# Quantum Improved Regular Kerr Black Holes

Chiang-Mei Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Zhongli, Taoyuan 320317, Taiwan  Center for High Energy and High
Field Physics (CHiP), National Central University, Zhongli, Taoyuan 320317,
Taiwan  Yi Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Zhongli, Taoyuan 320317, Taiwan  Akihiro Ishibashi
<EMAIL_ADDRESS>Department of Physics, Kindai University, Higashi-
Osaka, Osaka 577-8502, Japan  Research Institute for Science and Technology,
Kindai University, Higashi-Osaka, Osaka 577-8502, Japan  Nobuyoshi Ohta
<EMAIL_ADDRESS>Department of Physics, National Central University, Zhongli,
Taoyuan 320317, Taiwan  Research Institute for Science and Technology, Kindai
University, Higashi-Osaka, Osaka 577-8502, Japan

###### Abstract

We study the quantum improvement of Kerr black holes with mass-dependent scale
identifications in asymptotically safe gravity. We find that a physically
sensible identification can only be a function of $Mr$ and the area
$A=4\pi(r^{2}+a^{2})$ where $M$ is the mass of the black hole and $a$ is an
angular momentum parameter. We then discuss various properties of the rotating
quantum black holes for a simple choice of the identification. We show that
the resulting regular rotating black holes have the following nice properties:
(i) admitting a consistent black hole thermodynamics at the horizon, (ii)
resolving the ring singularity, (iii) partially eliminating closed time-like
curves present in the classical Kerr black holes.

##  I Introduction

The theory of general relativity has a universal and robust nature that
gravitation produces singularities; the examples include the singularity in
black holes and the big bang singularity in cosmology [1, 2, 3]. The existence
of singularity invalidates the applicability of general relativity. Therefore,
many possible ways were proposed to avoid physical troubles caused by the
singularities. The cosmic censorship conjecture presumes that a physical world
should not have a causal connection with singularity [4]. The singularities in
black holes should also never be “naked” which are always “protected” by a
horizon. Those are the proposals in classical theory. Alternatively the
existence of singularity could be just a consequence of incompleteness of
general relativity which is a classical theory of gravity, and a complete
theory of quantum gravity may resolve the problem of singularity.

Another serious problem is that closed time-like curves (CTCs) can exist in
solutions of general relativity. The existence of CTCs implies a theoretical
possibility of traveling backward in time and then causes problems in
causality. Indeed, it has been shown that CTCs do exist in Kerr black holes
[5]. The chronology protection conjecture was proposed that theory beyond
general relativity, such as quantum gravity, should prevent time travel [6].

Several regular black holes have been proposed, for the review see [7, 8],
including Hayward [9] and Bardeen [10] black holes. However, the focus in the
previous proposals was mainly on the singularity resolution, but the
preservation of black hole thermodynamics was not considered.

Among various approaches, the asymptotic safety scenario is an interesting
proposal to formulate the quantum gravity using functional renormalization
group [11, 12]. For reviews, see [13, 14]. The quantum improvement, at
solution level, may be made by replacing the coupling “constants” in the
classical solutions with running couplings as functions of energy scale. The
essential issue in this approach is a suitable choice of identification of the
energy scale with some length scale in the considered solutions. This is often
referred to as the “scale identification.” Many proposals have been studied
for various black hole solutions [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31].

Quantum effects of gravity are expected to play a significant role in the core
region near the black hole singularity. In the asymptotically safe gravity, it
turns out that quantum effects provide a “repulsive” force at the core of
black hole which stabilizes the unstable phase of small black holes and gives
the possibility of resolving the singularity [15, 17]. However a problem in
this approach is that we do not have clear physical criterion how to make the
identification of the energy scale and the length scale. In Ref. [32], we have
considered this problem for the Kerr-(A)dS black holes and found that in order
to have consistency of the first law of black hole thermodynamics, the
identification should be such that the gravitational coupling is a function of
the horizon area

$A_{h}=4\pi(r_{h}^{2}+a^{2}),$ (1)

where $r_{h}$ is the horizon radius and $a$ is the angular momentum parameter.
Since the first law of thermodynamics is the fundamental law of energy
conservation, it must be valid not only semiclassically but also quantum
mechanically. Note that this is a general consequence from the first law if we
allow coordinate dependence of the gravitational constant, and hence
independent of the asymptotic safety. It is natural to extend the
identification away from the horizon as a function of the area
$A=4\pi(r^{2}+a^{2})$ at the fixed radius $r$. Unfortunately it is clear that
this kind of the gravitational coupling depending only on $A$ cannot resolve
the ring singularity at $r=0$ because it never vanishes or diverges to cancel
the singularity whatever the choice of the identification. This problem arises
because in [32] we considered only the mass-independent identification. In
this paper, we generalize our previous analysis to include the dependence on
the mass parameter, and show that then a more general scale identification
which preserves thermodynamics is possible and the gravitational coupling can
be a function of $Mr_{h}$ as well as the area $A_{h}$ on the horizon. Indeed,
a simple mass-dependent identification was already considered in Refs. [15,
19] but the thermodynamics was a nontrivial issue there. Actually the former
variable should also be a function of the area on the horizon. It is then
natural to extend this away from the horizon, keeping the dependence on both
variables. We then consider a simple physically sensible choice of scale
identification in the asymptotically safe gravity, which gives regular
rotating black holes with the following nice properties: (i) admitting a
consistent black hole thermodynamics at horizon, (ii) resolving the ring
singularity, (iii) partially eliminating CTCs.

We note that in a recent paper [33], the problem is studied if known rotating
regular black holes can satisfy the thermodynamics. Most of the works on
rotating regular black holes consider the case when the mass or equivalently
the gravitational constant is a function of coordinate, and the thermodynamics
are not satisfied. However our study [32] has shown that the running coupling
or mass should be a function of horizon area in order to preserve the
consistency of the first law of thermodynamics. So it is to be expected that
the thermodynamics cannot be satisfied if one consider only the coordinate
dependence.

When we make quantum improvement, there are several choices of coordinate
systems such as Boyer-Linquist (BL) or Eddington-Finkelstein (EF) coordinates,
which may give different solutions upon quantum improvement. At the classical
level, they are related by a coordinate transformation and so they describe
the same solution. However, if we include the angular dependence in the
running gravitational constant, they are no longer the same solution. Here we
show that the absence of the singularity on the horizon or some other places
requires that such an angular dependence should not be included. Then the
solutions are related by the coordinate transformation and we can study the
problem in any coordinate, and find the above results in the BL coordinate
system. Requiring the consistency with the thermodynamics, we find a quite
general formula for the entropy

$S=\int\frac{dA_{h}}{4(G+M\partial_{M}G)},$ (2)

where $G$ is the gravitational constant dependent on $Mr_{h}$ and area
$A_{h}$.

As a bonus of our quantum improvement, we find that the CTC present in the
Kerr solution may be also removed in the region where the quantum effects are
significant. The precise discussions are given in Sec. IV.

This paper is organized as follows. In the next section, we briefly review the
asymptotic safe quantum gravity. In Sec. III, we start our study of scale
identification for Kerr black holes with possible mass dependence. In Sec.
III.1, after briefly reviewing some basic aspects of the classical Kerr
geometry, we consider a scale identification as a function of the radial and
angular coordinates in the BL coordinate, and show that the running
gravitational coupling should not depend on the angle in order to avoid
singularity on the horizon (this result itself was given before in [34, 32]).
In Sec. III.2, we consider the solution in the EF coordinates where there is
no curvature singularity even if we keep the angular dependence. It turns out
that in this case, we encounter another problem of “parallelly propagated
curvature singularity,” so the angular dependence is excluded. We then show in
Sec. III.3, that the consistency with the thermodynamics requires that the
gravitational coupling should be a function of $Mr_{h}$ and the horizon area
$A_{h}$, and give a general formula for the entropy. It is natural to extend
the dependence away from the horizon, keeping the dependence on $Mr$ and the
area $A=4\pi(r^{2}+a^{2})$ at fixed radius $r$. We then discuss the properties
of the quantum improved black holes in Sec. IV. First in Sec. IV.1, we show
that a simple choice of the scale identification can resolve the ring-
singularity existed in the classical Kerr geometry. We also study geodesic
complete/incompleteness in Sec. IV.2 and the resolution of CTCs in Sec. IV.3.
We summarize our results in Sec. V. In appendix, we list the algebraically
complete set of invariant curvature scalars for quantum improved Kerr black
holes (Appendix A), and the analysis of quantum improved Kerr-(A)dS black
holes (Appendix B).

##  II Asymptotically Safe Gravity

The asymptotically safe scenario for quantum generalization of the general
relativity with cosmological constant, in the units
$c=\hbar=1,G=l_{P}^{2}=1/M_{P}^{2}$:

$S=-\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}(R-2\Lambda),$ (3)

proposes an energy-scale $k$ dependent Newton constant $G(k)$ and cosmological
constant $\Lambda(k)$ [11]. Assuming that the cosmological constant is already
at the fixed point and negligibly small,111 This assumption may be justified
if we consider the wave function renormalization [35]. the associated
renormalization group equations lead to the solutions for the Newton coupling
[26]

$G(k)=\frac{G_{0}}{1+\omega G_{0}k^{2}},$ (4)

where $\omega$ is a constant of order 1 representing the quantum effects, and
$G_{0}=G(k=0)$. In order to see what physical consequence this may have on the
black hole geometry, we have to identify the energy scale $k$ with some length
scale in the black hole geometry. This is an essential issue for constructing
quantum improved solutions. A simple possibility is to assume that the energy
scale is dependent on the radius and angular momentum parameter $a$ but
independent of the mass parameter. The first law of black hole thermodynamics
for the Kerr-(A)dS black holes then requires that the scale identification
near the horizon should be a function of the horizon area $A_{h}$ [32]. It is
natural to generalize the scale identification away from the horizon by
replacing the horizon radius $r_{h}$ [see below Eq. (6)] with the radial
coordinate $r$, but then the energy scale determined by the area remains
finite at the classical singularity ($r=0,\theta=\pi/2$ for Kerr black holes)
and cannot resolve the singularity.

In order to resolve the ring singularity for Kerr black holes, we have to
extend our assumption. Note that it is also possible to include the dependence
on the mass parameter in the scale identification. We will show in Sec. III
that there is a more general scale identification, corresponding to quantum
improvement of the classical rotating Hayward black holes [36], which is
consistent with black hole thermodynamics on the horizon and, when extended
away from the horizon, can resolve the classical ring singularity. The results
can be generalized to include a cosmological constant; we will discuss this in
Appendix B.

##  III Running Coupling for Kerr Black Holes

When we make scale identification, we find that the quantum improvements in
different coordinate system give different solutions. In this section, we
first discuss the identification in the BL coordinates and argue that the
identification should not include the angular coordinate in order not to
introduce the singularity on the horizon. It turns out that such a singularity
does not appear if we use the EF coordinates. However in this case there
appears another problem, so the identification should not have angular
dependence either.

###  III.1 Scale identification in the BL coordinates

Let us start with the Kerr black hole in the BL coordinates. The line-element
of the classical solution is

$ds^{2}=-\frac{\Delta}{\Sigma}\left(dt-a\sin^{2}\theta
d\varphi\right)^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\left(adt-(r^{2}+a^{2})d\varphi\right)^{2},$
(5)

where

$\Delta=r^{2}-2GMr+a^{2},\qquad\Sigma=r^{2}+a^{2}\cos^{2}\theta,$ (6)

and $a$ is the angular momentum parameter. In the semiclassical case, i.e. $G$
is a constant, the Hawking temperature and the entropy are given by

$T=\frac{\kappa}{2\pi}=\frac{r_{h}^{2}-a^{2}}{4\pi
r_{h}(r_{h}^{2}+a^{2})},\qquad
S=\frac{A_{h}}{4G}=\frac{\pi(r_{h}^{2}+a^{2})}{G},$ (7)

where $r_{h}$ is the radius of the horizon determined by $\Delta(r_{h})=0$ and
$A_{h}$ is the area of horizon (1). The quantity $\kappa$ here is the surface
gravity, which is defined as follows: We choose $\Omega_{H}$ in the Killing
vector $\chi=\partial_{t}+\Omega_{H}\partial_{\varphi}$ such that $\chi$ is
null ($\chi_{\mu}\chi^{\mu}=0$) at the horizon, and then the surface gravity
$\kappa$ is given by the formula

$\displaystyle\kappa^{2}=-\frac{(\nabla_{\mu}\chi_{\nu})(\nabla^{\mu}\chi^{\nu})}{2}.$
(8)

Note the second expression in (7) for the entropy is the standard Bekenstein-
Hawking formula. These thermal quantities satisfy the first law of
thermodynamics

$dM=TdS+\Omega dJ,$ (9)

where angular momentum and angular velocity of black holes are

$J=Ma,\qquad\Omega=\frac{a}{r_{h}^{2}+a^{2}}.$ (10)

Note that the classical Kerr geometry (and the quantum improved one, too)
admits two horizons, the (outer) event horizon with
$r_{h}=GM+\sqrt{(GM)^{2}+a^{2}}$ and the (inner) Cauchy horizon with
$r_{h}=GM-\sqrt{(GM)^{2}+a^{2}}$, and we can define the above thermodynamic
quantities in the same manner for both the horizons. Throughout this paper,
when we discuss the black hole thermodynamics, we are mainly concerned with
the outer event horizon, while when we discuss the region near the singularity
as below, we have in mind the area inside the Cauchy horizon.

The Kretschmann scalar is

$\mathcal{K}=\frac{48M^{2}G^{2}(r^{6}-15a^{2}\cos^{2}\theta
r^{4}+15a^{4}\cos^{4}\theta
r^{2}-a^{6}\cos^{6}\theta)}{(r^{2}+a^{2}\cos^{2}\theta)^{6}},$ (11)

which depends on the angle coordinate $\theta$. In the limit $r\to 0$, we have

$\lim_{r\to
0}\mathcal{K}=\begin{cases}\dfrac{48M^{2}G^{2}}{r^{6}}\to\infty,&\theta=\pi/2\\\
&\\\
-\dfrac{48M^{2}G^{2}}{a^{6}\cos^{6}\theta}<\infty,&\theta\neq\pi/2\end{cases}\,.$
(12)

This indicates that there is a ring singularity at
$r=0,\,\theta=\pi/2,\,\varphi=\mbox{arbitrary}$, and the Kretschmann scalar is
not a continuous function of $\theta$ on the disk $r=0$ at the ring
singularity.

In asymptotically safe gravity, the running coupling varies according to the
energy scale. We assume that it can be a function of the coordinates of both
$r$ and $\theta$ for Kerr black holes. However, we note some undesirable
properties in the quantum improvements by an angle-dependent running coupling
$G=G(r,\theta)$ in the metric (5). Namely, the scalar curvature becomes

$R=\frac{2M}{\Sigma}\left[r\partial_{r}^{2}G(r,\theta)+2\partial_{r}G(r,\theta)-\frac{Mr^{2}}{\Delta_{r}^{2}}\Bigl{(}\partial_{\theta}G(r,\theta)\Bigr{)}^{2}\right],$
(13)

indicating that a scalar curvature singularity occurs on the horizon when the
coupling depends on the angle $\theta$. To avoid the singularity, the
gravitational coupling must not depend on the angle $\theta$.

###  III.2 Scale identification in EF Coordinates

We note that there are other coordinate systems for the Kerr black hole which
are free from such a curvature singularity even if the gravitational coupling
depends on the angle [34]. For example, one can introduce the EF coordinates
by

$du=dt+\frac{r^{2}+a^{2}}{\Delta}dr,\qquad
d\tilde{\varphi}=d\varphi-\frac{a}{\Delta}dr.$ (14)

The classical Kerr black holes then become

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}du^{2}+2dudr-\frac{2a\sin^{2}\theta(r^{2}+a^{2}-\Delta)}{\Sigma}dud\tilde{\varphi}-2a\sin^{2}\theta
drd\tilde{\varphi}$ (15) $\displaystyle+$ $\displaystyle\Sigma
d\theta^{2}+\frac{(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}{\Sigma}\sin^{2}\theta d\tilde{\varphi}^{2}.$

As pointed out in [34], unlike in BL coordinates, the angle-dependent running
coupling $G=G(r,\theta)$ does not produce the curvature singularity at the
horizon in this coordinate system. Indeed, in EF coordinates, all the
algebraically complete set of four invariants [37] do not include any angle-
derivative of Newton coupling, and thus they remain as in the expressions
(65)-(68). However, the Ricci tensor does include $\partial_{\theta}G$:

$\displaystyle
R_{uu}=-\frac{M}{\Sigma^{3}}\Bigl{[}r\Sigma\Delta\partial_{r}^{2}G-2a^{2}(r^{2}\sin^{2}\theta-\Delta\cos^{2}\theta)\partial_{r}G+r\Sigma\partial_{\theta}^{2}G+r\cot\theta(\Sigma+4a^{2}\sin^{2}\theta)\partial_{\theta}G\Bigr{]},$
$\displaystyle
R_{ur}=\frac{M\left(r\Sigma\partial_{r}^{2}G+2a^{2}\cos^{2}\theta\partial_{r}G\right)}{\Sigma^{2}},\qquad
R_{u\theta}=\frac{M\left[r\Sigma\partial_{r}\partial_{\theta}G-(r^{2}-a^{2}\cos^{2}\theta)\partial_{\theta}G\right]}{\Sigma^{2}},$
$\displaystyle
R_{u\tilde{\varphi}}=\frac{Ma\sin^{2}\theta}{\Sigma^{3}}\Bigl{[}r\Sigma\Delta\partial_{r}^{2}G-2(r^{4}+r^{2}a^{2}-a^{2}\Delta\cos^{2}\theta)\partial_{r}G+r\Sigma\partial_{\theta}^{2}G+r\cot\theta(3\Sigma+4a^{2}\sin^{2}\theta)\partial_{\theta}G\Bigr{]},$
(16) $\displaystyle R_{r\tilde{\varphi}}=-a\sin^{2}\theta R_{ur},\qquad
R_{\theta\theta}=\frac{2Mr^{2}\partial_{r}G}{\Sigma},\qquad
R_{\theta\tilde{\varphi}}=-a\sin^{2}\theta R_{u\theta},$ $\displaystyle
R_{\tilde{\varphi}\tilde{\varphi}}=-\frac{M\sin^{4}\theta}{\Sigma^{3}}\left[ra^{2}\Sigma\Delta\partial_{r}^{2}G-2\left(\frac{r^{2}(r^{2}+a^{2})^{2}}{\sin^{2}\theta}-a^{4}\cos^{2}\theta\Delta\right)\partial_{r}G+ra^{2}\Sigma\partial_{\theta}^{2}G+ra^{2}\cot\theta(4r^{2}+4a^{2}+\Sigma)\partial_{\theta}G\right].$

Actually, the transformation (14) is not a valid coordinate transformation if
the running coupling depends on the angle $\theta$ (because then $\Delta$ also
depends on $\theta$). Therefore the black holes are different when we apply
the angle dependent identification to Kerr solution in BL and EF coordinates;
they are not related by the coordinate transformation.

So in this coordinate system, it appears that the angle dependence may not be
excluded. However, we encounter another problem here. Since our quantum
improved black hole is stationary and axisymmetric, according to the black
hole rigidity [38, 39], the event horizon is expected to be a Killing horizon
to which both the two Killing vector fields $\partial_{u}$ and
$\partial_{\tilde{\varphi}}$ are tangent,222 Note that we denote here the
stationary and axial Killing vector fields in terms of the EF coordinates
$(u,{\tilde{\varphi}})$. and hence should be determined by $\Delta(r_{h})=0$.
However, the linear combination $\partial_{u}+\Omega_{H}\partial_{\varphi}$
with the horizon angular-velocity $\Omega_{H}:=a/(r_{h}^{2}+a^{2})$, which is
null on the horizon, fails to be a Killing vector as $\Omega_{H}$ is now a
function of $\theta$ via $r_{h}$ with angular dependence through the horizon
condition $\Delta(r_{h})=0$. Even if one finds a null Killing vector field
$\chi$ tangent to the horizon null generators, one may encounter the following
physically unwanted problem. For simplicity, let us consider the Schwarzschild
case, i.e., $a=0$. In this case, $\chi=\partial_{u}$ itself is the horizon
null Killing vector field and one can compute the Hawking temperature via (8)
and obtain

$\displaystyle T=\frac{\kappa}{2\pi}=\frac{1}{8\pi
G(r_{h},\theta)M}-\frac{\partial_{r_{h}}G(r_{h},\theta)}{4\pi
G(r_{h},\theta)}.$ (17)

This is no longer a constant over the horizon, and hence the zero-th law of
black hole thermodynamics fails to hold. Furthermore, it is shown in general
circumstances that if the surface gravity is not constant over the Killing
horizon, then there must appear a “parallelly propagated curvature
singularity” along horizon null generators [40, 41]. The latter issue may be
overlooked when one examines the behavior of the curvature scalars only [see
Appendix A]. To remove these problems, the gravitational coupling has to be
independent of the angle.

###  III.3 Consistency in the BL coordinates

These observations lead us to consider that the scale-identified coupling
should be a function of the coordinate $r$ and the physical parameters $M$ and
$a$, namely $G=G(r,M,a)$ for rotating black holes in any coordinate system.
Note that in this case, (14) gives a coordinate transformation and we can use
any coordinate system to discuss the quantum improvement. So from now on, we
use the BL coordinate system to discuss the consistency with the
thermodynamics.

The horizon radius $r_{h}$ of quantum improved Kerr black holes is determined
by

$\Delta(r_{h})=r_{h}^{2}+a^{2}-2G(r_{h},M,a)Mr_{h}=0.$ (18)

In general, it is difficult to find the explicit expression for $r_{h}$ in
terms of $M$ and $a$, but we can have its first derivatives from the relation

$\displaystyle 0=d\Delta(r_{h})$ $\displaystyle=$ $\displaystyle
2r_{h}(\partial_{M}r_{h}\,dM+\partial_{a}r_{h}\,da)+2a\,da-2r_{h}G\,dM-2Mr_{h}\partial_{M}G\,dM-2Mr_{h}\partial_{a}G\,da$
(19) $\displaystyle-$ $\displaystyle
2M(G+r_{h}\partial_{r_{h}}G)(\partial_{M}r_{h}\,dM+\partial_{a}r_{h}\,da),$

leading to

$\displaystyle\partial_{M}r_{h}=\frac{r_{h}(G+M\partial_{M}G)}{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G},\qquad\partial_{a}r_{h}=-\frac{a-Mr_{h}\partial_{a}G}{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G}.$ (20)

In general, arbitrary identification does not guarantee to lead to a
thermodynamically consistent system, and the consistency is an essential
physical criterion for an acceptable scale identification. The mass-
independent identification has been discussed in [32], and it is shown that a
consistent running coupling should be a function of area at horizon which
tells us that the Newton coupling should be so; $G=G(r^{2}+a^{2})$. However,
it is clear that such a Newton coupling cannot give a necessary condition
$G(r=0)=0$ for resolving the singularity at $r=0$ [7]. Here we examine the
consistency for a more general identification depending on mass parameter as
well.

It is straightforward to compute the angular momentum, angular velocity and
temperature (note that the numerator of $T$ equals to denominator of
$\partial_{M}r_{h}$):

$J=Ma,\qquad\Omega=\frac{a}{r_{h}^{2}+a^{2}},\qquad T=\frac{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G}{2\pi(r_{h}^{2}+a^{2})}.$ (21)

Then the thermodynamical first law $\delta M=T\delta S+\Omega\delta J$ leads
to

$\displaystyle\delta S=\frac{\delta M-\Omega\delta J}{T}.$ (22)

This relation gives two first derivatives of the entropy $S$ with respect to
$M$ and $a$, respectively:

$\displaystyle\partial_{M}S=\frac{1-\Omega a}{T}=\frac{2\pi
r_{h}^{2}}{r_{h}-MG-Mr_{h}\partial_{r_{h}}G},$ (23)
$\displaystyle\partial_{a}S=-\frac{\Omega M}{T}=-\frac{2\pi Ma}{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G}.$ (24)

By applying the relations (20), we can rewrite $\partial_{M}S$ in terms of the
area of the horizon (1) as

$\partial_{M}S=\frac{2\pi
r_{h}\partial_{M}r_{h}}{G+M\partial_{M}G}=\frac{\partial_{M}A_{h}}{4(G+M\partial_{M}G)}.$
(25)

To ensure the consistency, i.e.
$\partial_{a}\partial_{M}S=\partial_{M}\partial_{a}S$, the following two
conditions should be satisfied:
(i) $G+M\partial_{M}G$ is a function of the area $A_{h}$ and (ii) the
derivative $\partial_{a}S$ can be expressed in the following form

$\partial_{a}S=\frac{\partial_{a}A_{h}}{4(G+M\partial_{M}G)}=\frac{2\pi(a+r_{h}\partial_{a}r_{h})}{G+M\partial_{M}G}.$
(26)

By comparing this relation with (24) and using (20), we get the equation

$Ma\partial_{M}G+r_{h}^{2}\partial_{a}G-ar_{h}\partial_{r_{h}}G=0,$ (27)

and its solution is

$G(r_{h},M,a)=G(Mr_{h},r_{h}^{2}+a^{2}).$ (28)

With this solution, $\Delta(r_{h})$ reduces to just a function of $Mr_{h}$ and
$A_{h}$. We can solve $\Delta(r_{h})=0$ for $Mr_{h}$ as a function of $A_{h}$.
Therefore, $G+M\partial_{M}G$ becomes a function of area for Eq. (28). It
follows that the entropy is also a function of $A_{h}$ which can be computed
by the general formula

$S=\int\frac{dA_{h}}{4(G+M\partial_{M}G)}.$ (29)

Here we have used a simple argument to derive the consistency equation (27).
It can also be derived by a brute force from $\partial_{a}$ of (23) equal to
$\partial_{M}$ of (24).

In summary, from the view points of the regularity of the horizon as well as
the consistency with the zero-th and first laws of stationary black hole
thermodynamics, it is necessary that the scale identification should be angle-
independent and be in the form of Eq. (28) as a function of the mass and the
area. Note that this result is quite general and valid beyond the asymptotic
safety program; once the Newton coupling or equivalently the mass of the black
holes are assumed to depend on the coordinates, we have to adopt Eq. (28).

##  IV Quantum Improved Regular Kerr Black Holes

It is natural to assume that the result for running coupling (28) on the
horizon may be extended away from the horizon, i.e. $G(r)=G(Mr,r^{2}+a^{2})$.
As a simplest identification, let us take

$k^{2}=\frac{1}{G_{0}M^{p}r^{p}},$ (30)

on dimensional grounds, where $p$ is a positive integer. We could also include
the dependence on the area, but it does not vanish or diverge at $r=0$ and
does not help to resolve the singularity. This is the simplest choice
consistent with (28) extended away from the horizon, and as we will see, this
choice can resolve the singularity. We expect that other choices of
identification which can resolve the singularity give the qualitatively
similar results.

This identification may appear to have a subtle property that the energy scale
is divergent as the mass approaches to zero (flat spacetime)

$\lim_{M\to 0}k^{2}=\lim_{M\to 0}\frac{1}{G_{0}M^{p}r^{p}}\to\infty.$ (31)

This is in accordance with the expectation that the quantum effects are large
for small black holes.

This turns out to give physically interesting results for the resolution of
the singularity and other properties of the black holes. From Eq. (4), we find
the Newton coupling

$G(r)=\frac{G_{0}M^{p}r^{p}}{M^{p}r^{p}+\omega}.$ (32)

In the following, we are going to analyze properties, in particular the
resolution of the ring singularity and the existence of CTCs, in the quantum
improved Kerr black holes.

###  IV.1 Resolving Singularity

The singularities of a spacetime are characterised in the associated geometric
invariants. In particular, for the rotating black holes the algebraically
complete set consists of four invariants [37], two of them, $R$ and $I_{6}$,
are real and the other two, $I$ and $K$, are complex (their definitions are
given in Appendix A). For the quantum improved Kerr black holes with running
coupling (32), these four invariants take the form

$\displaystyle R$ $\displaystyle=$ $\displaystyle-\frac{2\omega
G_{0}pM^{p+1}r^{p-1}\left[(p-1)M^{p}r^{p}-(p+1)\omega\right]}{(r^{2}+a^{2}\cos^{2}\theta)(M^{p}r^{p}+\omega)^{3}},$
(33) $\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}G_{0}^{2}p^{2}M^{2p+2}r^{2p-2}\Bigl{\\{}\left[(p-1)M^{p}r^{p}-(p+1)\omega\right](r^{2}+a^{2}\cos^{2}\theta)+4(M^{p}r^{p}+\omega)r^{2}\Bigr{\\}}^{2}}{12(r^{2}+a^{2}\cos^{2}\theta)^{4}(M^{p}r^{p}+\omega)^{6}},$
(34) $\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{G_{0}^{2}M^{2p+2}r^{2p-2}}{36(r^{2}+a^{2}\cos^{2}\theta)^{2}(r+ia\cos\theta)^{4}(M^{p}r^{p}+\omega)^{6}}\Bigl{\\{}\omega
p\left[(p-1)M^{p}r^{p}-(p+1)\omega\right](r+ia\cos\theta)^{2}$ (36)
$\displaystyle\quad+6r\left[M^{p}r^{p}+(p+1)\omega\right](M^{p}r^{p}+\omega)(r+ia\cos\theta)-12r^{2}(M^{p}r^{p}+\omega)^{2}\Bigr{\\}}^{2},$
$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}G_{0}^{3}p^{2}M^{3p+3}r^{3p-3}\Bigl{\\{}\left[(p-1)M^{p}r^{p}-(p+1)\omega\right](r^{2}+a^{2}\cos^{2}\theta)+4(M^{p}r^{p}+\omega)r^{2}\Bigr{\\}}^{2}}{12(r^{2}+a^{2}\cos^{2}\theta)^{5}(r+ia\cos\theta)^{2}(M^{p}r^{p}+\omega)^{9}}$
$\displaystyle\times$ $\displaystyle\\!\Bigl{\\{}\omega
p\left[(p\\!-\\!1)M^{p}r^{p}\\!-\\!(p\\!+\\!1)\omega\right](r\\!+\\!ia\cos\theta)^{2}\\!+\\!6r\left[M^{p}r^{p}\\!+\\!(p\\!+\\!1)\omega\right](M^{p}r^{p}\\!+\\!\omega)(r\\!+\\!ia\cos\theta)\\!-\\!12r^{2}(M^{p}r^{p}\\!+\\!\omega)^{2}\Bigr{\\}}.$

Near the “equatorial disk”, i.e. $r\to 0$, these quantities have the following
fall off:

$R\sim\left\\{\begin{array}[]{rl}r^{p-1},&\quad\theta\neq\pi/2\\\
r^{p-3},&\quad\theta=\pi/2\end{array}\right.,\qquad I_{6}\sim
I\sim\left\\{\begin{array}[]{rl}r^{2(p-1)},&\quad\theta\neq\pi/2\\\
r^{2(p-3)},&\quad\theta=\pi/2\end{array}\right.,\qquad
K\sim\left\\{\begin{array}[]{rl}r^{3(p-1)},&\quad\theta\neq\pi/2\\\
r^{3(p-3)},&\quad\theta=\pi/2\end{array}\right..$ (38)

Therefore, in order to avoid divergence in the limit $r\to 0$, the value of
$p$ cannot be less than three. In the case $p=3$, all invariants have
nonvanishing values for $\theta=\pi/2$, thus there is a discontinuity at the
edge of equatorial disk.333This case is analogous to the rotating Hayward
black holes by the change $M\to Mr^{3}/(r^{3}+\ell^{3})$ in Kerr solution.
However, the rotating Hayward black holes indeed do not have a consistent
thermodynamics unless the replacement $\ell\to\ell/M$ is made. Moreover, if
$p$ is an odd integer, the term $M^{p}r^{p}+\omega$ appearing in the
denominator in all four invariants can vanish in the region $r<0$ (both $M$
and $\omega$ are positive) and the curvature singularity occurs. To avoid the
singularity, we consider the case that $p$ is an even integer.

The minimal integer value for $p$ to ensure regular and smooth spacetimes is
$p=4$, and the associated invariant quantities are

$\displaystyle R$ $\displaystyle=$ $\displaystyle-\frac{8\omega
G_{0}M^{5}r^{3}(3M^{4}r^{4}-5\omega)}{(M^{4}r^{4}+\omega)^{3}(r^{2}+a^{2}\cos^{2}\theta)},$
(39) $\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\frac{4\omega^{2}G_{0}^{2}M^{10}r^{6}\left[(3M^{4}r^{4}-5\omega)(r^{2}+a^{2}\cos^{2}\theta)+4(M^{4}r^{4}+\omega)r^{2}\right]^{2}}{3(M^{4}r^{4}+\omega)^{6}(r^{2}+a^{2}\cos^{2}\theta)^{4}},$
(40) $\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{G_{0}^{2}M^{10}r^{6}\left[2\omega(3M^{4}r^{4}\\!-\\!5\omega)(r\\!+\\!ia\cos\theta)^{2}\\!+\\!3r(M^{4}r^{4}\\!+\\!5\omega)(M^{4}r^{4}\\!+\\!\omega)(r\\!+\\!ia\cos\theta)\\!-\\!6r^{2}(M^{4}r^{4}\\!+\\!\omega)^{2}\right]^{2}}{9(r^{2}+a^{2}\cos^{2}\theta)^{2}(r+ia\cos\theta)^{4}(M^{4}r^{4}+\omega)^{6}},$
(41) $\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{8\omega^{2}G_{0}^{3}M^{15}r^{9}\left[(3M^{4}r^{4}-5\omega)(r^{2}+a^{2}\cos^{2}\theta)+4(M^{4}r^{4}+\omega)r^{2}\right]^{2}}{3(r^{2}+a^{2}\cos^{2}\theta)^{5}(r+ia\cos\theta)^{2}(M^{4}r^{4}+\omega)^{9}}$
$\displaystyle\times\left[2\omega(3M^{4}r^{4}-5\omega)(r+ia\cos\theta)^{2}+3r(M^{4}r^{4}+5\omega)(M^{4}r^{4}+\omega)(r+ia\cos\theta)-6r^{2}(M^{4}r^{4}+\omega)^{2}\right].$
(42)

The scalar curvature $R$ in (39) changes its sign at the critical radius
$r=(5\omega/3M^{4})^{1/4}$. This indicates that the gravitational interaction
changes from attractive force for the larger $r$ with negative $R$ to
repulsive for the smaller $r$ with positive $R$.

###  IV.2 Geodesic Incompleteness

In this subsection, we study the geodesics in our quantum improved Kerr black
holes. A general analysis for regular black holes was made in [42, 43]. For
massless particles, the geodesic equations for their trajectory
$\\{t(\lambda),r(\lambda),\theta(\lambda),\varphi(\lambda)\\}$ satisfy a
constraint $ds^{2}=0$. In rotating black holes, there are two conserved
quantities associated with the Killing vectors $\xi^{\mu}=(1,0,0,0)$ and
$\eta^{\mu}=(0,0,0,1)$. These are energy and angular momentum:

$e=-\xi^{\mu}u^{\nu}g_{\mu\nu}=-g_{tt}\dot{t}-g_{t\varphi}\dot{\varphi},\qquad
l=\eta^{\mu}u^{\nu}g_{\mu\nu}=g_{\varphi
t}\dot{t}+g_{\varphi\varphi}\dot{\varphi}.$ (43)

Here dot denotes the derivative with respect to the affine parameter
$\lambda$.

Let us consider the simplest case when a massless probe particle moves along
the rotation axis, i.e. $\theta=\dot{\theta}=0$. This implies
$g_{t\varphi}=g_{\varphi\varphi}=0$ and $l=0$. For the quantum improved Kerr
metric, the constraint and converged quantities reduce to

$\left(1-\frac{2G(r)Mr}{r^{2}+a^{2}}\right)\dot{t}^{2}-\left(1-\frac{2G(r)Mr}{r^{2}+a^{2}}\right)^{-1}\dot{r}^{2}=0,\qquad
e=\left(1-\frac{2G(r)Mr}{r^{2}+a^{2}}\right)\dot{t},\qquad l=0,$ (44)

which lead to ($t^{\prime}\equiv dt/dr$)

$\dot{r}^{2}=e=\text{constant},\qquad t^{\prime
2}=(\dot{t}/\dot{r})^{2}=\left(1-\frac{2G(r)Mr}{r^{2}+a^{2}}\right)^{-2}.$
(45)

The behavior of $t^{\prime}$ in the $r\to 0$ is essential. Let us check this
behavior for two well-known “regular” black holes. For the rotating Hayward
black holes, we can view effectively $G(r)=G_{0}r^{3}/(r^{3}+\ell^{3})$ and
find

$\displaystyle t_{H}^{\prime 2}$ $\displaystyle=$ $\displaystyle
1+\frac{4G_{0}M}{\ell^{3}}\left(\frac{r^{4}}{a^{2}}-\frac{r^{6}}{a^{4}}+\frac{r^{8}}{a^{6}}-\frac{r^{10}}{a^{8}}+\cdots\right)-\frac{4G_{0}M}{\ell^{6}}\left(\frac{r^{7}}{a^{2}}-\frac{r^{9}}{a^{4}}+\frac{r^{11}}{a^{6}}-\cdots\right)+\frac{4G_{0}M}{\ell^{9}}\left(\frac{r^{10}}{a^{2}}-\cdots\right)+\cdots$
(46)
$\displaystyle\quad+\frac{12G_{0}^{2}M^{2}}{\ell^{6}}\left(\frac{r^{8}}{a^{4}}-2\frac{r^{10}}{a^{6}}+3\frac{r^{12}}{a^{8}}-\cdots\right)-\frac{24G_{0}^{2}M^{2}}{\ell^{9}}\left(\frac{r^{11}}{a^{4}}-2\frac{r^{13}}{a^{6}}+\cdots\right)+\cdots.$

Similarly, for the Bardeen black holes, we can consider
$G(r)=G_{0}r^{3}/(r^{2}+\ell^{2})^{3/2}$ effectively and obtain

$\displaystyle t_{B}^{\prime 2}$ $\displaystyle=$ $\displaystyle
1+\frac{4G_{0}M}{\ell^{3}}\left(\frac{r^{4}}{a^{2}}-\frac{r^{6}}{a^{4}}+\frac{r^{8}}{a^{6}}-\frac{r^{10}}{a^{8}}+\cdots\right)-\frac{6G_{0}M}{\ell^{5}}\left(\frac{r^{6}}{a^{2}}-\frac{r^{8}}{a^{4}}+\frac{r^{10}}{a^{6}}-\cdots\right)+\frac{15G_{0}M}{2\ell^{7}}\left(\frac{r^{8}}{a^{2}}-\cdots\right)+\cdots$
(47)
$\displaystyle\quad+\frac{12G_{0}^{2}M^{2}}{\ell^{6}}\left(\frac{r^{8}}{a^{4}}-2\frac{r^{10}}{a^{6}}+3\frac{r^{12}}{a^{8}}-\cdots\right)-\frac{36G_{0}^{2}M^{2}}{\ell^{8}}\left(\frac{r^{10}}{a^{4}}-2\frac{r^{12}}{a^{6}}+\cdots\right)+\cdots.$

These two types of black holes have very different behaviors. For the rotating
Hayward black holes, it was argued in [43] that the $r^{7}$ (odd power) term
flips the sign and this implies the geodesic is not analytic at $r=0$. In this
case, if one requires the analyticity in the unique continuation of
genedesics, one should extend the geodesic to negative value of $r$. On the
contrary, for the Bardeen black holes, there are only even powers of $r$, and
this implies that the geodesic can return back via the corresponding
antipodean point. Therefore, it is unnecessary to have an extension to
negative value of $r$.

For the quantum improved Kerr black holes with
$G(r)=G_{0}M^{p}r^{p}/(M^{p}r^{p}+\omega)$ we have

$\displaystyle t^{\prime 2}$ $\displaystyle=$ $\displaystyle
1+\frac{4G_{0}M^{p+1}}{\omega}\left(\frac{r^{p+1}}{a^{2}}-\frac{r^{p+3}}{a^{4}}+\frac{r^{p+5}}{a^{6}}-\cdots\right)-\frac{4G_{0}M^{2p+1}}{\omega^{2}}\left(\frac{r^{2p+1}}{a^{2}}-\frac{r^{2p+3}}{a^{4}}+\cdots\right)+\cdots$
(48)
$\displaystyle\quad+\frac{12G_{0}^{2}M^{2p+2}}{\omega^{2}}\left(\frac{r^{2p+2}}{a^{4}}-2\frac{r^{2p+4}}{a^{6}}+3\frac{r^{2p+6}}{a^{8}}-\cdots\right)-\frac{24G_{0}^{2}M^{3p+2}}{\omega^{3}}\left(\frac{r^{3p+2}}{a^{4}}-2\frac{r^{3p+4}}{a^{8}}+\cdots\right)+\cdots.$

It is obvious that odd powers of $r$, for example $r^{2p+1}$, always exist.
Similarly to the rotating Hayward black holes, the geodesics are not smooth at
$r=0$ for any value of $p$. The extension to negative value of $r$ is
inevitable, as far as the unique continuation is required.

###  IV.3 Closed Time-like Curves

As is well known, the Kerr black holes have CTCs which cause the violation of
causality inside the inner Cauchy horizon. The metric component
$g_{\varphi\varphi}$ ($G$ is a constant) is

$g_{\varphi\varphi}=\sin^{2}\theta\left(r^{2}+a^{2}+\frac{2GMra^{2}\sin^{2}\theta}{\Sigma}\right),$
(49)

which can be negative if $r<0$ and $\Sigma$ is small enough. For example, at
$\theta=\pi/2$, it reduces to

$g_{\varphi\varphi}\Big{|}_{\theta=\pi/2}=r^{2}+a^{2}+\frac{2GMa^{2}}{r}\quad\stackrel{{\scriptstyle|r|\ll
a}}{{\longrightarrow}}\quad
a^{2}\left(1+\frac{2GM}{r}\right)\quad\stackrel{{\scriptstyle|r|\ll
GM}}{{\longrightarrow}}\quad\frac{2GMa^{2}}{r}<0,\quad\textrm{if}\quad r<0,$
(50)

which could be negative for $r<0$ and $r\ll GM,a$.

For the quantum improved Kerr black holes with the identification (32) and
even integer $p$ (to avoid curvature singularity in $r<0$), we have

$g_{\varphi\varphi}=\sin^{2}\theta\left(r^{2}+a^{2}+\frac{2G_{0}M^{p+1}r^{p+1}a^{2}\sin^{2}\theta}{(M^{p}r^{p}+\omega)\Sigma}\right),$
(51)

which again can be negative if $r<0$ and $\Sigma$ is small enough. If
$g_{\varphi\varphi}$ is positive in the limit $\Sigma\ll 1$, then there are no
CTCs. To achieve $\Sigma\ll 1$, we should take $\theta=\pi/2$ and
$|rM|,\,|r/a|\ll 1$. Then the condition gives

$g_{\varphi\varphi}\Big{|}_{\theta=\pi/2}=r^{2}+a^{2}+\frac{2G_{0}M^{p+1}r^{p-1}a^{2}}{M^{p}r^{p}+\omega}\quad\stackrel{{\scriptstyle|r|\ll
1}}{{\longrightarrow}}\quad
a^{2}\left(1+\frac{2G_{0}M^{p+1}r^{p-1}}{\omega}\right)\quad\to\quad a^{2}>0.$
(52)

So $g_{\varphi\varphi}$ does not become negative in the region $r<0$ for small
$|r|$.

It is complicated to determine the region of negative $r$ when
$g_{\varphi\varphi}$ in (52) becomes negative. So just to get rough estimate
of the region, let us consider the negative third term of $g_{\varphi\varphi}$
in (52). It has a minimum (largest negative value) in the negative range of
$r$ at444 Note that we are considering even $p$ in order to avoid the
curvature singularity.

$r=-\frac{\tilde{\omega}^{1/p}}{M},\qquad\tilde{\omega}=(p-1)\omega.$ (53)

The value of $g_{\varphi\varphi}$ at this point has to be positive in order
for CTCs to be absent. This gives the necessary condition for the absence of
CTCs. It gives

$p\tilde{\omega}^{3/p}+pM^{2}a^{2}\tilde{\omega}^{1/p}-2(p-1)G_{0}M^{4}a^{2}\geq
0.$ (54)

It is clear that the CTC always exists in classical solution
($\tilde{\omega}=0$). The quantum effect can remove the CTC when

$1<\dfrac{\tilde{\omega}^{1/p}}{G_{0}M^{2}},\qquad\dfrac{|a|}{G_{0}M}<\dfrac{\tilde{\omega}^{1/p}}{G_{0}M^{2}}.$
(55)

Note that $\tilde{\omega}^{1/p}/G_{0}M^{2}$ is a measure for the impact of the
quantum gravitational effects on the geometry (see Eq. (4.8) in Ref. [15]),
while the left-hand side $|a|/G_{0}M$ of the second condition is the
extremality for the black hole rotation: when $|a|/G_{0}M=1$, the classical
metric describes an extremal Kerr black hole.

###  IV.4 Black Holes Thermodynamics

In this subsection, we will summarize the thermodynamical properties for
quantum improved black holes for an identification $G=G(Mr_{h})$. More general
case in which $G$ is a function of $Mr_{h}$ and the area $A_{h}$ is discussed
in Sec. III.3. In this restricted case, the radius of horizon $r_{h}$ is
determined by

$\Delta(r_{h})=r_{h}^{2}+a^{2}-2Mr_{h}G(Mr_{h})=0,$ (56)

leading to

$Mr_{h}G(Mr_{h})=\frac{r_{h}^{2}+a^{2}}{2}=\frac{A_{h}}{8\pi},$ (57)

which shows that $Mr_{h}$ is a function of horizon area $A_{h}$. With a new
variable $\rho\equiv Mr_{h}$, the running coupling can be written as a
function of $\rho$, i.e. $G(Mr_{h})=G(\rho)$ and then we have two useful
relations at horizon $r=r_{h}$: (i)
$G+M\partial_{M}G=G+\rho\partial_{\rho}G=\partial_{\rho}(\rho G)=d(\rho
G)/d\rho$, and (ii) $dA_{h}=4\pi d(r_{h}^{2}+a^{2})=8\pi d(\rho G)$. Now it is
straightforward to compute the entropy from our general formula (29):

$S=\int\frac{dA_{h}}{4(G+M\partial_{M}G)}=\int\frac{2\pi d(\rho
G)}{\partial_{\rho}(\rho G)}=2\pi\int d\rho=2\pi
Mr_{h}=\frac{A_{h}}{4G(Mr_{h})}.$ (58)

Interestingly the entropy preserves the “classical relation”, namely
$S=A_{h}/4G$, for identification $G=G(Mr_{h})$. Unlike the other possible
identification $G=G(r_{h}^{2}+a^{2})$ [32], here the quantum correction “does
not” create a logarithm correction term. Finally, owing to the relation
$G+r_{h}\partial_{r_{h}}G=G+\rho\partial_{\rho}G$, the temperature (21) can be
expressed as

$T=\frac{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G}{2\pi(r_{h}^{2}+a^{2})}=\frac{r_{h}^{2}-Mr_{h}(G+\partial_{\rho}G)}{2\pi
r_{h}(r_{h}^{2}+a^{2})}=\frac{r_{h}^{2}-a^{2}-2\rho\partial_{\rho}G}{4\pi
r_{h}(r_{h}^{2}+a^{2})}.$ (59)

For the identification (32), the temperature and entropy are

$T=\frac{r_{h}^{2}-a^{2}}{4\pi r_{h}(r_{h}^{2}+a^{2})}-\frac{\omega
pG_{0}M^{p}r_{h}^{p}}{2\pi
r_{h}(r_{h}^{2}+a^{2})\left(M^{p}r_{h}^{p}+\omega\right)^{2}},\qquad
S=\frac{A_{h}}{4G_{0}}+\frac{\omega A_{h}}{4G_{0}M^{p}r_{h}^{p}}.$ (60)

So the entropy deviates from the classical results though Eq. (58) looks like
a classical expression; $\omega$ represents the quantum contribution.

##  V Conclusion

In this paper we have studied the quantum improvement of Kerr black holes in
asymptotically safe gravity. The main purpose is to construct regular black
holes in which the ring singularity in classical Kerr solutions is resolved by
quantum effect. Resolution of singularity indeed has been largely discussed in
the literatures. But in those studies, the possible violation of the black
hole thermodynamics has been simply neglected. As pointed out in the
introduction, if the mass or equivalently the gravitational constant is a
function of coordinate only, our results show that the thermodynamics are not
satisfied. If we consider only the case that the running coupling does not
depend on the mass parameter, we find the condition $G=G(r^{2}+a^{2})$ [32] to
be necessary for the first law of black hole thermodynamics. The phase
structure in this case, is analyzed [44]. Unfortunately, this type of Newton
coupling does not approach to zero or diverge near the ring singularity and
hence cannot resolve singularity.

What we have found in this paper is that when the mass parameter is included
in the scale identification, there is more general identification
$G=G(Mr_{h},A_{h})$ that can preserve the consistency of black hole
thermodynamics. Though $Mr_{h}$ is actually also a function of the area on the
horizon, this form can be extended away from the horizon. We have then shown
that this can resolve the singularity. Although we have studied only a
particular “solution” for the Newton coupling which gives regular quantum-
improved Kerr black holes, we expect that even for more general solutions for
the Newton coupling, the qualitative feature would not change as far as the
choice of the scale identification can resolve the singularities The
regularity of our black hole solutions is confirmed by four geometric scalar
invariants (two real and 2 complex). All of them are continuous and never
divergent in all ranges of radial coordinate, from negative infinity to
positive infinity. We have also found that the geodesics are incomplete at
$r=0$, and therefore an extension to negative values of $r$ is necessary.
Moreover, it is remarkable that the occurrence of CTCs is also eliminated by
the quantum effects for the mass and angular momentum of black holes where the
quantum effects are significant.

###### Acknowledgements.

We thank Aaron Held for valuable discussions. The work of C.M.C. was supported
by the National Science and Technology Council of the R.O.C. (Taiwan) under
the grant 112-2112-M-008-020. The work of A.I. was supported in part by JSPS
KAKENHI Grants No. 21H05182, 21H05186, 20K03938, 20K03975, 17K05451, and
15K05092. The work of N.O. was supported in part by the Grant-in-Aid for
Scientific Research Fund of the JSPS (C) No. 20K03980 and Taiwan NSTC
112-2811-M-008-016.

##  Appendix A Geometric Invariant Quantities

For the rotating black holes, the algebraically complete set of invariants are
[37]

$R,\qquad I_{6}=\frac{1}{12}S_{\alpha\beta}S^{\alpha\beta},\qquad
I=\frac{1}{24}{\bar{C}}_{\alpha\beta\mu\nu}{\bar{C}}^{\alpha\beta\mu\nu},\qquad
K=\frac{1}{4}{\bar{C}}_{\alpha\mu\nu\beta}S^{\mu\nu}S^{\alpha\beta},$ (61)

where $S_{\mu\nu}$ and ${\bar{C}}_{\alpha\beta\mu\nu}$ are defined as

$S_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{4}g_{\alpha\beta}R,\qquad{\bar{C}}_{\alpha\beta\mu\nu}=\frac{1}{2}\left(C_{\alpha\beta\mu\nu}+i\,{}^{*}C_{\alpha\beta\mu\nu}\right),\quad{}^{*}C_{\alpha\beta\mu\nu}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}C^{\gamma\delta}{}_{\mu\nu},$
(62)

and $R,R_{\alpha\beta}$ and $C_{\alpha\beta\mu\nu}$ are the scalar curvature,
Ricci tensor and Weyl tensor, respectively,

$C_{\alpha\beta\mu\nu}=R_{\alpha\beta\mu\nu}+\frac{1}{2}\left(R_{\alpha\nu}g_{\beta\mu}-R_{\alpha\mu}g_{\beta\nu}+R_{\beta\mu}g_{\alpha\nu}-R_{\beta\nu}g_{\alpha\mu}\right)+\frac{1}{6}R\left(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}\right).$
(63)

The Kretschmann scalar is related to these invariants as

$\mathcal{K}=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}=C_{\alpha\beta\mu\nu}C^{\alpha\beta\mu\nu}+2R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^{2}=48\,\mathrm{Re}(I)+24I_{6}+\frac{1}{6}R^{2}.$
(64)

For the Kerr black holes in the BL coordinates (5) with the running coupling
$G=G(r)$, these invariant quantities are

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{2M(r\partial_{r}^{2}G+2\partial_{r}G)}{r^{2}+a^{2}\cos^{2}\theta},$
(65) $\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\frac{M^{2}\left[(r\partial_{r}^{2}G+2\partial_{r}G)(r^{2}+a^{2}\cos^{2}\theta)-4r^{2}\partial_{r}G\right]^{2}}{12(r^{2}+a^{2}\cos^{2}\theta)^{4}},$
(66) $\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{M^{2}\left[(r\partial_{r}^{2}G+2\partial_{r}G)(r+ia\cos\theta)^{2}-6(r\partial_{r}G+G)(r+ia\cos\theta)+12rG\right]^{2}}{36(r^{2}+a^{2}\cos^{2}\theta)^{2}(r+ia\cos\theta)^{4}},$
(67) $\displaystyle K$ $\displaystyle=$
$\displaystyle-\frac{M^{3}\left[(r\partial_{r}^{2}G+2\partial_{r}G)(r^{2}+a^{2}\cos^{2}\theta)-4r^{2}\partial_{r}G\right]^{2}}{12(r^{2}+a^{2}\cos^{2}\theta)^{5}(r+ia\cos\theta)^{2}}$
(68)
$\displaystyle\times\left[(r\partial_{r}^{2}G+2\partial_{r}G)(r+ia\cos\theta)^{2}-6(r\partial_{r}G+G)(r+ia\cos\theta)+12rG\right],$

and the Kretschmann scalar is

$\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle\frac{4M^{2}}{(r^{2}+a^{2}\cos^{2}\theta)^{6}}\Bigl{\\{}\Bigl{[}(r\partial_{r}^{2}G+2\partial_{r}G)(r^{2}+a^{2}\cos^{2}\theta)^{2}-2r(2r\partial_{r}G+3G)(r^{2}+a^{2}\cos^{2}\theta)+8r^{3}G\Bigr{]}^{2}$
(69)
$\displaystyle-12(r\partial_{r}G+G)^{2}(r^{2}+a^{2}\cos^{2}\theta)^{3}+4r^{2}(2r\partial_{r}G+3G)(2r\partial_{r}G+15G)(r^{2}+a^{2}\cos^{2}\theta)^{2}$
$\displaystyle-32r^{4}G(4r\partial_{r}G+15G)(r^{2}+a^{2}\cos^{2}\theta)+320r^{6}G^{2}\Bigr{\\}}.$

##  Appendix B Quantum Improved Kerr-(A)dS Black Holes

Here we generalize the analysis of thermodynamics in Sec. III to the case with
a fixed cosmological constant $\Lambda$. This case may not be directly related
to the asymptotically safe gravity (see, however Ref. [35]), but it is still
interesting to check the constraint on the running coupling with the
assumption $G=G(r,M,a)$. Here the cosmological constant is supposed to be
constant (not variable) in the first law of thermodynamics. The line-element
of the classical Kerr-(A)dS black hole solution in the BL coordinate is

$ds^{2}=-\frac{\Delta_{r}}{\Sigma}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\varphi\right)^{2}+\frac{\Sigma}{\Delta_{r}}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}}{\Sigma}\sin^{2}\theta\left(adt-\frac{r^{2}+a^{2}}{\Xi}d\varphi\right)^{2},$
(70)

where

$\Delta_{r}=(r^{2}+a^{2})\left(1-\frac{\Lambda}{3}r^{2}\right)-2GMr,\quad\Delta_{\theta}=1+\frac{\Lambda}{3}a^{2}\cos^{2}\theta,\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta,\quad\Xi=1+\frac{\Lambda}{3}a^{2}.$
(71)

In the semiclassical case, where $G$ is a constant, the Hawking temperature
and the entropy are given by

$T=\frac{\kappa}{2\pi}=\frac{r_{h}}{4\pi(r_{h}^{2}+a^{2})}\left(1-\frac{a^{2}}{r_{h}^{2}}-\frac{\Lambda
a^{2}}{3}-\Lambda r_{h}^{2}\right),\qquad S=\frac{A_{h}}{4G},$ (72)

where $A_{h}$ is the horizon area

$A_{h}=\frac{4\pi(r_{h}^{2}+a^{2})}{\Xi}.$ (73)

These quantities satisfy the first law of thermodynamics

$dE=TdS+\Omega dJ,$ (74)

where

$E=\frac{M}{\Xi^{2}},\qquad
J=\frac{Ma}{\Xi^{2}},\qquad\Omega=\frac{a(1-\Lambda
r_{h}^{2}/3)}{r_{h}^{2}+a^{2}}.$ (75)

Note again that these formulas hold not only for the outer event horizon, but
also for the inner Cauchy horizon, as well as for cosmological horizon when
$\Lambda>0$.

The Kretschmann scalar is

$\mathcal{K}=\frac{8}{3}\Lambda^{2}+\frac{48M^{2}G^{2}(r^{6}-15a^{2}\cos^{2}\theta
r^{4}+15a^{4}\cos^{4}\theta
r^{2}-a^{6}\cos^{6}\theta)}{(r^{2}+a^{2}\cos^{2}\theta)^{6}},$ (76)

which depends on the angle coordinate $\theta$. Taking the limit $r\to 0$ we
have

$\lim_{r\to
0}\mathcal{K}=\begin{cases}\dfrac{48M^{2}G^{2}}{r^{6}}\to\infty,&\theta=\pi/2\\\
&\\\
\frac{8}{3}\Lambda^{2}-\dfrac{48M^{2}G^{2}}{a^{6}\cos^{6}\theta},&\theta\neq\pi/2\end{cases}.$
(77)

This indicates a ring singularity at
$r=0,\,\theta=\pi/2,\varphi=\mbox{arbitrary}$, and the Kretschmann scalar is
not a continuous function in $\theta$ on the disk $r=0$ at the ring
singularity.

For the quantum improved rotating black holes with a cosmological constant,
the horizon radius is determined by

$\Delta_{r}(r_{h})=(r_{h}^{2}+a^{2})\left(1-\frac{\Lambda}{3}r_{h}^{2}\right)-2G(r_{h};M,a,\Lambda)Mr_{h}=0,$
(78)

and its first derivative, treating $r_{h}=r_{h}(M,a)$, gives

$\displaystyle 0=d\Delta_{r}(r_{h})$ $\displaystyle=$ $\displaystyle
2r_{h}(\partial_{M}r_{h}\,dM+\partial_{a}r_{h}\,da)+2a\,da-\frac{2\Lambda}{3}r_{h}(2r_{h}^{2}+a^{2})(\partial_{M}r_{h}\,dM+\partial_{a}r_{h}\,da)-\frac{2\Lambda}{3}r_{h}^{2}a\,da$
(79) $\displaystyle-$ $\displaystyle
2r_{h}G\,dM-2Mr_{h}\partial_{M}G\,dM-2Mr_{h}\partial_{a}G\,da-2M(G+r_{h}\partial_{r_{h}}G)(\partial_{M}r_{h}\,dM+\partial_{a}r_{h}\,da).$

This gives

$\displaystyle\partial_{M}r_{h}=\frac{r_{h}(G+M\partial_{M}G)}{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G-\frac{\Lambda}{3}r_{h}(2r_{h}^{2}+a^{2})},$
$\displaystyle\partial_{a}r_{h}=-\frac{a-Mr_{h}\partial_{a}G-\Lambda
r_{h}^{2}a/3}{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G-\frac{\Lambda}{3}r_{h}(2r_{h}^{2}+a^{2})}.$ (80)

For Kerr-(A)dS black holes, the internal energy, angular momentum, angular
velocity are given in (75) and the temperature is

$T=\frac{r_{h}-MG-
Mr_{h}\partial_{r_{h}}G-\frac{\Lambda}{3}r_{h}(2r_{h}^{2}+a^{2})}{2\pi(r_{h}^{2}+a^{2})}=\frac{r_{h}(G+M\partial_{M}G)}{2\pi(r_{h}^{2}+a^{2})\partial_{M}r_{h}},$
(81)

where we have used (80) in the second equality. The thermodynamical first law
$\delta E=T\delta S+\Omega\delta J$ gives

$\delta S=\frac{\delta E-\Omega\delta J}{T}$ (82)

leading to

$\partial_{M}S=\frac{r_{h}^{2}}{T\Xi(r_{h}^{2}+a^{2})},\qquad\partial_{a}S=-\frac{Ma(1+\Lambda
r_{h}^{2})}{T\Xi^{2}(r_{h}^{2}+a^{2})}.$ (83)

Using (81), we find

$\partial_{M}S=\frac{2\pi
r_{h}\partial_{M}r_{h}}{\Xi(G+M\partial_{M}G)},\qquad\partial_{a}S=-\frac{2\pi
Ma(1+\Lambda r_{h}^{2})\partial_{M}r_{h}}{\Xi^{2}r_{h}(G+M\partial_{M}G)}.$
(84)

Again we can rewrite $\partial_{M}S$ in terms of horizon area (73):

$\partial_{M}S=\frac{\partial_{M}A_{h}}{4(G+M\partial_{M}G)}.$ (85)

The consistency is ensured if $\partial_{a}S$ has the following form

$\partial_{a}S=\frac{\partial_{a}A_{h}}{4(G+M\partial_{M}G)}=\frac{2\pi}{G+M\partial_{M}G}\left(\frac{r_{h}\partial_{a}r_{h}}{\Xi}+\frac{a(1-\Lambda
r_{h}^{2}/3)}{\Xi^{2}}\right).$ (86)

Comparing this equation with (84) and using (80) and (78), we can get the
following equation for a consistent mass-dependent running coupling:

$Ma(1+\Lambda
r_{h}^{2})\partial_{M}G(r_{h},M,a)+r_{h}^{2}\Xi\partial_{a}G(r_{h},M,a)-ar_{h}\left(1-\frac{\Lambda}{3}r_{h}^{2}\right)\partial_{r_{h}}G(r_{h},M,a)=0.$
(87)

The solution of this equation is

$G(r_{h},M,a)=G\left(\frac{Mr_{h}}{\Xi^{2}},\frac{r_{h}^{2}+a^{2}}{\Xi}\right).$
(88)

For this case, it is not obvious that this solution is the function of the
horizon area. To confirm this relation, we first divide equation (78) by
$\Xi^{2}$:

$\frac{\Delta(r_{h})}{\Xi^{2}}=\left(\frac{r_{h}^{2}+a^{2}}{\Xi}\right)\left(\frac{1-\frac{\Lambda}{3}r_{h}^{2}}{\Xi}\right)-2\,G\left(\frac{Mr_{h}}{\Xi^{2}},\frac{r_{h}^{2}+a^{2}}{\Xi}\right)\frac{Mr_{h}}{\Xi^{2}}=0,$
(89)

where

$\frac{1-\frac{\Lambda}{3}r_{h}^{2}}{\Xi}=1-\frac{\Lambda}{3}\frac{r_{h}^{2}+a^{2}}{\Xi}=1-\frac{\Lambda}{12\pi}A_{h}.$
(90)

This is nothing but a function of $A_{h}$. Then we can easily see
$Mr_{h}/\Xi^{2}$ is also a function of $A_{h}$. The entropy for this system is
again given by the general formula (29).

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