′2 + r4f′′ (1 − αf′′) − 28αf2.
(56c)
15
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 [133],
S =
Z
d4x
√
−g W, (57)
where W is the Weyl scalar defined by contracting two Weyl tensors. The variation to the action
gives the gravitation-dependent field, Bμ
ν, which is called the Bach tensor. For the metric Eq.
(1), we obtain
24r4B0
0 = − 4rf
r
r2f(4) − f′′ + 3rf(3)
+ 2f′
+ r2 (rf′′ − 2f′)2 − 2r4f(3)f′ + 4f2 − 4,
(58a)
24r4B1
1 = −2r3f(3) (rf′ − 2f) + [r (rf′′ − 2f′) + 2f]2 − 4, (58b)
24r4B2
2 = 24r4B3
3 = − r2 (rf′′ − 2f′)2 + 2r4f(3)f′ − 4f2 + 4
+ 2rf
r
r2f(4) − 2f′′ + 2rf(3)
+ 4f′
,
(58c)
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 that of 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 spacetime without electromagnetic fields. Meanwhile, the resulting rotating RBH
matches [93,104,134] the Segr´e type, [(11)(11)]. So the interpretation works well in the aspect of
imperfect fluids for the models in Refs. [113, 120].
For the spacetime with electromagnetic fields, it is widely used [2, 25, 96] 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 Fμν is changed from one to four by the introduction of rotation, that
is, F01, F02, F13 and F23 are non-trivial, where the field strength is defined by Fμν ≡ ∂μAν −∂νAμ.
In the metric Eq. (41), these four components satisfy [135, 136] the relations,
F31 = a sin2 θF10, aF23 = (r2 + a2)F02. (59)
The gravitational field coupled to nonlinear electrodynamics is described by the action,
S =
1
16π
Z
d4x
√
−g[R − L(F)], (60)
F = FμνFμν. (61)
16
Using the Einstein field equations,
Gμν = 2LFFμαFν
α −
1
2
δμνL, (62)
one can determine L and LF , where LF ≡ dL/dF. To further determine Fμν one needs to
utilize the dynamic equations. The variation of the action with respect to Aμ yields the dynamic
equations,
∇μ (LFFμν) = 0, ∇μ
∗Fμν = 0, (63)
where ∗Fμν ≡ 1
2ημναβFαβ, and η0123 = −1/
√
−g. Then one obtains that the non-zero components
of Fμν satisfy the following equations,
∂
∂r
(r2 + a2) sin θLFF10
+
∂
∂θ
[sin θLFF20] = 0, (64a)
∂
∂r
[a sin θLFF10] +
∂
∂θ
1
a sin θ
LFF20
= 0, (64b)
∂
∂r
F20 −
∂
∂θ
F10 = 0, (64c)
∂
∂θ
a2 sin2 θF10
−
∂
∂r
(r2 + a2)F20
= 0. (64d)
Because LF 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 [137–139] the change of gauge fields under the NJA.
When the RN metric is transformed into the KN metric, the gauge potential Aμ changes [140] in
the following way.
In the RN metric, Aμ can be written as
Aμ =
q
r
δu
μ, (65)
and its contravariant counterpart takes the form,
Aμ = −
q
r
δμ
r = −
q
r
lμ, (66)
where lμ is the tetrad in Eq. (27). Under the transformation governed by Eqs. (19) and (28), the
gauge potential becomes
A˜μ = −
qr
ρ2 δμ
r , (67)
and its 1-form reads
A˜ =
qr
ρ2 (du − a sin2 θdθ), (68)
which can be written as
A˜ =
qr
ρ2
dt −
ρ2
Δ
dr − a sin2 θdθ
, (69)
due to du = dt − ρ2
Δ dr. Because the factor qr
Δ depends only on r, the term of dr can be removed
by a gauge transformation and the final formulation of the gauge potential can be simplified to
be
A˜ =
qr
ρ2
dt − a sin2 θdϕ
, (70)
17
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. (19)
may not be applicable to the gauge potentials in RBHs. For example, the gauge field of spherically
symmetric RBHs with the magnetic charge Qm is Aμ = Qm cos θδϕ
μ, then the gauge field of rotating
RBHs will become [139]
Aμ = −
Qma cos θ
ρ2 δtμ
+
Qm(r2 + a2) cos θ
ρ2 δϕ
μ, (71)
if one uses the above method. But LF calculated by Eq. (71) is different [141] from that by
Eq. (62), 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 [50, 125]. 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 [142].
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 [123]. 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 [143, 144]. In RBHs, the situation is improved
[27, 119, 133, 145]. For instance, the metric of conformal RBHs in Ref. [133] reads
ds2 =
1 +
L2
r2
2n
−fdt2 + f−1dr2 + r2dΩ2
, (72)
where L is the regularization parameter with the length dimension and f = 1−2M/r. This RBH
model can be produced by the following action,
Iconf = −
1
2
Z
d4x
√
−gϕ
1
6
Rϕ − □ϕ
, (73)
where ϕ is the scalar field. The equation of motion for the scalar field ϕ is
□ϕ =
1
6
Rϕ, (74)
and its solution takes [146] the form,
ϕ =
1 +
L2
r2
−n
c1
2M
ln
1 −
2M
r
+ c2
, (75)
where c1 and c2 are integration constants. This solution is divergent at the horizon, rH = 2M,
which implies c1 = 0, i.e., the solution can be simplified to be
ϕ = c2
1 +
L2
r2
−n
. (76)
18
Since n ≥ 1, ϕ is bounded by 0 ≤ ϕ ≤ c2. In other words, Eq. (73) has a non-trivial scalar hair
because of a non-minimal coupling [143].
Another example is shown in Ref. [27,52], where the action is cast in the Einstein gravity with
a minimally coupled (phantom) scalar field ϕ,
Iphatom =
Z
d4x
√
−g [R − ∂μϕ∂μϕ − 2V (ϕ)] , (77)
where V (ϕ) is potential, see Ref. [27, 52] 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(ρ) = 1 −
ρ0 (πb2 − 2bρ + πρ2)
2b3 +
ρ0 (b2 + ρ2)
b3 tan−1
ρ
b
, (78)
where we have used the condition 2bc = −πρ0 with ρ0 > 0 and b > 0 to replace c in the original
formula in Ref. [27, 52]. In addition, the potential takes the form,
V (ϕ) = −
ρ0
2b3
h
2
√
2ϕ + 3 sin
√
2ϕ
+ (π −
√
2ϕ) cos
√
2ϕ
− 2π
i
, (79)
thus we obtain
ϕV ′(ϕ) =
ρ0ϕ
2b3
h
(
√
2π − 2ϕ) sin
√
2ϕ
− 2
√
2 cos
1 +
√
2ϕ
i
. (80)
Because ϕV ′(ϕ) is not always positive, it is not restricted [143] by the no-hair theorem. Thus the
non-trivial scalar-hair solution exists,
ϕ = ±
√
2 tan−1
ρ
b
+ ϕ0, (81)
where ϕ0 is integration constant, and it is bounded by ϕ0 − π/
√
2 < ϕ < ϕ0 + π/
√
2.
4 Energy conditions of regular black holes
The energy conditions are important to the study of RBHs. On one hand, they are related to
the formation of RBHs, and on the other hand, they are regarded as 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, 147] that RBHs can be constructed when the singularity at their
centers is replaced by a 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 [100, 148],
dΘ
dτ
= −Rμνuμuν, (82)
19
where τ is proper time, uμ is four-velocity and Θ 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. (82). Then, choosing uμ = (1, 0, 0, 0), we arrive at
dΘ
dτ
= −R00 = −4πG
ρ +
X3
i=1
pi
!
, (83)
wPhere ρ is energy density and pi are three components of pressure. The violation of SEC, ρ + 3
i=1 pi < 0, implies an increasing Θ along the proper time, i.e., the interaction is repulsive.
Nevertheless, it was also discovered [56, 96, 149, 150] that RBHs can have an anti-de Sitter or
a flat core. For instance, the RBH constructed in Ref. [149] is described by
ds2 = −fdt2 + f−1dr2 + r2dΩ2, f = 1 +
r4
r4 + 2qQ2
m
−
2M
r
+
Q2
m
r2
, (84)
where q is a positive parameter describing the non-minimal coupling of Yang-Mills fields, Qm is a
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 ∼ 1 +
r2
2q
+ O(r3), R ∼ −
6
q
+ O (r) (85)
as r approaches zero. Moreover, a spherically symmetric RBH model with a flat core is highlighted
in Ref. [150, 151], where the shape function reads [152]
f = 1 −
2M
r
e−a/r. (86)
It is obvious that f → 1 and R → 0 as r approaches 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 [53] on the introduction
of the Tolman mass which can be regarded as a kind of integral SEC [153, 154],
mT =
1
4π
Z
√
−gR00d3x =
Z
r2R00dr. (87)
The integral SEC breaks in the core (r ∈ [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. (84) and Eq.
(86), we can conclude that the two models violate the integral SEC in their cores.
In summary,g it is the impressionist SEC that plays the key role in constructing RBHs, not the
pointillist one [155]. 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 [156].
gAccording to Ref. [155], the energy conditions can be divided into two categories: One restricts average
behaviors across regions of spacetime, and the other restricts behaviors at specific points. Here the “impressionist
SEC” means the former, and the “pointillist SEC” means the latter.
20
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 [51, 135, 157–160] 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) [155]. 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 [100].
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,
WEC : σ′ ≥ 0 ∪ rσ′′ ≤ 2σ′,
NEC : rσ′′ ≤ 2σ′,
DEC : σ′ ≥ 0 ∪ −2σ′ ≤ rσ′′ ≤ 2σ′,
(88)
where the prime denotes the derivatives w.r.t. r. The relation between these three energy conditions
can be represented by
NEC ⊂ WEC ⊂ DEC. (89)
Among these three independent differential inequalities in Eq. (88), σ′ ≥ 0 implies that σ is a
monotone increasing function of r; while rσ′′ ≤ 2σ′ provides σ ≤ σ0r3, where σ0 ≡ limr→0 σ/r3; the
last one rσ′′ ≥ −2σ′ gives a solution, rσ ≥ 0, under the boundary conditions, σ|r=0 = 0 = σ′|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(ξ), which causes the
inequalities to be unsolvable without extra specific constraints. Therefore, we cannot extract any
valuable information from these differential inequalities.
In addition, some RBH models violate [87,161,162] the three energy conditions. For instance,
the well-known Bardeen and Hayward BHs break the DEC, and the RBH generated by the nonminimal
coupled Wu-Yang monopole brakes [39, 40, 75, 163, 164] the WEC, etc. A remedy is
proposed in Ref. [165] by deforming the shape function. For a generic σ function, its deformed
formulation reads
σ =
Mμν−3r3
(rμ + qμ)ν , (90)
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 μ and ν
take the values in the following regions,
2
ν
< μ ≤
1
2
r
49ν + 96
ν
−
7
2
when
2
5
< ν ≤ 3; (91a)
2
ν
< μ ≤
3
ν
when ν > 3, (91b)
where Mμν−3 is introduced for balancing the dimension, and this parameterization is not unique.
21
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 in the context of Einstein’s gravity coupled with
nonlinear electrodynamics.
5.1 What is the entropy of regular black holes?
It was reported [37, 166–171] that RBHs have a deviation term in entropy, i.e., the entropy
of RBHs breaks the area law, S ̸= A/4, but the opposite opinion was also reported [26, 59,
62, 172–174]. 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 [118] as an example, whose shape function is
f = 1 −
2M
r
r3
r3 + 2Ml2 , (92)
where l is the length scale introduced for regularization, we can obtain the entropy from the first
law of thermodynamics, dM = TdS,
S =
Z r+
r−
dM
T
= SBH + ΔS, (93)
where r+ and r− are outer and inner horizons, respectively, SBH is the Bekenstein-Hawking entropy,
SBH = π
r2+
− r2−
, (94a)
and ΔS is a deviation term,
ΔS =
πl4
r2+
− r2−
(r2−
− l2) (r2+
− l2)
+ 2πl2 ln
r2+
− l2
r2−
− l2
. (94b)
We note ΔS > 0 because of r+ ≥ r− > l, otherwise, there will be no horizons. As a result, if
Eq. (93) is still used to calculate the entropy, the area law is violated, S ̸= A/4, even though a
pressure term,
P = −
3
8πl2 , (95)
is added [175,176]. We note that the pressure in Eq. (95) is introduced in a dS spacetime, not in an
AdS spacetime due to the minus sign meaning an outward pressure from the center of BHs. For a
varying cosmological constant, its corresponding pressure can be introduced whether a spacetime
is an AdS or a dS one because the thermodynamic relations of BHs can remain consistent from
a mathematical standpoint, see, for instance, Refs. [175–177]. Here the Hayward BHs have a
dS core. On one hand, the dS core is related to a length scale regarded as the regularization
parameter in Eq. (92). On the other hand, it is related to the pressure expressed by the so-called
22
dS radius in Eq. (95). Considering the uniqueness of the cosmological constant, we can deduce
the equality of the two length scales in Eqs. (92) and (95).
As a matter of fact, if we interpret the metric, Eq. (1), as a spacetime produced by a field of
Dirac magnetic monopoles [2,178], 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 =
1
16π
Z
d4x
√
−g [R − L(F)] , (96)
where F is the contraction of the electromagnetic tensors, F ≡ FμνFμν = 2q/r4, and q is the
magnetic charge of the monopole. By solving one of Einstein’s equations [26],
L
2
−
2Mσ′(r)
r2 = 0, (97)
and replacing r by r = [F/(2q)]1/4, we can determine the Lagrangian L(F).
To calculate the entropy, we apply the path-integral method in the zero-loop approximation
[179],
Z =
Z
DgDA e−I ≈ e−Icl , (98)
where the full Euclidean action Icl consists of four parts,
Icl = IEH + IGHY − I0 + IM. (99)
IEH is the Einstein-Hilbert action,
IEH = −
1
16π
Z
M
d4x
√
−g R, (100)
IGHY is the Gibbons-Hawking-York boundary term,
IGHY = −
1
8π
Z
∂M
d3x
√
−hK, (101)
I0 is the subtraction term,
I0 = −
1
8π
Z
∂M
d3x
√
−hK0, (102)
and IM is the matter action of a nonlinear electrodynamic source, where R is the Ricci curvature
of bulk space, K and K0 are extrinsic curvatures of surface and background reference, respectively.
For the metric Eq. (1), we obtain
IEH =
β
2
(rH −M) − πr2H
, (103)
IGHY =
M
2
β (rσ′ + 3σ) − rβ ∼
3M
2
β − rβ, (104)
I0 = −rβ
r
1 −
2Mσ
r
∼ −rβ +Mβ + O
r−1
, (105)
23
where β is the inverse of BH temperature, 1/T , and the last two asymptotic relations at the
boundary r → ∞ are derived based on the assumption that the RBH is asymptotic to the
Schwarzschild BH at infinity [96], i.e., limr→∞ σ = 1 and limr→∞ rσ′ = 0. Meanwhile, the action
of matter can be determined with the help of one component of Einstein’s equations Eq. (97),
IM = βM −
βrH
2
. (106)
Substituting Eqs. (103)-(106) into Eq. (99), we arrive at the total Euclidean action,
Icl = βM − πr2H
. (107)
On the other hand, the thermodynamic law for the canonical ensemble is F = M − TS, where
F = TIcl is Helmholtz free energy. Thus, we can read S = πr2H
from Eq. (107), 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. [180].
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. (96). If we reinterpret the source as dyons, the path-integral
method is not applicable because no actions for dyons can be constructed [26]. 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 [181]. More
precisely, unexpected extra termsh 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 [25], thus the first mechanical law reads [181]
dM =
κ
8π
dA + ΨHdq + KMdM + Kqdq, (108)
where κ is the surface gravity, ΨH 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 [26] introduced a parameter α in the
action of nonlinear electrodynamics. According to their method, the first thermodynamic law for
the Bardeen BH, Eq. (47), can be cast in the following form,
dE = TdS + ΨHdQm + Πdα, (109)
hExtra terms indicate additional terms for a parameter, e.g., for mass M in Eq. (108), KMdM is an extra term,
while for charge q, Kqdq is an extra term.
24
where the three thermodynamic variables,
E = M, Qm =
p
Mq/2, α = q3/M = 8Q6
m/M4, (110)
are not independent of each other in the phase space. In other words, the dimension of thermodynamic
phase space is two because α is a redundant dimension. The other correspondences
between thermodynamic and mechanical variables are
T ←→
κ
2π
, S ←→
A
4
. (111)
Nevertheless, Eq. (109) is still problematic because
A
4
= S ̸=
Z
dE
T
=
Z
dM
T
, (112)
where the integral is calculated under the condition, dQm = 0 = dα. The reason of S ̸=
R
dM/T
or T ̸= (∂E/∂S)Q,α is that the relations, dQm = 0 = dα, imply that M is also a constant, see Eq.
(110), which leads to a trivial integral,
R
dM/T ≡ 0.
If only one parameter is fixed, say dQm = 0, we have
A
4
=
Z
dM
T
1 +
32Q6
mΠ
M5
̸=
Z
dM
T
, (113)
namely, the simple thermodynamic relation, S =
R
dM/T , is broken. This result gives rise to a
tendency to abandon [37, 166–171, 182–184] the area-entropy law,
S =
Z
dE
T
̸=
A
4
. (114)
In the above treatment, the entropy S cannot be determined if the first law of thermodynamics
is not applied. 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 law of thermodynamics is. We
list its most important features below:
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. Every thermodynamic variable should be independent of the first thermodynamic law, i.e.,
it can be determined without the first thermodynamic law, but the thermodynamic formula,
S =
R
dE/T , must hold. Here are some counterexamples: Temperature is not independent
in Ref. [185], deviation of internal energy is not independent in Ref. [186], etc.
3. Every thermodynamic variable should be independent of each other, e.g., dM = TdS +
K1dα + K2dβ + . . ., is ill-defined if α = M and β = TM that make α and β dependent on
M in the thermodynamic phase space.
25
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 applied [97,185,187] the following
form in terms of the Gliner vacuum,
dU = TdS − P+dV, (115)
where the total internal energy reads
U =
r+
2
, (116)
P+ and V are the thermodynamic pressure and volume, respectively,
P+ =
Gr
r
8π
r=r+
, V =
4
3
πr3+
. (117)
6 Thermodynamic geometry of regular black holes
Thermodynamic geometry gives a different way to understand BHs and it is also a very
powerful tool for exploring the microstructures of RBHs. The study of Ruppeiner geometry and
Weinhold geometry can help us to further understand the thermodynamic properties of RBHs.
6.1 What is thermodynamics of regular black hole?
6.1.1 Thermodynamic phase transition and shift of critical points
In recent years, there have been a lot of studies on thermodynamic behaviors of RBHs, and
these behaviors are similar to those in thermodynamics of SBHs, such as gas/liquid phase transitions
[182,188,189], van der Waals-like fluid properties [190–193], triple points [194,195], and heat
engines [196–200]. For a given equation of state, the critical point (Tc, Pc, Vc) of phase transitions
can be determined by the following condition,
∂P
∂V
T
= 0,
∂2P
∂V 2
T
= 0. (118)
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 [201,
202] 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 [59] I
V dP = −
I
ΨdQm +
I
extra term, (119)
which makes Maxwell’s equal area law invalid in the P − V plane. Thus, the critical point (Tc
, Pc) of the first-order small/large BH transition does not coincide with the inflection point of
isotherms.
26
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 [203] by
Johnson, where BHs act as a working substance. The heat engine flows in a cycle as shown in
Fig. 1 of Ref. [203]. Using equations of state of AdS BHs, for instance, we can compute the total
mechanical work as follows,
W = QH − QC, (120)
where QH is the net input heat flow and QC is the net output heat flow. Therefore, the efficiency
of heat engines is defined by
η ≡
W
QH
= 1 −
QC
QH
. (121)
Recently, some progress has been made [196, 198, 204, 205] in the studies of heat engines for
RBHs, where the relation between efficiency and entropy (pressure) has been obtained [198] 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 [206–208] a new perspective on thermodynamic properties of
RBHs.
6.2 How to eliminate the singularity of thermodynamic geometry for
regular black holes?
6.2.1 Construction of thermodynamic geometry
A similar situation has happened in the studies of thermodynamic geometry, where such
geometry has been shown to be a powerful tool for understanding the thermodynamic properties
and microstructures of SBHs. The Ruppeiner metric is defined [209] by the Hessian matrix of
thermodynamic entropy,
gR
μν ≡ −
∂2S(X)
∂Xμ∂Xν , (122)
and the Weinhold metric is defined [210] by the Hessian matrix of BH mass,
gW
μν =
∂2M(X)
∂Xμ∂Xν , (123)
where Xμ 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,
ds2
R =
1
T
ds2
W. (124)
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,
dM =
T
1 − Π
dS −
P
1 − Π
dV, (125)
27
where Π is related to additional terms. In this case, if we constructed thermodynamic geometry
from the problematic first law Eq. (125), the thermodynamic metric would also contain additional
terms associated with Π that has no thermodynamic counterpart. Correspondingly, for RBHs the
conformal relationship between Ruppeiner geometry and Weinhold geometry becomes
ds2
R =
1 − Π
T
ds2
W. (126)
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 [190, 211, 212] in different gravity theories with the
help of thermodynamic geometry. As is known, RBHs are very special BH models and have been
widely concerned [194, 200, 213, 214] 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., CV = 0. This property makes the thermodynamic line element, Eq. (126), singular
and the corresponding thermodynamic information is unavailable from thermodynamic geometry.
A new normalized scalar curvature is defined [215],
RN = RCV , (127)
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 an optional item 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 SBHs 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. For instance,
the coordinate singularity of Hayward BHs [105] can be removed in the Painlev´e-Gullstand
coordinate system [216] and such a coordinate system is not unique, that is, the Kruskal–Szekeres
coordinate and Eddington-Finkelstein coordinate systems [12] have the same effect. We note that
we just eliminate the singularity on an event horizon but not the event horizon itself.
28
The second issue arises due to the construction of rotating RBHs. Bambi and Modesto utilized
[117] the NJA to create the metrics of rotating RBHs. On one hand, the NJA depends on theories
of gravity. Taking a RBH model in the Chern-Simons gravity [217] as an example, the NJA can
provide a rotating metric, but the Pontryagin density, ∗RR, does not vanish [218], 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 SBHs
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 is expected to play a fundamental role in the study of
RBHs. Because the interior of a black hole cannot be directly observed, we have to investigate the
dynamical and thermodynamic phenomena occurred outside horizons, in particular, the differences
of phenomena with and without singularities. These differences may provide a chance to test RBHs
from observational consequences mentioned in Introduction, e.g., the gravitational wave, X-ray,
and BH shadows, etc. The theoretical predictions together with currently available experimental
data may shed some light on the existence of RBHs. |
