# Observational signatures of rotating black holes in the semiclassical
gravity with trace anomaly

Zhenyu Zhang1 Yehui Hou1∗ Minyong Guo2

###### Abstract

In a recent work by Fernandes [1], an exact stationary and axisymmetric
solution was discovered in semiclassical gravity with type-A trace anomaly,
identified as a quantum-corrected version of the Kerr black hole. In this
study, we explore the observational signatures of this black hole solution.
Our investigation reveals that there exist prograde and retrograde light
rings, whose radii increase monotonically with the coupling parameter
$\alpha$. When $\alpha$ is negative, the shadow area for the quantum-corrected
black hole is smaller than that of the Kerr black hole, whereas when $\alpha$
is positive, the area is larger. For a near-extremal black hole, its high-spin
feature (the NHEKline) is found to be highly susceptible to disruption by
$\alpha$. Furthermore, we discuss the images of the quantum-corrected black
hole in the presence of a thin accretion disk and compare them to those of the
Kerr black hole. Our study highlights the importance of near-horizon emission
sources in detecting the effects of quantum corrections by black hole images.

1Department of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871,
P.R. China

2 Department of Physics, Beijing Normal University, No.19, Xinjiekouwai St,
Beijing 100875, P. R. China

$\ast$ Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

Semiclassical gravity is an approach that considers the backreaction of
quantum fields while treating spacetime classically. One of the quantum
effects in this scheme is the trace anomaly, which refers to the breaking of
symmetry in a conformally invariant classical theory due to one-loop quantum
corrections [2]. As a result of the trace anomaly, the renormalized stress
tensor of quantum fields has a non-zero trace, serving as a source term in the
semiclassical Einstein equations. The trace anomaly may also induce higher-
order curvature terms such as the Gauss-Bonnet term, which arises from the
type-A anomaly [3].

One of the topics of solutions in semiclassical gravity is the corrected
versions of classical black holes that account for quantum effects. However,
the derivations are challenging because the renormalized stress tensor is
often unknown, requiring additional assumptions to solve the problem. More
than a decade ago, considering only the type-A anomaly, [4] was the first to
find the static and spherically symmetric black hole solution in a four-
dimensional spacetime within the framework of such semiclassical gravity. This
spherically symmetric black hole has been widely studied in subsequent works
[5, 6, 7, 8, 9] because of its intriguing features resulting from quantum
effects. Furthermore, having an exact stationary and axisymmetric solution to
the semiclassical Einstein equations that is sourced by the type-A trace
anomaly is crucial for modeling the actual black hole in space.

Very recently, by adopting a Kerr-Schild ansatz with stationary and
axisymmetric configuration, the author in [1] analytically solved the
semiclassical Einstein equation and obtained a Kerr black hole solution with
the type-A trace anomaly. Compared to the classical Kerr black hole, this new
solution replaces the ADM mass with a mass function

$\mathcal{M}(r,\theta)=\frac{2M}{1+\sqrt{1-\frac{8\alpha r\xi
M}{\Sigma^{3}}}}\,$ (1.1)

in the ingoing Kerr-like coordinates, where $M$ represents the ADM mass,
$\Sigma=r^{2}+a^{2}\cos^{2}\theta$ with $a$ denoting the spin parameter,
$\xi=r^{2}-3a^{2}\cos^{2}\theta$, and $\alpha$ representing the coupling
constant of the type-A anomaly. This rotating black hole solution includes
quantum corrections and reduces to the classical Kerr spacetime when
$\alpha=0$. Furthermore, when $a=0$, the solution reduces to the static and
spherically symmetric solution in semiclassical gravity. This new solution
presents several unique characteristics. For instance, the event horizon
geometry is non-spherically symmetric, and there exists another Killing
horizon outside of it. Additionally, under specific coupling constants, the
spin parameter may surpass the traditional Kerr bound. This suggests that
black holes may possess higher spins than their classical counterparts [10].

From an astrophysical perspective, the rotating black hole with the type-A
anomaly may have some observational features. With the Event Horizon Telescope
collaboration already capturing images of supermassive black holes at the
centers of galaxies [11, 12, 13], studying black hole images, particularly the
shadow of this black hole, becomes essential [14, 15]. The size and shape of a
shadow can reflect the geometric structure and physical properties of the
central black hole, thus having the potential to test the coupling parameter
of the quantum-corrected black hole.

In this study, we investigate the observable features of the recently
discovered black hole [1]. Specifically, our focus lies on the analysis of
light rings (LRs) [16, 17, 18, 19], shadows [20, 21, 22, 23, 24], and images
when illuminated by an external light source [25, 26, 27, 28, 29, 30, 31].
Initially, we examine the effective potential of particles in the equatorial
plane and derive the equation governing the positions of the LRs. Due to the
absence of an analytical solution, we numerically calculate the LRs as a
function of the coupling constant for various spin values. The existence of
LRs implies the presence of a critical curve and a shadow on the observer’s
screen when illuminated by celestial light. To explore the shadow images, we
employ the backward ray-tracing method. Furthermore, we utilize a simplified
thin disk model to compare the intensity images with those of the Kerr black
hole, aiming for a more realistic scenario.

The paper is organized as follows. In Sec. 2, we provide a review of the
quantum-corrected Kerr black hole, examining particle trajectories and the
existence of light rings within this spacetime. In Sec. 3, we investigate the
shadow images of the quantum-corrected Kerr black hole when illuminated by a
celestial source. Then, in Sec. 4, we introduce a thin disk model that serves
as a more realistic light source, presenting the obtained results of black
hole images. Finally, we summarize our findings and conclude this study in
Sec. 5. We work in the geometrized unit with $8\pi G=c=1$ in this paper.

##  2 The metric and the light rings

To begin, let us briefly review the semiclassical Einstein gravity and the
quantum-corrected Kerr black holes in [1]. In this framework, the background
geometry remains classical while the quantum fields influence the geometry
through their expectation value of the renormalized stress tensor, denoted as
$\langle T_{\mu\nu}\rangle$, in the Einstein equations. Due to the quantum
anomaly, the trace of $\langle T_{\mu\nu}\rangle$ is non-zero and dependent
only on the local curvature. If we only consider the type-A anomaly, we have

$g^{\mu\nu}\langle T_{\mu\nu}\rangle=\frac{\alpha}{2}\,\mathcal{G}\,,$ (2.1)

where
$\mathcal{G}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
is the Gauss-Bonnet scalar, and $\alpha$ represents the coupling constant. By
combining Eq. (2.1) with the semiclassical Einstein equations,
$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\langle T_{\mu\nu}\rangle$, one arrives at:

$R=\frac{\alpha}{2}\mathcal{G}\,.$ (2.2)

In general, it is impossible to solve the semiclassical Einstein equations
when the renormalized stress tensor remains undetermined. However, by adopting
a Kerr-Schild ansatz and directly solving Eq. (2.2), the author of [1] found a
stationary and axisymmetric solution, which is interpreted as a quantum-
corrected Kerr black hole. The line-element was written in the ingoing Kerr-
like coordinates $(\nu,r,\theta,\varphi)$,

$\mathrm{d}s^{2}=-\mathopen{}\mathclose{{}\left(1-\frac{2\mathcal{M}r}{\Sigma}}\right)\mathopen{}\mathclose{{}\left(\mathrm{d}\nu-a\sin^{2}{\theta}\mathrm{d}\varphi}\right)^{2}+2\mathopen{}\mathclose{{}\left(\mathrm{d}\nu-a\sin^{2}{\theta}\mathrm{d}\varphi}\right)\mathopen{}\mathclose{{}\left(\mathrm{d}r-a\sin^{2}{\theta}\mathrm{d}\varphi}\right)+\Sigma\mathopen{}\mathclose{{}\left(\mathrm{d}\theta^{2}+\sin^{2}{\theta}\mathrm{d}\phi^{2}}\right)\,,$
(2.3)

where $\mathcal{M}$ is the mass function in Eq. (1.1). Since $\mathcal{M}$ is
dependent on $\theta$, the resulting spacetime does not satisfy the
circularity conditions [32]. Fig. 1 provides various examples of $\mathcal{M}$
under typical coupling constants. It is worth noting that a significant
difference exists between the cases with positive and negative coupling
constants. Moreover, the radius of the event horizon is also a function of
$\theta$, rendering it non-spherically symmetric and requiring numerical
solving. Besides, it has been observed that another Killing horizon is present
at $r^{2}-2\mathcal{M}r+a^{2}=0$, which does not coincide with the event
horizon but is located quite close to it.

Figure 1: $\mathcal{M}$ as function of $r$ for differnet values of $\alpha$,
evaluated at the pole (Left) and the equatorial plane (Right). The spin is fix
at $a=0.8$.

In this work, we focus on potential observational features of the quantum-
corrected Kerr black hole. Specifically, we would like to solve the
propagations of light and determine the associated black hole images. Under
the WKB approximation, i.e., the wave length of the light is much smaller than
the gravitational radius, the light propagations can be described by null
geodesics. The constrained Hamiltonian for geodesics is given by

$\mathcal{H}=\frac{1}{2}g_{\mu\nu}p^{\mu}p^{\nu}=-\frac{1}{2}m^{2}\,,$ (2.4)

where $p^{\mu}$ is the four momentum, $m=0,1$ denotes null and timelike
geodesics, respectively. The equations of motion are

$\displaystyle\frac{\partial\mathcal{H}}{\partial
p_{\mu}}=\dot{x}^{\mu}\,,\quad\frac{\partial\mathcal{H}}{\partial
x^{\mu}}=-\dot{p}_{\mu}\,,$ (2.5)

where the dot denotes the derivative with respect to the affine parameter. As
the spacetime is non-circular, it seems impossible to recast the equations of
motion into first-order equations as can be done in Kerr spacetime. Moreover,
the global transformation between the ingoing Kerr coordinates and the Boyer-
Lindquist coordinates is rather complicated [33].

Nonetheless, we can still gain some insights from the simplified scenario of
geodesics on the equatorial plane. When restricted to $\theta=\pi/2$, the
metric Eq. (2.3) can be easily transformed to

$\displaystyle\mathrm{d}\nu=\mathrm{d}t+\frac{r^{2}-a^{2}}{\Delta}\mathrm{d}r\,,\quad\mathrm{d}\varphi=\mathrm{d}\phi+\frac{a}{\Delta}\mathrm{d}r\,,$
(2.6)
$\mathrm{d}s^{2}=-\mathopen{}\mathclose{{}\left(1-\frac{2\mathcal{M}_{0}r}{\Sigma}}\right)\mathrm{d}t^{2}+\frac{\Sigma}{\Delta}\mathrm{d}r^{2}+\mathopen{}\mathclose{{}\left(r^{2}+a^{2}+\frac{2\mathcal{M}_{0}ra^{2}}{\Sigma}}\right)\mathrm{d}\phi^{2}-\frac{4\mathcal{M}_{0}ra}{\Sigma}\mathrm{d}t\mathrm{d}\phi\,,$
(2.7)

where
$\Delta=r^{2}-2\mathcal{M}_{0}r+a^{2},\mathcal{M}_{0}=\mathcal{M}(r,\pi/2)=2M/\mathopen{}\mathclose{{}\left(1+\sqrt{1-\frac{8\alpha
M}{r^{3}}}}\right)$. In this case, the geodesics are determined by an one-
dimensional equation of $r$, which takes

$\displaystyle\bigg{(}\frac{\mathrm{d}r}{\mathrm{d}\lambda}\bigg{)}^{2}=V_{eff}(r)=E^{2}-m^{2}+\frac{2\mathcal{M}_{0}m^{2}}{r}+\frac{a^{2}(E^{2}-m^{2})-L^{2}}{r^{2}}+\frac{2\mathcal{M}_{0}(L-aE)^{2}}{r^{3}}$
(2.8)

with $E=-p_{t},L=p_{\phi}$ the conserved energy and angular momentum, and
$V_{eff}$ acting as an effective potential. Let us now focus on the case of
photons with $m=0$. The key property of photons is that their conserved energy
can be absorbed into the affine parameter by rescaling
$\lambda\rightarrow\lambda E$. Then, by defining $l\equiv L/E$, the potential
is simplified to

$V_{eff}(r)=1+\frac{a^{2}-l^{2}}{r^{2}}+\frac{2\mathcal{M}_{0}(l-a)^{2}}{r^{3}}\,.$
(2.9)

The unstable photon orbits, known as light rings (LRs), are circular orbits of
photons that play a crucial role in the observational properties of black
holes. LRs can be obtained by setting

$V_{eff}=0\,,\quad\partial_{r}V_{eff}=0\,.$ (2.10)

From these equations, we obtain

$\displaystyle
l=-\frac{a\bigg{(}r^{3}+9r-8\alpha+6r^{2}\sqrt{1-\frac{8\alpha}{r^{3}}}\,\bigg{)}}{r^{3}-9r+8\alpha}\,,$
$\displaystyle
1+\frac{a^{2}-l^{2}}{r^{2}}+\frac{4(l-a)^{2}}{r^{3}\sqrt{1-\frac{8\alpha}{r^{3}}}}=0\,,$
(2.11)

where and thereafter, we set $M=1$, without loss of generality. Eq. (2)
provides the equation for the LRs,

$1+\frac{16a^{2}\bigg{(}r^{3}-8\alpha+3r^{2}\sqrt{1-\frac{8\alpha}{r^{3}}}\,\bigg{)}^{2}}{r^{3}\big{(}r^{3}-9r-8\alpha\big{)}^{2}\bigg{(}1+\sqrt{1-\frac{8\alpha}{r^{3}}}\,\bigg{)}}+\frac{a^{2}}{r^{2}}-\frac{a^{2}\bigg{(}r^{3}+9r-8\alpha+6r^{2}\sqrt{1-\frac{8\alpha}{r^{3}}}\,\bigg{)}^{2}}{r^{2}(r^{3}-9r-8\alpha)^{2}}=0\,.$
(2.12)

Figure 2: LRs as functions of $\alpha$, under different spin parameters. The
red and orange colors denote the prograde orbit and retrograde orbit,
respectively.

As Eq. (2.12) is very complicated, we solve the LRs numerically. There are two
real roots, $r_{p}$, $r_{m}$, corresponding to prograde and retrograde photon
orbits respectively. We have confirmed that these LRs are unstable under
radial perturbations, that is, $\partial^{2}_{r}V_{eff}>0$. Consequently,
similar to the Kerr spacetime, the LRs in the quantum-corrected Kerr solution
imply a $shadow$ region in the image of such black hole.

Fig. 2 illustrates how the radii of LRs change as the coupling constant
$\alpha$ increases, while the spin parameter remains fixed at different
values. It’s worth noting that although Eq. (2.12) has solutions throughout
the range of $-1\leq\alpha\leq 8$, only the colored parts of the curves
correspond to the actual black hole solution, i.e., where the event horizon
encloses the curvature singularities. The gray portions of the curves
correspond either to cases where the event horizon is complex or where the
singularity crosses the event horizon. For the case of $a=0.9$ (bottom right),
there are two distinct intervals for black hole solutions:
$-0.0649\leq\alpha\leq 0.3487$ and $2.25\leq\alpha\leq 6.784$, consistent with
the domain of black hole solutions found in [1]. In the case of $a=1$ (bottom
left), the black hole touches the Kerr bound, and a black hole still exists
for $\alpha=0$ (red dots) and $4.5698\leq\alpha\leq 6.47976$. Anyway, it is
apparent that both $r_{p}$ and $r_{m}$ increase with $\alpha$. This implies
that a larger value of $\alpha$ could potentially result in a larger shadow in
the black hole image. This finding aligns with previous research studies [5,
6, 34].

##  3 Shadows cast by a celestial light source

It is known that the LRs suggest the existence of a family of unstable photon
orbits around the black hole, which significantly affect the observational
characteristics of the black hole. Photons launched near these orbits complete
multiple loops around the black hole before reaching the observer, leading to
the formation of a critical curve in the observer’s screen [35]. To
investigate these characteristics of the quantum-corrected Kerr black hole, we
utilize a celestial sphere as the light source to illuminate the black hole
[36]. The black hole is located at the origin, whose size is much smaller than
that of the celestial sphere and the distance between the observer and the
origin. This setup allows the celestial sphere to outline the critical curve
accurately and reveal the unstable photon orbits. When illuminated by the
celestial sphere, any spacetime information that lies behind the LRs remains
invisible on the screen. Light rays that penetrate the interior of the
prograde LR $r_{p}$ are inevitably captured by the black hole, creating a
shadow region on the screen.

Once the celestial sphere model is established, we utilize the backward ray-
tracing method to generate the black hole images. The numerical strategy
involves setting up a camera model at the observer and integrating the
equations of motion along the null geodesics moving backward from the
observer. More specifically, we employ a fisheye camera that incorporates the
stereographic projection of the momentum $p_{\mu}$ of photons onto the screen.
With the values of $p_{\mu}$ on the screen determined, we may now proceed with
solving the trajectories of photons in the metric Eq. (2.3) by performing
backward integration of the Hamiltonian equations,

$\displaystyle\frac{\partial\mathcal{H}}{\partial
p_{\mu}}=\dot{x}^{\mu}\,,\quad\frac{\partial\mathcal{H}}{\partial
x^{\mu}}=-\dot{p}_{\mu}\,,$ (3.1)

where the dot denotes the derivative concerning the affine parameter. During
the ray-tracing process, light rays that reach the celestial sphere are
colored based on their positions. Rays that reach the horizon are colored
black, creating a shadow region on the screen.

Figure 3: Black hole images of $a=0.99$ with $\alpha\geq 0$. Left: $\alpha=0$,
which corresponds to a near extreme Kerr black hole. Middle: $\alpha=0.02$.
Right: $\alpha=6.2$. The observer is placed at $r_{o}=300$,
$\theta_{o}=\pi/2$. Figure 4: Left: shadow curves of black holes with $a=0.99$
and $\alpha\geq 0$. Right: the variation of the area ratio $\eta$ with respect
to postive $\alpha$ for different black hole spin $a$. The incline angle of
the observer is fixed at $\theta_{o}=\pi/2$. Figure 5: Black hole images of
$a=1.06$ and $\alpha=6.275$, which violates the Kerr bound $a\leq 1$. The
observer is placed at $r_{o}=300$. And the incline angles of the observer are
$\theta_{o}=\pi/9,\pi/3,\pi/2$, respectively.

Let us first consider the case of $\alpha\geq 0$. Fig. 3 displays black hole
images for three different positive values of $\alpha$, with an identical
black hole spin $a=0.99$ for an edge-on observer ($\theta_{o}=\pi/2$). It is
evident that the black hole shadow increases in size due to the impact of a
positive $\alpha$. As $\alpha$ becomes sufficiently large, the shadow shape
transforms into an ellipse as demonstrated in the right panel of Fig. 3.
Moreover, regarding the case of large $\alpha$, the lensed images surrounding
the shadow become narrower, indicating a different gravitational lensing
process compared to that in Kerr spacetime. To quantify the impact of quantum
corrections on the shadow, we introduce a parameter $\eta\equiv
S_{\text{BH}}/S_{\text{Kerr}}$, representing the area ratio between the shadow
of a quantum-corrected black hole and that of a Kerr black hole with identical
spin. The right panel of Fig. 4 shows the variation of $\eta$ concerning
$\alpha$ under different black hole spins. The area ratio $\eta$ progressively
increases as the quantum-corrected parameter grows, with a more substantial
impact on higher-spin black holes. This finding aligns with the behavior of
the LRs, whose radii are monotonically increasing functions of $\alpha$.
Furthermore, as illustrated in Fig. 5, we present an example of a violation of
the Kerr bound, where $a=1.06$ and $\alpha=6.275$, which is close to the
maximum value found in [1]. However, we do not observe a remarkable feature
compared to the right panel of Fig. 3, where $a=0.99$ and $\alpha=6.2$. It is
possible that the quantum corrections suppress the high-spin effect, even when
it surpasses the Kerr bound.

Figure 6: Black hole shadows of $a=0.8$ with $\alpha\leq 0$. Left: $\alpha=0$,
which corresponds to a Kerr black hole. Middle: $\alpha=-0.09$. Right:
$\alpha=-0.15$. The observer is placed at $r_{o}=300$, $\theta_{o}=\pi/2$.
Figure 7: Left: shadow curves of black holes with $a=0.8$ and $\alpha\leq 0$.
Right: the variation of the area ratio $\eta$ with respect to negative
$\alpha$. The black hole spin is fixed at $a=0.8$ and the incline angle of the
observer is fixed at $\theta_{o}=\pi/2$.

Next, we proceed to the case of $\alpha\leq 0$. For clarity, we present the
black hole shadows for three different values of $\alpha$, but with the same
spin $a=0.8$ for an edge-on observer. The results are shown in Fig. 6. Here,
we choose a smaller spin than that of the case with $\alpha\geq 0$, since the
high spin limits the range of values for $\alpha$, as shown in Fig. 2 of [1].
As $\alpha$ decreases, the left-hand side of the shadow curve exhibits a
concave shape, as seen in the right panel of Fig. 6 and the left panel of Fig.
7. Due to such effect, the shadow area may decrease as $\alpha$ decreases. To
verify this, we plot the variation of the area ratio $\eta$ concerning
$\alpha$ in this case in Fig. 7. The range of $\alpha$ follows the domain of
existence for a black hole. By combining Fig. 7 with Fig. 4, we conclude that
the area ratio $\eta$ is a monotonically increasing function concerning the
quantum-corrected parameter $\alpha$, similar to the radii of LRs.

Moreover, in the case of a nearly extreme Kerr black hole, an edge-on observer
can observe a vertical line segment within the left contour of the shadow, as
depicted in the left panel of Fig.3. This vertical line, known as the NHEKline
[37], originates from photons emitted in the near-horizon-extreme-Kerr (NHEK)
geometry of the Kerr spacetime [38, 37, 39]. The existence of the NHEK
geometry is attributed to the degeneracy of the inner and outer horizons [38].
However, the presence of the quantum-corrected parameter $\alpha$ breaks the
global degeneracy of the horizons, thereby destroying the NHEK geometry. As
demonstrated in the left panel of Fig.4, NHEKlines can be blurred by even a
small value of $\alpha$.

##  4 Black hole images with a thin accretion disk

The millimeter-wave images of supermassive black holes are typically thought
to be dominated by accretion disks [11, 12, 13]. In this section, we use the
disk model employed in [28] to generate images of the black hole-disk system.
The disk is geometrically and optically thin, placed on the equatorial plane.
For simplicity, we determine the disk structure in the Boyer-Lindquist
coordinates Eq. (2.7). The accretion flows are divided into two parts by the
innermost stable circular orbit (ISCO), whose radius is determined by solving
the following equations:

$\displaystyle
V_{eff}|_{r=r_{\text{ISCO}}}=0\,,\quad\partial_{r}V_{eff}|_{r=r_{\text{ISCO}}}=0\,,\quad\partial^{2}_{r}V_{eff}|_{r=r_{\text{ISCO}}}=0\,,$
(4.1)

where $V_{eff}$ is given by Eq. (2.8) with $m=1$. The above equations are
quite complicated, and the radius of the ISCO can only be solved numerically.
Outside the ISCO, the accretion flows move along timelike circular orbits,

$u^{\mu}=u^{t}_{\text{out}}\mathopen{}\mathclose{{}\left(1\,,0\,,0\,,\Omega_{s}}\right)\,,$
(4.2)

where

$\displaystyle
u^{t}_{\text{out}}=\sqrt{-\frac{1}{g_{\phi\phi}\Omega_{s}^{2}+2g_{t\phi}\Omega_{s}+g_{tt}}}\,\,\Bigg{|}_{\theta=\pi/2}\,,\quad\Omega_{s}=\frac{-\partial_{r}g_{t\phi}+\sqrt{\mathopen{}\mathclose{{}\left(\partial_{r}g_{t\phi}}\right)^{2}-\partial_{r}g_{\phi\phi}\partial_{r}g_{tt}}}{\partial_{r}g_{\phi\phi}}\,\,\Bigg{|}_{\theta=\pi/2}\,.$
(4.3)

Inside the ISCO, the fluid falls from the ISCO to the event horizon, with the
same conserved quantities as that for the ISCO. The components of the four-
velocity take the form:

$\displaystyle
u^{t}_{\text{in}}=(-g^{tt}E_{\text{ISCO}}+g^{t\phi}L_{\text{ISCO}})\,\,\big{|}_{\theta=\pi/2}\,,\quad
u^{\phi}_{\text{in}}=(-g^{t\phi}E_{\text{ISCO}}+g^{\phi\phi}L_{\text{ISCO}})\,\,\big{|}_{\theta=\pi/2}\,,$
$\displaystyle
u^{r}_{\text{in}}=-\sqrt{-\frac{g_{tt}u^{t}_{\text{in}}u^{t}_{\text{in}}+2g_{t\phi}u^{t}_{\text{in}}u^{\phi}_{\text{in}}+g_{\phi\phi}u^{\phi}_{\text{in}}u^{\phi}_{\text{in}}+1}{g_{rr}}}\,\,\Bigg{|}_{\theta=\pi/2}\,,\quad
u^{\theta}_{\text{in}}=0\,,$ (4.4)

where $E_{\text{ISCO}}$ and $L_{\text{ISCO}}$ is the energy and angular
momentum at the ISCO.

Light rays propagating around the black hole can undergo multiple crossings
with the disk, leading to the accumulation of intensity. The expression for
the observed intensity can be formulated as follows:

$I_{\nu_{o}}=\sum_{n=1}^{N_{max}}g^{3}_{n}J_{n}\,,$ (4.5)

where $\nu_{o}$ is the observed frequency on the screen, $n=1...N_{max}$ is
the number of times that the ray crosses the disk and $N_{max}$ is the maximal
crossing number, $g_{n}$ is the corresponding redshift factor, and $J_{n}$
denotes the emissivity at the $n$-th intersection point. The emissivity is
chosen as a function of $r$,

$J=\text{exp}\mathopen{}\mathclose{{}\left(-\frac{1}{2}z^{2}-2z}\right)\,,\quad
z=\text{log}\frac{r}{r_{H}}\,,$ (4.6)

where $r_{H}$ is the horizon radius on the equatorial plane.

Figure 8: Black hole images illuminated by the thin disk model. The spin is
set to be $a=0.99$, and the observer is positioned at $r_{o}=200$,
$\theta_{o}=80^{\circ}$. The top row corresponds to a typical Kerr black hole
with $\alpha=0$, while the bottom row represents the quantum-corrected Kerr
black hole with $\alpha=0.02$. In the left column, we display the intensity
maps generated by the thin disk. The middle column shows the number of times
the light rays intersect the disk. Finally, the right column exhibits the
intensity maps generated solely by the inner-ISCO region of the disk.

The left column of Figure 8 illustrates the intensity maps obtained from the
thin disk imaging. In this case, the spin parameter is fixed at $a=0.99$, and
the observer is situated at $r_{o}=200$, $\theta_{o}=80^{\circ}$. Within these
intensity maps, the bright, narrow curves are referred to as “photon rings”,
which emerge due to the strong gravitational lensing effect. The shapes of
these photon rings align with the shadows cast by the celestial light source
mentioned earlier. Moreover, the completely dark regions central in the images
result from the direct imaging of horizons, known as the “inner shadows” [40].

When comparing the two images in the left column of Figure 8, the most
prominent distinction is that the quantum-corrected black hole exhibits a
smaller inner shadow. To make a further investigation, we plot the maximal
number of times light rays crossed the disk, $N_{max}$, as depicted in the
middle column of Figure 8. Two notable differences are observed: First, the
area occupied by light rays that do not intersect the disk is smaller in the
image of a quantum-corrected black hole. As light rays accumulate intensity
while passing through the disk, the inner shadow of a quantum-corrected black
hole naturally appears diminished.

Second, for the quantum-corrected black hole, the regions corresponding to
$N_{max}=3$ and $N_{max}=4$ have extended “tentacles” outward, which suggests
that light rays are more likely to oscillate in their $\theta$-direction near
the equatorial plane of the quantum-corrected black hole. We have analyzed the
trajectories of light rays associated with the tentacles and observed that
they predominantly intersect the accretion disk between the ISCO and the
horizon. This suggests that the non-circular spacetime effects resulting from
quantum corrections are more pronounced in the vicinity of the horizon. In
other words, the contribution of a near-horizon light source plays a crucial
role in the observable effects of a quantum-corrected black hole. To verify
this, we present the imaging results in the right column of Fig. 8, where only
the light source inside the ISCO is present. It is evident that the image of
the quantum-corrected black hole differs significantly from that of a typical
Kerr black hole. We would like to stress that though the conventional disk
model considers the ISCO as its inner boundary, there are also studies
indicating that the accretion flow within the ISCO exhibits nontrivial
dynamics and significant radiative capabilities [41]. Therefore, we may expect
the presence of observable effects of quantum corrections in realistic
accretion systems of black holes 111It is worth noting that we have employed a
phenomenological formula (4.6) for the emissivity when generating the images.
Realistic millimeter-wave emissions are dominated by thermal or non-thermal
synchrotron radiations, which are determined by the thermodynamics of the
accretion flow..

##  5 Summary and discussion

In this work, we studied the LRs, shadows and disk images of the Kerr black
hole in the semiclassical gravity with type-A trace anomaly [1]. Considering
that the mass function ${\mathcal{M}}$ depends both on $r$ and $\theta$, the
spacetime should be non-circular. Consequently, the Hamiltonian for geodesics
does not allow for the separation of variables. To determine the radii of LRs,
we performed numerical calculations based on the effective potential of
photons. Our results demonstrated that the radius of the retrograde LR are
always larger than that of the prograde LR, regardless of the value of the
coupling constant.

Furthermore, we employed a celestial sphere to illuminate the black hole and
utilized the backward ray-tracing technique to determine the shape and area of
the shadow on the observer’s screen. To characterize the variations in the
shadow’s area, we introduced the parameter
$\eta=S_{\text{BH}}/S_{\text{Kerr}}$, representing the ratio of the shadow’s
area for a quantum-corrected black hole to that of a Kerr black hole with the
same spin. Our findings revealed that as the coupling constant gradually
increased within its range of values, $\eta$ also increased. Moreover, an
intriguing discovery was that the NHEKline was highly susceptible to
disruption by the coupling constant. This phenomenon may arise from the fact
that the presence of the trace anomaly breaks the degeneracy between the inner
and outer horizons, resulting in the absence of the NHEK geometry.

After that, we employed a thin disk model to study the black hole images in a
more realistic manner. Our findings revealed that the images of the quantum-
corrected black hole could deviate from those of the typical Kerr black hole.
The quantum correction leads to a higher frequency of light ray oscillations
around the equatorial plane, specifically between the ISCO and the horizon. As
a result, the influence of a source near the horizon becomes crucial in
determining the observable effects of a quantum-corrected black hole.

## Acknowledgments

We thank Prof. Bin Chen for the valuable discussions. The work is partly
supported by NSFC Grant No. 12275004, 12205013 and 11873044. MG is also
endorsed by ”the Fundamental Research Funds for the Central Universities” with
Grant No. 2021NTST13.

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