Regular Black Holes:
Entropy Products and Central Charges

Hanif<EMAIL_ADDRESS>
Faculty of Physics, Shahid Bahonar University of Kerman,
PO Box 76175, Kerman, Iran and

School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box 19395-5531, Tehran, Iran

In this paper for variety types of regular black hole solutions, we
investigate the entropy product of inner and outer horizons. Similar to
singular black holes, for the regular ones we find that universality (mass
independence) of the entropy product is true for some solutions and it fails
for some others. In the case of regular black holes that respect the
universality, we read central charges of the dual CFTs from the entropy
product, according to the Thermodynamics method introduced in [1]. For these
solutions we also calculate central charges, using the asymptotic symmetry
group formalism. The results of these two approaches are the same, which means
that universality of the entropy product provides a simple method to find
central charges of the dual CFTs.

###### Contents

  1. 1 Introduction
  2. 2 Universality of entropy product and dual CFTs
  3. 3 RBHs with non-universal entropy product
    1. 3.1 The Bardeen BH
    2. 3.2 The Hayward BH
    3. 3.3 The rotating Bardeen and Hayward BHs
    4. 3.4 The Frolov-Zelnikov BH
    5. 3.5 Five dimensional RBHs
      1. 3.5.1 Static RBH
      2. 3.5.2 Magnetically charged Myers-Perry solution
      3. 3.5.3 Electrically charged rotating Bardeen BH
  4. 4 RBHs with universal entropy product
    1. 4.1 The Kerr-like RBH
    2. 4.2 Reissner-Nordström outside a de Sitter core
    3. 4.3 The topological star
    4. 4.4 BHs in Einstein gravity coupled to extended nonlinear electrodynamics
      1. 4.4.1 Bardeen class
      2. 4.4.2 Hayward class
      3. 4.4.3 new class
  5. 5 Central charges and entropy product
    1. 5.1 Central charge of The Kerr-like RBH
    2. 5.2 Central charge of the charged RBHs
  6. 6 Conclusions

##  1 Introduction

According to the singularity theorem [2], formation of singularities are
unavoidable (under some circumstances) in general relativity. The existence of
singularity is the failure of general relativity and to protect it, Penrose
proposed that singularities are covered by horizons (the weak cosmic
censorship conjecture). It is believed that singularity appears in the
classical theory of gravity and by taking into account the quantum effects,
they can be avoided [3, 4]. Inspired by this idea, Bardeen proposed [5] a
regular black hole (RBH) solution in which the singularity is replaced by a de
Sitter core. After the seminal work of Bardeen, a large number of RBHs are
constructed [6]-[20]. Recently it is also proposed some models for RBHs with a
Minkowski core [14, 15, 21, 22].

Black holes (BH)s are good locations to explore the relation of gravity and
quantum mechanics. It has been observed [23]-[35] that the horizons entropy
product for many BH solutions is universal (mass independent) as
$S_{+}S_{-}=4\pi^{2}N$, where $N$ is related to the quantized a charges of the
solution, like angular momentum and electric/magnetic charge. It is shown [29]
that the condition $T_{+}S_{+}=T_{-}S_{-}$, where $T_{+}$ ($T_{-}$) is the
Hawking temperature of the outer (inner) horizon, is equivalent to the mass
independence of $S_{+}S_{-}$ . This condition also implies that central
charges of the left-moving and right-moving sectors in the dual conformal
field theory (CFT) are the same ($c_{L}=c_{R}$). It is discussed in [1] that
for the solutions with universal entropy product, one can read the central
charge(s) of the dual CFT(s) as

$c_{i}=\frac{6}{4\pi^{2}}\,\frac{\partial\left(S_{+}S_{-}\right)}{\partial
N_{i}}\,,$ (1.1)

which is the Thermodynamics method for finding the central charge.

There are also BH solutions in which the entropy product is not universal. For
instance, it is observed [36] that the universality of entropy product fails
for some BH solutions of higher curvature gravity. The entropy product is also
mass dependent for the BHs with NUT charge [37, 38]. Although in the case of
5D Myers-Perry and the BTZ black hole the universality is true, it fails for
the Myers-Perry BHs in $D\geq 6$ and for Kerr-AdS BHs in $D\geq 4$ [29].

In the case of RBHs it is shown [39] that the universality fails for the ABG
(Ayón-Beato and García) BH. In [18] it is shown that for RBHs of Einstein
gravity coupled to extended nonlinear electrodynamics, by choosing adequate
parameters $S_{+}S_{-}$ can be mass independent. However for large number of
RBHs (maybe because of the complex form of the metric), the entropy product is
not studied yet.

This paper is organized as follows. In section 2 we briefly review the
Thermodynamics method for finding central charge(s) of the dual CFT(s). Then
we investigate the entropy product for RBHs where the universality fails for
some of them (solutions in section 3 ) and it works for some others (section
4). In section 5 using Thermodynamics method, we find the central charges of
the dual CFTs for RBHs with universal entropy product, studied in section 4.
We then recalculate the central charges using the asymptotic symmetry group
formalism to check the result of Thermodynamics method. We conclude the paper
with a discussion on the universality of entropy product and its importance in
the BH solutions.

Throughout the paper we set the Newton’s gravitational constant to unity, i.e.
$G=1$.

##  2 Universality of entropy product and dual CFTs

In this section we review some BH solutions with universal entropy product.
For these solutions, it is possible to find central charges of the dual CFTs,
using the Thermodynamics method.

$\bullet$ The Kerr-Newman BH
The Kerr-Newman BH [40] is determined by three conserved charges $M$ (mass),
$J$ (angular momentum) and $Q$ (electric charge). In [25] it has been shown
that the entropy product of inner and outer horizons for this solution is
universal as

$S_{+}S_{-}=\pi^{2}(4J^{2}+Q^{4})\,.$ (2.1)

The dual CFT for this solution is studied in [41, 42] where it is shown that
the $J$-picture central charge $c^{J}$ and the $Q$-picture central charge
$c^{Q}$ are give by

$c^{J}=12\,J\,,\qquad c^{Q}=6Q^{3}\,.$ (2.2)

Due to the universality of entropy product, one can also find these central
charges by using Thermodynamics method (1.1) as

$\displaystyle c^{J}=\frac{3}{2}\,\frac{\partial}{\partial
J}\,\Big{(}4J^{2}+Q^{4}\Big{)}=12J\,,$ $\displaystyle
c^{Q}=\frac{3}{2}\,\frac{\partial}{\partial
Q}\,\Big{(}4J^{2}+Q^{4}\Big{)}=6\,Q^{3}\,,$ (2.3)

which are in agreement with (2.2).

$\bullet$ The Myers-Perry BH
The Myers-Perry BH [43] in five dimensions is characterized by its two angular
momenta $J_{\phi}$, $J_{\psi}$ and mass $M$. Applying the Kerr/CFT analysis,
it is shown that [44, 45, 46] for the Myers-Perry BH, there are two dual CFT
descriptions associated with the rotations along $\phi$ and $\psi$ directions.
The central charges of these dual CFTs are

$c_{\phi}=6J_{\psi}\,,\qquad c_{\psi}=6J_{\phi}\,.$ (2.4)

It is also shown that [23, 29] the horizons entropy product for this solution
is universal, i.e.

$S_{+}S_{-}=4\pi^{2}J_{\phi}J_{\psi}\,.$ (2.5)

Now using Thermodynamics method (1.1), one can easily find the central charges
(2.4) from the above entropy product.

There are also more examples for the relation between entropy product and
central charges that we refer the reader to [29, 1, 47, 35]. These examples
tell us that the universality of entropy product implies the existence of dual
CFTs. In the following we check the validity of Thermodynamics method in the
case of RBH solutions. To this end, we should find solutions with universal
entropy product, so in the next sections we investigate the universality for
RBHs.

##  3 RBHs with non-universal entropy product

As we mentioned, the product of horizon entropies is mass independent if the
condition $T_{+}S_{+}=T_{-}S_{-}$ is satisfied [29]. In this section and the
next one, by explicit calculations we investigate this universality condition
for RBHs. We consider variety types of regular solutions in four and higher
dimensions, with/without angular momentum and charge. We also consider RBHs
with both de Sitter and Minkowski cores. We find BH horizons by solving
$g^{rr}=0$ which may have negative or complex roots. By outer (inner) horizon
we mean the largest (smallest) positive real root.

###  3.1 The Bardeen BH

Bardeen BH is the first known RBH solution [5]. The matter source which
supplies this solution was unknown for many years till Ayón-Beato and Garsía
showed that the Bardeen BH is a magnetic solution of Einstein equations
coupled to a nonlinear electrodynamics [6, 7]. The metric of Bardeen BH is in
the form

$\displaystyle
ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (3.1) $\displaystyle\hskip
45.52458ptf=1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}\,,$

where $M$ is the mass and $q$ is the magnetic monopole charge. In Bardeen BH,
the singularity at $r=0$ is replaced by a de Sitter core. For this solution,
there are three horizons (roots of $f=0$) which are too long to be written
here. We denote the outer and inner horizons by $r_{+}$ and $r_{-}$
respectively. The entropy and temperature of Bardeen BH on the outer and inner
horizons are

$S_{\pm}=\pi r_{\pm}^{2}\,,\qquad
T_{\pm}=\pm\frac{M{r_{\pm}}\left({{r_{\pm}}}^{2}-2{q}^{2}\right)}{2\pi\left({q}^{2}+{{r_{\pm}}}^{2}\right)^{5/2}}\,.$
(3.2)

It is straightforward to check that $T_{+}S_{+}\neq T_{-}S_{-}$, which means
that entropy product is not universal for the Bardeen BH.

###  3.2 The Hayward BH

Another RBH solution with a de Sitter core is the Hayward BH [8]. For this
solution, the metric is in the static spherically symmetric form (3.1) with
the function

$f=1-\frac{2Mr^{2}}{r^{3}+q^{3}}\,.$ (3.3)

Similar to the Bardeen BH, we find the entropy and temperature on the outer
and inner horizons as

$S_{\pm}=\pi r_{\pm}^{2}\,,\qquad
T_{\pm}=\pm\frac{Mr_{\pm}(r_{\pm}^{3}-2q^{3})}{2\pi(r_{\pm}^{3}+q^{3})}\,,$
(3.4)

one can check that the condition $T_{+}S_{+}=T_{-}S_{-}$ does not satisfied
here, so the universality of entropy product is not valid for the Hayward BH.

###  3.3 The rotating Bardeen and Hayward BHs

Extension of the Bardeen and Hayward RBHs to the solutions containing angular
momentum is done in [9]. The line element for these solutions is

$ds^{2}=-F(r,\theta)dt^{2}+\frac{dr^{2}}{G(r,\theta)}+\Sigma(r,\theta)d\theta^{2}+H(r,\theta)d\phi^{2}-2K(r,\theta)dtd\phi\,,$
(3.5)

with the functions

$\displaystyle F(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{\Pi(r)-a^{2}\sin^{2}\theta}{\Sigma(r,\theta)},\quad
G(r,\theta)=\frac{\Pi(r)}{\Sigma(r,\theta)}\,,\quad\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle K(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{2rm(r)a\sin^{2}\theta}{\Sigma(r,\theta)}\,,\qquad
H(r,\theta)=\frac{(r^{2}+a^{2})^{2}-\Pi(r)a^{2}\sin^{2}\theta}{\Sigma(r,\theta)}\,\sin^{2}\theta\,,$
(3.6)

the function $\Pi(r)$ in the above is introduced as
$\Pi(r)=r^{2}+a^{2}-2rm(r)$ , where $a$ is the rotation parameter. In order to
remove the singularity, $m(r)$ is given in the form

$m(r)=M\left(\frac{r^{p}}{r^{p}+r_{0}^{p}}\right)^{3/p}\,.$ (3.7)

The rotating Bardeen and Hayward BHs are constructed by setting $p=2$ and
$p=3$ in (3.7), respectively. Note that for $r>>r_{0}$ the above mass reaches
the Kerr BH mass $M$. Moreover in the case of $r_{0}=0$ one finds $m(r)=M$ and
(3.5) reduces to the Kerr solution, in other words, one can interpret $r_{0}$
as the deviation parameter from the Kerr BH. Horizons of the solution are
roots of $\Pi(r)=0$. In order to find entropy and temperature on the inner and
outer horizons, it is more convenient to solve $\Pi(r_{+})=0$, $\Pi(r_{-})=0$
and find $a$, $M$ in terms of $r_{-}$ and $r_{+}$ . Noticing this point, we
calculate entropy and temperature for the rotating Hayward BH ($p=3$). The
result is

$\displaystyle S_{+}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\pi\,{r_{+}}^{4}\left(r_{-}+r_{+}\right)\left({r_{0}}^{3}+{r_{-}}^{3}\right)}{\left(r_{-}+r_{+}\right)\left({r_{-}}^{2}+{r_{+}}^{2}\right){r_{0}}^{3}+{r_{-}}^{3}{r_{+}}^{3}}\,,\qquad
S_{-}=\frac{\pi\,{r_{-}}^{4}\left(r_{-}+r_{+}\right)\left({r_{0}}^{3}+{r_{+}}^{3}\right)}{\left(r_{-}+r_{+}\right)\left({r_{-}}^{2}+{r_{+}}^{2}\right){r_{0}}^{3}+{r_{-}}^{3}{r_{+}}^{3}}\,,$
$\displaystyle T_{+}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{(r_{-}-r_{+})\left[2r_{0}^{6}(r_{+}+r_{-})^{2}-r_{0}^{3}r_{+}^{2}(4r_{-}^{3}+4r_{-}^{2}r_{+}+2r_{-}r_{+}^{2}+r_{+}^{3})-r_{-}^{3}r_{+}^{5}\right]}{4\pi
r_{+}^{3}(r_{+}+r_{-})(r_{0}^{3}+r_{-}^{3})(r_{0}^{3}+r_{+}^{3})}\,,$
$\displaystyle T_{-}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{(r_{-}-r_{+})\left[2r_{0}^{6}(r_{+}+r_{-})^{2}-r_{0}^{3}r_{-}^{2}(4r_{+}^{3}+4r_{+}^{2}r_{-}+2r_{+}r_{-}^{2}+r_{-}^{3})-r_{+}^{3}r_{-}^{5}\right]}{4\pi
r_{-}^{3}(r_{+}+r_{-})(r_{0}^{3}+r_{-}^{3})(r_{0}^{3}+r_{+}^{3})}\,.$ (3.8)

In the case of rotating Bardeen BH ($p=2$), $S_{\pm}$ are messy and we do not
show them here. We checked that for both rotating Bardeen and Hayward
solutions $T_{+}S_{+}\neq T_{-}S_{-}$, means that the entropy product is not
universal for them. In the case of small values for deviation parameter
$r_{0}$, by keeping the first order in Taylor expansion around $r_{0}=0$, it
is possible to find perturbed (around the Kerr) solution and then calculate
the entropy and temperature. For instance we find entropy and temperature for
the Bardeen BH with small $r_{0}$ as

$\displaystyle S_{+}=\frac{\pi
r_{-}(r_{-}+r_{+})(3r_{0}^{2}-2r_{+}^{2})}{3r_{0}^{2}+2r_{-}r_{+}}\,,$
$\displaystyle S_{-}=\frac{\pi
r_{+}(r_{-}+r_{+})(3r_{0}^{2}-2r_{-}^{2})}{3r_{0}^{2}+2r_{-}r_{+}}\,,$
$\displaystyle
T_{+}=\frac{(r_{+}-r_{-})\left[r_{0}^{2}(r_{-}+2r_{+})+\frac{2}{3}r_{-}r_{+}^{2}\right]}{4\pi
r_{+}r_{-}(r_{+}+r_{-})\left(r_{0}^{2}-\frac{2}{3}r_{+}^{2}\right)}\,,$
$\displaystyle
T_{-}=\frac{(r_{+}-r_{-})\left[r_{0}^{2}(r_{+}+2r_{-})+\frac{2}{3}r_{+}r_{-}^{2}\right]}{4\pi
r_{+}r_{-}(r_{+}+r_{-})\left(r_{0}^{2}-\frac{2}{3}r_{-}^{2}\right)}\,,$ (3.9)

it is straightforward to check that, due to violation of
$T_{+}S_{+}=T_{-}S_{-}$, the entropy product of the rotating Bardeen solution
with small $r_{0}$ is not universal. In other words, adding rotation to the
Bardeen and Hayward BHs does not lead to the universality of entropy product.

###  3.4 The Frolov-Zelnikov BH

In [10] an evaporating (time evolving) RBH with a de Sitter core is
introduced. The metric is in the form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
fdt^{2}+\frac{dr^{2}}{f}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ $\displaystyle f$ $\displaystyle=$ $\displaystyle
1-\frac{2M(t)r^{2}}{r^{3}+2M(t)\ell^{2}+\ell^{3}}\,,$ (3.10)

in the static case, this solution differs from the Hayward BH by an extra term
$\ell^{3}$ in the denominator of function $f$. This term guarantees the metric
smoothness at $r=0$ in the limit $M(t)\to 0$. For this solution, one can find
the horizons by solving $f=0$ (in the static case $M(t)=M$) as

$\displaystyle r_{+}$ $\displaystyle=$
$\displaystyle\frac{\ell}{2}+M+\frac{1}{2}\sqrt{4M(M-\ell)-3\ell^{2}},$ (3.11)
$\displaystyle r_{-}$ $\displaystyle=$
$\displaystyle\frac{\ell}{2}+M-\frac{1}{2}\sqrt{4M(M-\ell)-3\ell^{2}},\qquad
r_{0}=-\ell\,.$

For Hayward and Frolov solutions $\ell$ is positive, so the third root
($r_{0}=-\ell$) is a negative valued radius, which is not relevant to our
study. Now it is possible to find the entropy and temperature as

$\displaystyle S_{+}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}(\ell+2M)\big{(}2M-\ell+\sqrt{4M(M-\ell)-3\ell^{2}}\big{)}\,,$
$\displaystyle S_{-}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}(\ell+2M)\big{(}2M-\ell-\sqrt{4M(M-\ell)-3\ell^{2}}\big{)}\,,$
$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{Mr_{+}\big{(}\frac{r_{+}^{3}}{2}-\ell^{3}-2M\ell^{2}\big{)}}{\pi(r_{+}^{3}+2\ell^{2}M+\ell^{3})^{2}}\,,$
$\displaystyle T_{-}$ $\displaystyle=$
$\displaystyle\frac{Mr_{-}\big{(}\frac{r_{-}^{3}}{2}-\ell^{3}-2M\ell^{2}\big{)}}{\pi(r_{-}^{3}+2\ell^{2}M+\ell^{3})^{2}}\,,$
(3.12)

it is easy to check that $T_{+}S_{+}\neq T_{-}S_{-}$ which means that the
entropy product is mass dependent in the case of the Frolov-Zelnikov BH.

###  3.5 Five dimensional RBHs

In this part we investigate the entropy product for some five dimensional BHs.
We consider regular solutions with both Minkowski and de Sitter cores.

####  3.5.1 Static RBH

Five dimensional static RBH is introduced in [13]. The line element is in the
form

$\displaystyle ds^{2}\\!$ $\displaystyle=$
$\displaystyle\\!-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\big{(}d\theta^{2}\\!+\sin^{2}\\!\theta
d\phi^{2}\\!+\cos^{2}\\!\theta d\psi^{2}\big{)},$ (3.13) $\displaystyle\hskip
56.9055ptf(r)=1-\frac{M}{r^{2}}e^{-k/r^{2}}\,.$

This solution which satisfies the equations of motion of 5D Einstein gravity
coupled to nonlinear electrodynamics, is a RBH with Minkowski core at $r=0$.
Here $M$ is the black hole mass and $k$ is related to the magnetic monopole
charge ($q$) via $q^{2}=Mk$. The solution (3.13) reduces to 5D Schwarzschild-
Tangherlini BH when $k=0$ and one can also find the Minkowski space-time by
setting $M=0$. For this RBH temperature on the outer and inner horizons are
[13]

$T_{+}=\frac{1}{4\pi r_{+}}\left(1-\frac{k}{r_{+}^{2}}\right),\quad
T_{-}=\frac{-1}{4\pi r_{-}}\left(1-\frac{k}{r_{-}^{2}}\right),$ (3.14)

and the entropies are

$S_{+}=\frac{\pi^{2}}{2}r_{+}^{3}\,,\quad S_{-}=\frac{\pi^{2}}{2}r_{-}^{3}.$
(3.15)

This solution is not universal since $T_{+}S_{+}\neq T_{-}S_{-}$.

####  3.5.2 Magnetically charged Myers-Perry solution

Applying the Newman-Janis algorithm on the static solution of the previous
subsection, a 5D rotating RBH is constructed in [13]. This solution is a
magnetically charged Myers-Perry BH with Minkowski core. The metric is

$\displaystyle ds^{2}$
$\displaystyle\\!\\!=-\left(1-\frac{\mathcal{M}}{\rho^{2}}\right)dt^{2}+\frac{r^{2}\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}-\frac{2\mathcal{M}a\sin^{2}\theta}{\rho^{2}}dtd\phi-\frac{2\mathcal{M}b\cos^{2}\theta}{\rho^{2}}dtd\psi$
$\displaystyle+$
$\displaystyle\\!\\!\\!\sin^{2}\\!\theta\left(\\!r^{2}\\!+a^{2}\\!+\frac{\mathcal{M}a^{2}\sin^{2}\\!\theta}{\rho^{2}}\right)\\!d\phi^{2}+\cos^{2}\\!\theta\left(\\!r^{2}\\!+b^{2}\\!+\frac{\mathcal{M}b^{2}\cos^{2}\\!\theta}{\rho^{2}}\right)\\!d\psi^{2}+\frac{2\mathcal{M}ab\sin^{2}\\!\theta\cos^{2}\\!\theta}{\rho^{2}}d\phi
d\psi,$

where $\rho$, $\mathcal{M}$ and $\Delta$ are defined as

$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+b^{2}\cos^{2}\theta,\quad\mathcal{M}=Me^{-k/r^{2}},\quad\Delta=(r^{2}+a^{2})(r^{2}+b^{2})-\mathcal{M}r^{2}.$
(3.17)

The above solution which is a RBH with Minkowski core at $r=0$, is
characterized by four parameters: mass $M$, rotation parameters $a,b$ and $k$
which is related to the magnetic charge. Setting $k=0$ in (3.5.2) one can
recover the 5D Myers-Perry BH, so $k$ can be interpreted as a deviation from
the Myers-Perry geometry. Moreover by inserting $a=b=0$ the solution reduces
to Schwarzschild-Tangherlini BH.

Although one may find horizons by solving $\Delta=0$, it is more convenient to
find $M$ and $a$ in terms of $r_{\pm}$. For the regular solution with small
charge parameter $k$, we calculate the entropy and temperature on the
horizons. The result is

$\displaystyle S_{+}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}(k-r_{+}^{2})(b^{2}+r_{+}^{2})(b^{2}+r_{-}^{2})}{2r_{+}(k+b^{2})}\,,$
$\displaystyle S_{-}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}(k-r_{-}^{2})(b^{2}+r_{+}^{2})(b^{2}+r_{-}^{2})}{2r_{-}(k+b^{2})}\,,$
$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{r_{+}^{4}(r_{-}^{2}-b^{2}-2k)-k^{2}(b^{2}+r_{-}^{2})}{2\pi
r_{+}^{3}(b^{2}+r_{-}^{2})(k-r_{+}^{2})}\,,$ $\displaystyle T_{-}$
$\displaystyle=$
$\displaystyle\frac{r_{-}^{4}(r_{+}^{2}-b^{2}-2k)-k^{2}(b^{2}+r_{+}^{2})}{2\pi
r_{-}^{3}(b^{2}+r_{+}^{2})(r_{-}^{2}-k)}\,.$ (3.18)

Now it is straightforward to check that the condition $T_{+}S_{+}=T_{-}S_{-}$
does not hold and so the entropy product for the above solution is not
universal.

####  3.5.3 Electrically charged rotating Bardeen BH

5D Bardeen BH with electric charge and its rotating version are introduced in
[11, 12]. The line element for electrically charged rotating Bardeen BH is
[12]

$\displaystyle\\!\\!\\!\\!ds^{2}=-dt^{2}+\frac{m(r)}{\rho^{2}}\left(dt-a\sin^{2}\theta
d\phi+b\cos^{2}\theta d\psi\right)^{2}$ (3.19)
$\displaystyle\\!\\!\\!\\!+\frac{r^{2}\rho^{2}}{\Delta}dr^{2}\\!+\rho^{2}d\theta^{2}\\!+(r^{2}\\!+\\!a^{2})\\!\sin^{2}\\!\theta
d\phi^{2}\\!+(r^{2}\\!+\\!b^{2})\\!\cos^{2}\\!\theta d\psi^{2}\\!\\!,$

where $\rho$, $m(r)$ and $\Delta$ are as the following

$\displaystyle\\!\\!\\!\rho^{2}=r^{2}\\!+a^{2}\cos^{2}\\!\theta+b^{2}\sin^{2}\\!\theta,\quad
m(r)=\mu\left(\\!\\!\frac{r^{3}}{r^{3}\\!+\\!q_{e}^{3}}\\!\right)^{\\!4/3}\\!,$
$\displaystyle\\!\\!\\!\Delta=\big{(}r^{2}+a^{2}\big{)}\left(r^{2}+b^{2}\right)-m(r)r^{2}.$
(3.20)

The above solution is characterized by four parameters $a$, $b$, $\mu$ and
$q_{e}$ where $a$, $b$ are rotation parameters around $\phi$, $\psi$ axis and
$\mu$ and $q_{e}$ are related to the mass and electric charge of the BH
respectively. Setting $q_{e}=0$ in (3.19), the Myers-Perry BH is recovered so
$q_{e}$ determines the deviation of charged rotating Bardeen BH from the
Myers-Perry solution.

Horizons of this solution are the roots of $\Delta=0$ and due to the
complicated form of the metric, one finds messy terms for the entropies and
temperatures on the horizons. In the following, for the sake of simplicity, we
consider single rotating Bardeen BH with small value of charge which is
obtained by setting $b=0$ and doing Taylor expansion for $m(r)$. Noticing the
ranges of angles in this solution as $\theta\in[0,\pi/2]$ and
$\phi,\psi\in[0,2\pi]$, we calculate the entropy and temperature on the inner
and outer horizons. The result is

$\displaystyle S_{+}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}r_{+}r_{-}^{3}(r_{-}+r_{+})(4q_{e}^{3}-3r_{+}^{3})}{8q_{e}^{3}(r_{-}^{2}+r_{-}r_{+}+r_{+}^{2})}\,,$
$\displaystyle S_{-}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}r_{-}r_{+}^{3}(r_{-}+r_{+})(4q_{e}^{3}-3r_{-}^{3})}{8q_{e}^{3}(r_{-}^{2}+r_{-}r_{+}+r_{+}^{2})}\,,$
(3.21) $\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{-3r_{-}^{4}-3r_{-}^{3}r_{+}+2r_{-}^{2}r_{+}^{2}+2r_{-}r_{+}^{3}+2r_{+}^{4}}{3\pi
r_{-}^{3}r_{+}^{4}(r_{-}+r_{+})}q_{e}^{3},$ $\displaystyle T_{-}$
$\displaystyle=$
$\displaystyle\frac{-3r_{+}^{4}-3r_{+}^{3}r_{-}+2r_{-}^{2}r_{+}^{2}+2r_{+}r_{-}^{3}+2r_{-}^{4}}{3\pi
r_{+}^{3}r_{-}^{4}(r_{-}+r_{+})}q_{e}^{3},$

one can easily check that $T_{+}S_{+}\neq T_{-}S_{-}$ which means the
universality of entropy product fails in this solution.

##  4 RBHs with universal entropy product

Until now, we observed that the universality of entropy product fails for the
RBHs. However there are some regular solutions which respect the universality.

###  4.1 The Kerr-like RBH

In [14, 15] it has been introduced a regularized Kerr BH with modified mass
$m\to m(r)=me^{-\ell/r}$. The metric is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta^{\prime}}{\Sigma}(dt-a\sin^{2}\theta\,d\phi)^{2}+\frac{\Sigma}{\Delta^{\prime}}\,dr^{2}+\Sigma\,d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}[(r^{2}+a^{2})\,d\phi-a\,dt]^{2}\,,$
$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta\,,\qquad\qquad\Delta^{\prime}=r^{2}+a^{2}-2mr\,e^{-\ell/r}\,.$
(4.1)

This RBH is characterized by its mass $m$, rotation parameter $a$ and the
regularization parameter $\ell$. Setting $\ell=0$, (4.1) reduces to the Kerr
metric, so $\ell$ can be also viewed as the deviation form Kerr. This solution
at $r\to\infty$ reaches the Kerr BH, while the ring singularity of Kerr is
replaced by an asymptotically Minkowski spacetime [15]. For the small values
of $\ell$, we keep the first order in Taylor expansion of the metric. We also
rewrite $a$ and $m$ in terms of $r_{\pm}$. Now the entropy and temperature for
this perturbed solution takes to the following form

$\displaystyle S_{+}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\pi(r_{+}\\!+\\!r_{-})(\ell\\!-\\!r_{+}),\quad
S_{-}=\pi(r_{+}\\!+\\!r_{-})(\ell\\!-\\!r_{-}),$ (4.2) $\displaystyle
T_{+}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\frac{r_{-}-r_{+}}{4\pi(r_{-}\\!+r_{+})(\ell\\!-r_{+})},\quad
T_{-}=\frac{r_{-}-r_{+}}{4\pi(r_{-}\\!+r_{+})(\ell\\!-r_{-})},$

it is easy to check that $T_{+}S_{+}=T_{-}S_{-}$ is satisfied, which means
that the entropy product is universal. We also calculate the angular momentum
for the solution with small $\ell$ as

$J=\frac{1}{2}(r_{+}+r_{-})\,\sqrt{r_{-}r_{+}-\ell(r_{+}+r_{-})}=ma\,.$ (4.3)

Noticing (4.2), (4.3) and by ignoring the $\ell^{2}$, $\ell^{3}$, … terms, one
finds the universality of entropy product as

$S_{+}S_{-}=4\pi^{2}J^{2}\,.$ (4.4)

In the next section we will use this result to find the central charge of the
dual CFT for the regular Kerr-like BH.

###  4.2 Reissner-Nordström outside a de Sitter core

In [16] a regular solution is constructed by connecting a de Sitter space to
the core of Reissner-Nordström BH. The metric is written in the static
spherically symmetric form

$ds^{2}=-B(r)dt^{2}+A(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\,.$
(4.5)

In order to remove the singularity at $r=0$, it is supposed a sphere with
radius $r_{0}$, centered at $r=0$, which is filled by a static charged perfect
fluid distribution with spherical symmetry. Solving the equations of motion,
functions $A(r)$ and $B(r)$ are obtained as [16]

$B(r)=A^{-1}(r)=\left\\{\begin{array}[]{l
l}\displaystyle{1-\frac{r^{2}}{R^{2}}},&{r\leq r_{0}},\\\
\displaystyle{1-\frac{2m}{r}+\frac{q^{2}}{r^{2}}},&{r\geq
r_{0}}.\end{array}\right.$ (4.6)

The above solution is determined by its mass ($m$), charge ($q$), radius of de
Sitter space at the core ($R$) and radius of matter distribution ($r_{0}$). In
the region $r>r_{0}$ the solution is given by the Reissner-Nordström metric
which has two horizons at $r_{\pm}=m\pm\sqrt{m^{2}-q^{2}}$. It has been shown
in [16] that $r_{0}\leq r_{-}$, in other words the region of matter
distribution lies inside the inner horizon and the spacetime for $r\geq r_{-}$
is given by the Reissner-Nordström metric. The entropy and temperature on the
inner and outer horizons for this RBH is just like that of the Reissner-
Nordström

$S_{+}=\pi r_{+}^{2}\,,\quad S_{-}=\pi r_{-}^{2}\,,\qquad
T_{+}=\frac{r_{+}-r_{-}}{4\pi r_{+}^{2}}\,,\quad T_{-}=\frac{r_{+}-r_{-}}{4\pi
r_{-}^{2}}\,.$ (4.7)

The universality is true, since the condition $T_{+}S_{+}=T_{-}S_{-}$ is
satisfied. Moreover the entropy product for this solution is mass independent

$S_{+}S_{-}=\pi^{2}q^{4}\,.$ (4.8)

###  4.3 The topological star

The topological star is a solution which is obtained from dimensional
reduction of a 5D solution in Einstein-Maxwell theory [17]. The starting point
is the static spherically symmetric metric in five dimension with a magnetic
flux $F$

$\displaystyle ds^{2}\\!$ $\displaystyle=$
$\displaystyle\\!-f_{S}(r)dt^{2}+f_{B}(r)dy^{2}+\frac{dr^{2}}{f_{S}(r)f_{B}(r)}+r^{2}d\theta^{2}$
(4.9) $\displaystyle+$ $\displaystyle r^{2}\sin^{2}\\!\theta d\phi^{2},\qquad
F=P\sin\theta d\theta\wedge d\phi\,,$

coordinate $y$ in the above, parametrizes a circle of perimeter $2\pi R_{y}$
and the functions $f_{S}(r)$, $f_{B}(r)$ and $P$ are given by

$\displaystyle f_{B}(r)=1-\frac{r_{B}}{r}\,,\qquad
f_{S}(r)=1-\frac{r_{S}}{r}\,,$ $\displaystyle
P=\pm\frac{1}{\kappa_{5}^{2}}\sqrt{\frac{3\,r_{S}\,r_{B}}{2}}\,,$ (4.10)

where $\kappa_{5}$ is the gravitational coupling. By Kaluza-Klein reduction
along $y$, one finds the following solution

$\displaystyle ds_{5}^{2}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!e^{2\Phi}ds_{4}^{2}+e^{-4\Phi}dy^{2},\qquad
e^{2\Phi}=f_{B}^{-\frac{1}{2}},$ (4.11) $\displaystyle ds_{4}^{2}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!f_{B}^{\frac{1}{2}}\big{[}-\\!f_{S}dt^{2}\\!+\frac{dr^{2}}{f_{B}f_{S}}+r^{2}d\theta^{2}\\!+r^{2}\sin^{2}\\!\theta
d\phi^{2}\big{]}\\!.$

In the case of $r_{B}=0$ the above metric reduces to the Schwarzschild BH.
Note also that the coefficient $f_{B}^{1/2}$ is imaginary for $r<r_{B}$ which
means that the spacetime ends at $r=r_{B}$. In other words the singularity at
$r=0$ is excluded from the four dimensional BH. There are two horizons (roots
of the term $f_{B}f_{S}$) for this solution at $r_{B}$ and $r_{S}$ with the
entropy and temperature

$S_{S}=\frac{\pi r_{S}^{2}}{G}\,,\quad S_{B}=\frac{\pi r_{B}^{2}}{G}\,,\qquad
T_{S}=\frac{1}{4\pi r_{S}}\,,\quad T_{B}=\frac{r_{S}}{4\pi r_{B}^{2}}\,.$
(4.12)

The universality is held since the condition $T_{+}S_{+}=T_{-}S_{-}$ is
respected. For this solution the mass and magnetic charge take to the form

$M=\frac{2\pi}{\kappa_{4}^{2}}\left(2r_{S}+r_{B}\right),\qquad
Q_{m}=\frac{1}{\kappa_{4}}\sqrt{\frac{3}{2}\,r_{B}\,r_{S}}\,,$ (4.13)

where $\kappa_{4}$ is the 4D gravitational constant which is related to
$\kappa_{5}$ as $\kappa_{4}=\frac{\kappa_{5}}{\sqrt{2\pi R_{y}}}$. One can
check that the entropy product for this regular solution is mass independent
as 222Considering (4.13) and Noticing that in the case of Schwarzschild black
hole ($r_{B}=0$) we find $r_{S}=2M$, it is obvious that
$\kappa_{4}=\sqrt{8\pi}$. In addition, we rescale the magnetic charge as
$Q_{m}\to 4\sqrt{\frac{\pi}{3}}\,Q_{m}$.

$S_{S}\,S_{B}=\pi^{2}\,Q_{m}^{4}.$ (4.14)

###  4.4 BHs in Einstein gravity coupled to extended nonlinear electrodynamics

A physical source for the RBHs is the nonlinear electrodynamics [6, 7]. For
the Einstein gravity coupled to the nonlinear electrodynamics

$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}\,[R-\mathcal{L}(\mathcal{F})]\,,$
(4.15)

where $F_{\mu\nu}=dA_{\nu}$ is the field strength tensor,
$\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$ and Lagrangian density $\mathcal{L}$ is a
function of $\mathcal{F}$, it has been shown [18] that one can construct
variety classes of RBHs by choosing different forms of
$\mathcal{L}(\mathcal{F})$. In the following we review three classes of these
RBHs.

####  4.4.1 Bardeen class

Considering the Lagrangian density

$\mathcal{L}=\frac{4\mu\big{(}\alpha\mathcal{F}\big{)}^{5/4}}{\alpha\big{(}1+\sqrt{\alpha\mathcal{F}}\big{)}^{1+\mu/2}}\,,$
(4.16)

where $\mu>0$ and $\alpha>0$ are some constants, the general form for the two-
parameter (mass and magnetic charge) BH solutions is

$ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\,,\qquad
f=1-\frac{2M}{r}-\frac{2\alpha^{-1}q^{3}r^{\mu-1}}{\big{(}r^{2}+q^{2}\big{)}^{\mu/2}}\,,$
(4.17)

in the above, $q$ is related to the black hole magnetic charge and the ADM
mass can be read off from the asymptotic behavior of the metric

$f=1-\frac{2(M+\alpha^{-1}q^{3})}{r}+...~{},$ (4.18)

which implies that $M_{ADM}=M+M_{em}$ where $M_{em}=\alpha^{-1}q^{3}$. In
other words, the BH mass receives contributions from the Schwarzschild mass
$M$ and the nonlinear effects that denoted by $M_{em}$. Since by setting $M=0$
and $\mu=3$ the Bardeen BH (3.1) is recovered, (4.17) is called ”Bardeen
class”. It has been observed [18] that for a solution with $M=0$ and $\mu=2$
the entropy product is universal as

$S_{+}S_{-}=\pi^{2}q^{4}\,.$ (4.19)

In the next section we will use the above result to find the central charge of
dual CFT.

####  4.4.2 Hayward class

The nonsingular BHs of the Hayward class are solutions to Einstein gravity
coupled to a nonlinear electrodynamics with Lagrangian density [18]

$\mathcal{L}=\frac{4\mu(\alpha\mathcal{F})^{(\mu+3)/4}}{\alpha\big{[}1+(\alpha\mathcal{F})^{\mu/4}\big{]}^{2}}\,,$
(4.20)

similar to the Bardeen class, for these BHs the metric is in the static
spherically symmetric form (4.17) with function

$f=1-\frac{2M}{r}-\frac{2\alpha^{-1}q^{3}\,r^{\mu-1}}{r^{\mu}+q^{\mu}}\,.$
(4.21)

Setting $M=0$ and $\mu=3$ one finds the Hayward solution, for this reason they
called ”Hayward class”. It has been found [18] that in the case of vanishing
Schwarzschild mass ($M=0$) and $\mu=2$ the entropy product is universal as

$S_{+}S_{-}=\pi^{2}q^{4}\,.$ (4.22)

####  4.4.3 new class

By choosing the Lagrangian density as [18],

$\mathcal{L}=\frac{4\mu\mathcal{F}}{\big{[}1+(\alpha\mathcal{F})^{1/4}\big{]}^{\mu+1}}\,,$
(4.23)

a ”new class” of RBHs can be obtained. The metric is given by (4.17) with new
function

$f=1-\frac{2M}{r}-\frac{2\alpha^{-1}q^{3}r^{\mu-1}}{(r+q)^{\mu}}\,.$ (4.24)

For a solution with $M=0$ and $\mu=2$ the entropy product is universal as
(4.22).

##  5 Central charges and entropy product

In sections 3 and 4 we studied the universality of entropy product in the case
of RBHs. We found that universality is true for some of them and it fails for
some others. According to Thermodynamics method (reviewed in section 2) when
the entropy product is universal, one can easily read the central charge of
the dual CFT from (1.1). In the following, for RBHs in section 4 which respect
the universality, we find central charges form (1.1). Then we compare this
result with the central charge calculated from the asymptotic symmetry group
(ASG) method [48, 49], to check the validity of Thermodynamics method.

###  5.1 Central charge of The Kerr-like RBH

In subsection (4.1) we observed that for the small deviation parameter $\ell$,
entropy product of the Kerr-like RBHs is universal as

$S_{+}S_{-}=4\pi^{2}J^{2},$ (5.1)

Now using (1.1), it is easy to read off the central charge from this entropy
product

$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{6}{4\pi^{2}}\,\frac{\partial\big{(}4\pi^{2}J^{2}\big{)}}{\partial
J}=12J$ (5.2) $\displaystyle=$ $\displaystyle
6(r_{+}+r_{-})\sqrt{r_{-}r_{+}-\ell(r_{+}+r_{-})},$

where in the second line we use (4.3). We are going to verify the above result
by doing the Kerr/CFT analysis [49]. The first step is to find the near
horizon metric of the extremal solution. Remember that we have written
parameters $a$ and $m$ in the Kerr-like BH (4.1) in terms of $r_{\pm}$, so one
obtains the extremal solution by setting $r_{-}=r_{+}$. Following the
procedure as in [30, 47], we find the near horizon extremal metric. The result
is

$ds^{2}=\alpha(\theta)\big{[}-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\big{]}+\beta(\theta)d\theta^{2}+\gamma(\theta)(d\phi+f^{\phi}rdt)^{2}\\!,$
(5.3)

where $f^{\phi}$ and functions $\alpha(\theta)$, $\beta(\theta)$ and
$\gamma(\theta)$ are

$\displaystyle f^{\phi}=\frac{r_{+}(r_{+}-2\ell)}{\ell-
r_{+}}\,,\quad\gamma(\theta)=\frac{4r_{+}(\ell-
r_{+})^{2}\sin^{2}\theta}{r_{+}-(2\ell-r_{+})\cos^{2}\theta}\,,$
$\displaystyle\alpha(\theta)=\beta(\theta)=-2r_{+}\left[\big{(}\ell-\frac{r_{+}}{2}\big{)}\cos^{2}\theta-\frac{r_{+}}{2}\right]\,.$
(5.4)

By choosing the adequate boundary conditions given in [49] and doing the
calculations, we find the central charge as

$c=\frac{3}{2\pi}f^{\phi}\\!\\!\int\\!\\!\\!d\theta
d\phi\sqrt{\beta(\theta)\gamma(\theta)}=12\sqrt{r_{+}^{3}(r_{+}\\!-2\ell)}=12J_{\\!ext},$
(5.5)

where $J_{ext}$ is the angular momentum of the extremal solution. To compare
the central charges (5.2) and (5.5) it is important to note that in the
Kerr/CFT analysis we have inserted the extremality on the solution. Now
applying the extremality condition $r_{-}=r_{+}$ on (5.2), it is obvious that
the result is in complete agreement with (5.5) which means the validity of
Thermodynamics method. It is worth mentioning that using the Thermodynamics
method, one finds central charge for the generic (non-extremal) solution while
in the Kerr/CFT analysis, the central charge is obtained for the extremal
solution.

###  5.2 Central charge of the charged RBHs

For the regular Reissner-Nordström and topological star in 4.2 and 4.3 we
found that the entropy product is universal

$S_{+}S_{-}=\pi^{2}q^{4}\,,$ (5.6)

where $q$ is the charge (electric or magnetic) of the solution. The same
relation is obtained in the case of RBHs of Einstein gravity coupled to
nonlinear electrodynamics (subsection 4.4). Now it is easy to find the central
charge of the dual CFT using Thermodynamics method (1.1) as

$c=\frac{6}{4\pi^{2}}\,\frac{\partial\big{(}\pi^{2}q^{4}\big{)}}{\partial
q}=6q^{3}\,.$ (5.7)

One can also find central charges for charged solutions using the ASG
formalism. In fact this is done in [50], the result is

$c=6q^{3},$ (5.8)

which is in agreement with the result of Thermodynamics method.

##  6 Conclusions

In order to extend our intuition about the entropy product law and its
relation to dual CFTs of the BHs, we investigated the entropy product for
RBHs. We considered variety types of regular solutions including the rotating
Bardeen and Hayward BHs, the Frolov-Zelnikov solution, the Kerr-like RBH, the
regular Reissner-Nordström solution, topological star, 5D charged rotating
RBHs and regular solutions in extended nonlinear electrodynamics.

It seems that there is no rule for the RBHs that makes their entropy product
universal. In other words, regardless of the charges or asymptotic behavior at
$r\to 0$ limit or even the theory that RBH satisfies its equations of motion,
the universality of entropy product is true for some RBHs but fails for some
others! The similar pattern is observed for the singular BHs. For instance,
the universality is true for the Kerr BH and 5D Myers-Perry but it fails for
the Myers-Perry in $D\geq 6$. The same situation is for the BTZ (which is
universal) and Kerr-AdS in $D\geq 4$ (non-universal) BHs. One can deduce from
these observations that in generic, the universality is a characteristic for
the “solutions” and not for the “theories” 333There is an exception here [35]:
In the case of theories in which the action contains parity violating terms
(such as the Chern-Simons), one finds that $c_{R}\neq c_{L}$ which means
$T_{+}S_{+}\neq T_{-}S_{-}$..

So what is the advantage of entropy product universality? One may find answer
in the relation between entropy product and dual CFTs of the BHs. It has been
discussed [1] that when the entropy product is universal, the central charges
of the dual CFTs can be easily read off from the entropy product according to
Thermodynamics method (1.1).

In the case of RBHs with universal entropy product, we found the central
charges using the Thermodynamics method. We then found the central charges
from the ASG method. These two results are the same which shows the validity
of Thermodynamics method for RBHs. In other words, universality of the entropy
product provides an easy way to find central charges of the dual CFTs. It is
worth to mention that, in the ASG method one takes the extremal limit of the
solutions and so the central charges are obtained for the extremal BHs. While
the Thermodynamics method yields central charges of the generic (non-extremal)
BH solutions.

## Acknowledgment

I would like to thank IPM, school of Physics, because of useful discussions
and hospitality during my visit. I would also like to thank S. Sadeghian for
reading the manuscript and her valuable comments.

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