# Shadows and photon rings of a spherically accreting Kehagias-Sfetsos black
hole

Mohaddese Heydari-Fard1 , Malihe Heydari-Fard2 and Nematollah Riazi1
_1 Department of Physics, Shahid Beheshti University, Evin 19839, Tehran,
Iran_
_2 Department of Physics, The University of Qom, 3716146611, Qom, Iran_
Electronic address<EMAIL_ADDRESS>address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

By considering Kehagias-Sfetsos black hole in the framework of the Hořava-
Lifshitz gravity, we study the optical appearance of such black holes
surrounded by spherical accretion flow. For the static/infalling spherical
accretion flow, we compute the observed specific intensity as a function of
impact parameter. We also investigate the effect of the Hořava parameter and
accreting matter on the luminosity of shadows and photon rings. It is found
that an increase in the Hořava parameter decreases the shadow size, while the
shadows and photon rings luminosities increase. Moreover, we constrain the
Hořava parameter from the observational data reported by the Event Horizon
Telescope for M87* and Sgr A*.
Keywords: Black hole shadow, Spherical accretion, Modified theories of gravity

##  1 Introduction

The Event Horizon Telescope (EHT) collaboration released the first image of a
black hole shadow [1]–[7]. The Image is formed by null geodesics in the strong
gravity regime. The photons with low angular momentum fall into the black hole
and form a dark area for a distant observer, while photons with large angular
momentum coming from infinity will be deflected by the gravitational potential
of the black hole. However, the photons with critical value of angular
momentum revolve around the black hole an infinite time and surround the dark
interior, which are called the photon ring and the black hole shadow,
respectively. In a seminal work, Synge calculated the angular radius of the
shadow of Schwarzschild black hole [8]. Then, Bardeen studied the shadow of
Kerr black hole and argued that the black hole angular momentum causes the
deformation of its shadow [9].

By modelling M87* with the Kerr geometry in general relativity (GR), the
observation was found to be in agreement with the predictions of GR. However,
due to the EHT systematic uncertainties it is still possible to test the
alternative theories of gravity by simulating the black hole image and
observing deviations from the Kerr solution. To this end, one can explore the
distortion in the black hole image which contains valuable information about
the structure of space-time around a black hole solution. This motivated many
authors over the recent years to study the black hole shadow in the context of
modified theories of gravity [10]–[23].

On the other hand, the astrophysical black holes are expected to be surrounded
by sources of the luminous accretion material which makes it possible to
investigate the observational appearance of the black hole from the accretion
flow. Indeed, before the discovery of the black hole shadow by EHT, the
possible observational characteristics of the black hole shadow by considering
different accretion flows were studied. Luminet was the first to investigate
the optical properties of the Schwarzschild black hole in 1979, and
constructed the simulated shadow image of the Schwarzschild black hole
surrounded by an emitting thin accretion disk [24]. The simulated image
obtained by Luminet is remarkably similar to the black hole shadow image
captured by EHT [1]. He found that the emergence of the shadow and ring
depends on the position and profile of the accretion flow and the inner edge
of disk can have a remarkable signature in the image. Thereafter, Falcke et
al. by considering the radiation of a hot optically thin accretion flow around
a supermassive black hole in the center of our galaxy, created a ray-tracing
code to obtain the images of Sgr A*, and showed that the black hole shadow is
equivalent to the gravitational lensing effect [25]. For a geometrically thick
and optically thin accretion disk, the gravitational lensing and the shadow of
the Schwarzschild black hole was studied by Cunha et al. [26]. Gralla et al.
by considering the Schwarzschild black hole with both thin and thick accretion
disks, investigated the trajectory of light rays and ring that surrounds the
black hole shadow. It was found that the shadow size depends on the position
of the accretion disk as well as the emission profile of the model [27].
However, when the Schwarzschild black hole is surrounded by spherically
symmetric accretion flow, Narayan et al. showed that the location of the outer
edge of the shadow is independent of the inner radius at which the accreting
gas stops radiating [28]. Therefore, the size of the shadow depends on the
space-time geometry and is not affected by the details of the accretion flow.
Also, the optical appearance of black holes surrounded by various accretions
in modified gravity theories, have been extensively studied in [29]–[50].

Amongst the many modifications of GR that have been suggested is the Hořava-
Lifshitz gravity which is motivated by the need to include the quantum effects
in the low-mass limit. The theory is a renormalizable four-dimensional theory
of gravity which reduces to the Einstein’s gravity with non-vanishing
cosmological constant in the IR limit but with improved UV behavior. A class
of static and spherically symmetric black hole solutions of the theory with a
cosmological constant was obtained in [51]. Among them, the Ads type solution
has an asymptotic behavior which differs from the Schwarzschild-Ads solution
in GR; namely in the IR limit the theory of GR is not always recovered.
However, in the context of the modified Hořava model, a static spherically
symmetric solution with asymptotically flat behavior, which is a counterpart
of the Schwarzschild black hole in GR, has been obtained by Kehagia and
Sfetsos [52]. This solution is usually known as the KS black hole. Then, in
the slow rotation approximation, the black hole solution in the IR regime has
been obtained in [53]–[54]. In the literature, many physical aspects of KS
black hole have already been studied [55]–[66]. Moreover, for cosmological
implications of Hořava-Lifshitz gravity see for instance Refs.[67]–[79].

The shadows and rings of the KS black hole surrounded by a thin accretion disk
have been studied in [80]. However, the optical appearance of the KS black
hole surrounded by spherical accretion flow has not yet been studied. So, in
the present work, we consider the KS black hole surrounded with
static/infalling accretion flows and discuss the effects of Hořava parameter
and spherical accretion on the observed appearance of the black hole.

The paper is organized as follows. In section 2, after a brief review of KS
black holes, we discuss the photon trajectories in the space-time of such
black holes and investigate the effects of the Hořava parameter on them. Then
we present the shadow images of KS black hole with static and infalling
spherical accretion flows in section 3 and section 4, respectively. The paper
ends with concluding remarks in section 5.

##  2 KS black holes and trajectories of surrounding photons

### A. KS geometry

In the ADM formalism of Hořava-Lifshitz gravity the four-dimensional metric is
parameterized as [67]

$ds^{2}=-N^{2}c^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (1)

where $N$, $N_{i}$ and $g_{ij}$ are the lapse function, the shift function and
three-dimensional spatial metric, respectively.

The action of the IR-modified Hořava gravity is

$\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle\int
dtdx^{3}\sqrt{g}N\left[\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})-\frac{\kappa^{2}}{2\nu^{4}}C_{ij}C^{ij}+\frac{\kappa^{2}\mu}{2\nu^{2}}\epsilon^{ijk}R^{(3)}_{il}\nabla_{j}R^{(3)l}_{k}\right.$
(2) $\displaystyle-$
$\displaystyle\left.\frac{\kappa^{2}\mu^{2}}{8}R^{(3)}_{ij}R^{(3)ij}+\frac{\kappa^{2}\mu^{2}}{8(3\lambda-1)}\left(\frac{4\lambda-1}{4}(R^{(3)})^{2}-\Lambda_{W}R^{(3)}+3\Lambda_{W}^{2}\right)+\frac{\kappa^{2}\mu^{2}\tilde{\omega}}{8(3\lambda-1)}R^{(3)}\right]$

where $\mu$, $\nu$, $\lambda$, $\kappa$, $\tilde{\omega}$ and $\Lambda_{W}$
are constant parameters. $R^{(3)}$ is the three-dimensional curvature scalar
for $g_{ij}$ and the extrinsic curvature, $K_{ij}$, is given by

$K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (3)

where the dot represents a derivative with respect to $t$ and $\nabla_{i}$
denotes the covariant derivative with respect to the spatial metric $g_{ij}$.
$C^{ij}$ is the Cotton tensor, reads as

$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{(3)j}_{l}-\frac{1}{4}R^{(3)}\delta^{j}_{l}\right).$
(4)

Now, we consider the static and spherically symmetric metric as

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2},$ (5)

where $g_{tt}(r)$ and $g_{rr}(r)$ are functions of the radial coordinate $r$.
In the specific case of $\lambda=1$, which reduces to the Einstein-Hilbert
action in IR limit, the solution of the vacuum field equations can be obtained
as follows

$-g_{tt}(r)=1/g_{rr}(r)=f(r)=1+(\tilde{\omega}-\Lambda_{W})r^{2}-\sqrt{r[\tilde{\omega}(\tilde{\omega}-2\Lambda_{W})r^{3}+\beta]},$
(6)

where $\beta$ is an integration constant. By considering
$\beta=4\tilde{\omega}M$ and $\Lambda_{W}=0$, the KS asymptotically flat
solution is given by [52]

$f(r)=1+\tilde{\omega}r^{2}\left[1-\left(1+\frac{4M}{\tilde{\omega}r^{3}}\right)^{1/2}\right],$
(7)

where $M$ is the mass of the black hole and $\tilde{\omega}$ is the Hořava-
Lifshitz parameter. By rearranging the parameter $\tilde{\omega}$ as
$\tilde{\omega}=\frac{1}{2\omega^{2}}$, one can rewrite the metric function in
the following form [80]

$f(r)=1+\frac{r^{2}}{2\omega^{2}}\left(1-\sqrt{1+\frac{8M\omega^{2}}{r^{3}}}\right).$
(8)

The radius of the outer and inner horizons can be found by solving $f(r)=0$

$r_{\pm}=M[1\pm\sqrt{1-\omega^{2}/M^{2}}].$ (9)

As is clear, for the existence of black hole solution a constraint
$\omega/M\leq 1$ should be imposed and for the extremal black hole,
$r_{+}=r_{-}$, which corresponds to the case $\omega/M=1$. Also, in the limit
of $\omega\rightarrow 0$ the above metric becomes the static solution in GR
which is described by the Schwarzschild metric. Note the parameter $\omega$
always takes the positive values. Thus, for the range $0<\omega<1$, the
behavior of the inner horizon $r_{-}$ and the event horizon $r_{+}$, is
plotted in the left panel of Fig. 1. The points at the beginning of each curve
denote the corresponding values to the Schwarzschild solution.

### B. Null trajectory around KS black hole

The trajectory of null geodesics in the space-time of KS black hole can be
obtained using the Euler-Lagrange equations. Without loss of generality, we
restrict ourselves to the equatorial plane, $\theta=\frac{\pi}{2}$. Since the
metric coefficients of KS black hole are independent of $t$ and $\varphi$
coordinates, there are two constants of motion correspond to the energy $E$
and angular momentum $L$ of photons

$E=f(r)\dot{t},$ (10) $L=r^{2}\dot{\varphi}.$ (11)

Now, taking the condition $2{\cal L}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$
for null geodesics and using above equations, we find the equations of photon
motion around a KS black hole as

$\dot{t}=\frac{E}{f(r)},$ (12) $\dot{\varphi}=\frac{L}{r^{2}},$ (13)
$\dot{r}^{2}=E^{2}-V_{\rm eff}(r),$ (14)

where the effective potential is as follows

$V_{\rm
eff}(r)=\frac{L^{2}}{r^{2}}f(r)=\frac{L^{2}}{r^{2}}\left[1+\frac{r^{2}}{2\omega^{2}}\left(1-\sqrt{1+\frac{8M\omega^{2}}{r^{3}}}\right)\right].$
(15)

We have displayed the effective potential for different values of the Hořava
parameter, $\omega$, in the right panel of Fig. 1. As one can see, the peak of
the potential increases with $\omega$. Note that for non-radial geodesics, it
is convenient to set $L=1$ and thus we plot the figure for this value of the
angular momentum.

Next, we focus on the photon motion in the vicinity of KS black hole.
Combining equations (13) and (14) the differential equation governing the
light rays trajectory can be obtained as

$\left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{b^{2}}-r^{2}f(r)$ (16)

where the impact parameter is defined as $b\equiv\frac{L}{E}$. In particular,
for the photons with critical value of the impact parameter, $b=b_{\rm ph}$,
an unstable circular orbit occurs at the maxima of the effective potential at
$r=r_{\rm ph}$, known as the photon sphere [81]. To study circular orbits with
constant radius $r=r_{\rm ph}$ from equation (14) we have

$V_{\rm eff}(r_{\rm ph})=E_{\rm ph}^{2},\hskip 14.22636ptV^{{}^{\prime}}_{\rm
eff}(r_{\rm ph})=0,$ (17)

where prime denotes differentiation with respect to the radial coordinate $r$.
Use of equation (15) leads to the following relation

$rf^{\prime}(r)-2f(r)=0$ (18)

which gives the radius of unstable photon circular orbits as

$r_{\rm
ph}=2\sqrt{3}M\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{-4\omega^{2}}{3\sqrt{3}M^{2}}\right)\right].$
(19)

The dependence of the radius of the photon sphere $r_{\rm ph}$ on the
parameter $\omega$ is also plotted in the right panel of Fig. 1, showing that
$r_{\rm[h}$ is a decreasing function of the Hořava parameter. Moreover, we see
that in the limiting case $\omega\rightarrow 0$, $r_{\rm ph}=3M$ which is the
radius of the unstable circular photon orbit for the Schwarzschild black hole.
The impact parameter of the photon sphere is also given by

$b_{\rm ph}=\frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})}}=\left(\frac{1}{r_{\rm
ph}^{2}}+\frac{1-\sqrt{1+\frac{8M\omega^{2}}{r_{\rm
ph}^{3}}}}{2\omega^{2}}\right)^{-\frac{1}{2}},$ (20)

which for an asymptotically flat space-time with a metric in the form (5) is
equal to the radius of the black hole shadow. The results of the radii of the
event horizon $r_{+}$, photon sphere $r_{\rm ph}$ as well as the impact
parameter of the photon sphere $b_{\rm ph}$ are also presented in Table. 1.

Figure 1: Left panel: The photon radii $r_{\rm ph}$, $r_{-}$ and $r_{+}$ as a function of Hořava parameter for $M=1$. Right panel: The effective potential $V_{\rm eff}$ as a function of radius for different values of Hořava parameter. Table 1: The values of the event horizon radius $r_{+}$, photon radius, $r_{\rm ph}$ and impact parameter, $b_{\rm ph}$ for different values of $\omega$. The first column with $\omega\rightarrow 0$ corresponds to the Schwarzschild black hole. | $\omega\rightarrow 0$ | $\omega=0.1$ | $\omega=0.3$ | $\omega=0.5$ | $\omega=0.7$ | $\omega=0.9$
---|---|---|---|---|---|---
$r_{\rm ph}/M$ | 3 | 2.9956 | 2.9592 | 2.8820 | 2.7524 | 2.5393
$b_{\rm ph}/M$ | 5.1962 | 5.1923 | 5.1609 | 5.0951 | 4.9868 | 4.9117
$r_{+}/M$ | 2 | 1.9949 | 1.9539 | 1.8660 | 1.7141 | 1.4359
$r_{-}/M$ | 0 | 0.0050 | 0.0461 | 0.1339 | 0.2859 | 0.5641

From equation (16), it is clear that the trajectory of light rays depends on
the impact parameter $b$. The photons with small angular momentum, $b<b_{\rm
ph}$, finally enter the black hole singularity, while photons with large
angular momentum, $b>b_{\rm ph}$, will be deflected and pushed back to the
distant observer. For critical values of impact parameter $b=b_{\rm ph}$, the
photons swirl around the black hole on the photon sphere, with an infinite
time. These photon trajectories are shown with green, gray and red curves in
Fig. 2, respectively. The black disk also shows the event horizon surface. We
have plotted the light trajectories for $\omega=0.1$ and $\omega=0.9$. As one
can see, the event horizon radius and the radius of the photon sphere are
smaller for a larger $\omega$. This is because the larger values of $\omega$
weakens the strength of gravity so that the instability area around KS black
hole decreases and thus the photon radius takes smaller values.

Figure 2: The trajectory of light rays around a KS black hole with
$\omega=0.1$ (left panel) and $\omega=0.9$ (right panel). The red, gray and
green curves correspond to the trajectory of light rays with $b=b_{\rm ph}$,
$b>b_{\rm ph}$ and $b<b_{\rm ph}$, respectively.

Now, we aim to investigate the constraints on the Hořava parameter using the
EHT observations of the shadow of M87* and Sgr A*. As one can see from
equation (20), the shadow size is dependent on the Hořava parameter and thus
from EHT observational data we can impose bounds on it. For a distant
observer, the angular diameter $\Omega$ of the black hole shadow is given by
[82]

$\Omega=\frac{2b_{\rm ph}}{D},$ (21)

where $D$ is the distance between the black hole and distant observer, and
$b_{\rm ph}$ is obtained from equation (20). The above equation can be
rewritten as

$\left(\frac{\Omega}{\rm\mu as}\right)=\left(\frac{6.191165\times
10^{-8}}{\pi}\frac{\gamma}{D/\rm Mpc}\right)\left(\frac{b_{\rm
ph}}{M}\right),$ (22)

with $\gamma$ is the mass ratio of the black hole to the Sun. From the reports
released by EHT for the shadow of M87* the distance and mass correspond to
$D=16.8$ Mpc and $\gamma=6.5\times 10^{9}$, while for the shadow of Sgr A*
these values are given by $D=8.27$ kpc and $\gamma=4.3\times 10^{6}$,
respectively [1]–[7]. In Fig. 3, we have plotted the diameter of the shadow of
KS black hole as a function of Hořava parameter, using the data of M87* (red
curve) and of Sgr A* (black curve). It is found that the $\Omega$ decreases
with increase of $\omega$. The cyan and yellow regions are also the shadow
diameters of M87* and Sgr A* reported by the EHT observations. As we observe,
the KS black hole is able to describe the shadow size of Sgr A*, provided that
the Hořava parameter is constrained to the values in range $0<\omega<0.8695$.
We also constrain the parameter $\omega$ of the KS black hole as
$0<\omega<0.4504$ for M87*.

Figure 3: Shadow diameter of the KS black hole as a function of Hořava
parameter, $\omega$. The yellow and cyan regions are the experimental data of
Sgr A* ($51.8\pm 2.3$ $\rm\mu as$) and M87* ($42\pm 3$ $\rm\mu as$) as
reported by the EHT, respectively.

##  3 KS black hole with gas at rest

Now, we are going to study the optical appearance of KS black hole surrounded
by spherical accretion flow, which is assumed to be optically thin. We
consider two cases of accretion: a static spherical accretion model and an
infalling spherical accretion.

First, we discuss the shadow image and photon sphere in the background of KS
black hole surrounded by static spherical accretion flow. By integrating
specific emissivity along the photon path $\gamma$, one can obtain the
observed intensity of photons for a distant observer as follows [83]–[84]

$I(\nu_{\rm o})=\int_{\gamma}g^{3}j(\nu_{\rm e})dl_{p},$ (23)

where $g\equiv\frac{\nu_{\rm o}}{\nu_{\rm e}}$ is gravity redshift factor,
$\nu_{\rm e}$ is the radiated photon frequency, $j(\nu_{\rm e})$ is the
emissivity per unit volume in the rest frame of the emitter, and $dl_{\rm p}$
is the infinitesimal proper length. For spherically symmetric space-time (5)
the redshift factor is $g=f(r)^{1/2}$. By considering monochromatic emission
with rest-frame frequency $\nu_{\rm s}$ and the emission radial profile as
$1/r^{2}$ [84], the specific emissivity is given by

$j(\nu_{\rm e})\propto\frac{\delta(\nu_{\rm e}-\nu_{\rm s})}{r^{2}}.$ (24)

Also, the proper length measured in the rest frame of the emitter for the KS
black hole is

$dl_{\rm
p}=\sqrt{\frac{1}{f(r)}dr^{2}+r^{2}d\varphi^{2}}=\sqrt{\frac{1}{f(r)}+r^{2}\left(\frac{d\varphi}{dr}\right)^{2}}dr.$
(25)

Then, using equations (23)-(25) and equation (16), the photon intensity
observed by a distant observer can be expressed as

$I(\nu_{\rm
o})=\int_{\gamma}\frac{f(r)}{r^{2}}\sqrt{1+\frac{b^{2}f(r)}{r^{2}-b^{2}f(r)}}dr.$
(26)

By substituting $f(r)$ from equation (8), the specific intensity of KS black
hole shadow and photon rings can be calculated. The observed specific
intensity $I_{\rm o}$ is a function of both the impact parameter $b$ and the
Hořava parameter $\omega$. We plot the behavior of intensity as a function of
impact parameter for different values of $\omega$ in the left panel of Fig. 4.
As the figure shows, by increasing the Hořava parameter $\omega$ the light
intensity of the shadow increases too. According to the figure the observed
light intensity sharply increases with the increase of impact parameter and
reaches its maximum value at $b_{\rm ph}$, due to the fact that the light rays
round the black hole several times on the photon sphere, then the light
intensity gradually decreases and takes its minimum value.

Furthermore, the KS black hole shadow cast in the $(x,y)$ plane for the static
spherical accretion is plotted in the Fig. 5. In each panel, the central disk
with a faint of luminosity is the black hole shadow and the brightest ring
around the shadow shows the photon ring. As can be seen, with the increase of
the parameter $\omega$, the radius of the shadow decreases with a smaller
photon ring which is in agreement with Table. 1. Also, according to the
figure, by increasing $\omega$ the specific intensity increase since, with the
increase of $\omega$ the strength of gravitational field of KS black hole
decreases which leads to lower light rays being trapped by KS black hole and
thus in this region higher luminosity of shadow and photon rings being
observed. We also conclude that only the KS black holes with larger values of
$\omega$ have significant deviation from the Schwarzschild black hole with
$\omega=0$ in GR.

Figure 4: The intensity profile as a function of impact parameter for KS black
hole with gas at rest (left panel) and with radially infalling gas (right
panel) for different values of $\omega$ and $M=1$.

Figure 5: The shadow and photon rings for KS black hole with gas at rest with
different values of $\omega$ and $M=1$.

##  4 KS black hole with infalling gas

In this section we consider the KS black hole surrounded by radial infalling
spherical accretion flow. This model is more realistic than the static
accretion model since most of accretions are moving in reality. In this
scenario, the redshift factor is related to the velocity of accretion flow as

$g=\frac{k_{\alpha}u_{\rm o}^{\alpha}}{k_{\beta}u_{\rm e}^{\beta}},$ (27)

where $k_{\mu}$ is the photon four-velocity, $u^{\mu}_{\rm o}=(1,0,0,0)$ is
the distant observer four-velocity, and $u^{\mu}_{\rm e}$ is the four-velocity
of the infalling spherical accretion flow. Based on equations (12)-(14), we
know that $k_{t}=1/b$ is a constant and $k_{r}$ can be obtained by the
equation $k_{\mu}k^{\mu}=0$. Therefore,

$\frac{k_{r}}{k_{t}}=\pm\sqrt{\frac{1}{f(r)}\left(\frac{1}{f(r)}-\frac{b^{2}}{r^{2}}\right)},$
(28)

where the sign + (-) corresponds to the case that the photon approaches (away
from) the black hole. Also, the four-velocity of the infalling accretion is
given by

$\displaystyle u_{\rm e}^{t}$ $\displaystyle=\frac{1}{f(r)},$ (29)
$\displaystyle u_{\rm e}^{r}$ $\displaystyle=-\sqrt{1-f(r)},$ $\displaystyle
u_{\rm e}^{\theta}$ $\displaystyle=u_{\rm e}^{\varphi}=0.$

Thus, the redshift factor in equation (27) can be obtained as

$g=\frac{1}{u_{\rm e}^{t}+\left(\frac{k_{r}}{k_{t}}\right)u_{\rm e}^{r}}.$
(30)

In addition, the proper length is

$dl_{\rm p}=k_{\alpha}u^{\alpha}_{\rm e}ds=\frac{k_{t}}{g\mid k_{r}\mid}dr,$
(31)

where $s$ is the affine parameter along the photon path. Assuming that the
specific emissivity has the same form as equation (24), the specific intensity
$I(\nu_{\rm o})$ in the case of infalling spherical accretion can be expressed
as

$I(\nu_{\rm o})\propto\int\frac{g^{3}k_{t}dr}{r^{2}|k_{r}|}.$ (32)

Similar to the static model, the observed intensity for the infalling
spherical accretion as a function of impact parameter is plotted in the right
panel of Fig. 4. As can be seen, the maximum value of intensity is at $b_{\rm
ph}$ but the peak of intensity for infalling case is smaller than the static
one. The numerical results of intensity shows that the observed intensity
increases with increasing $\omega$. We also present the two-dimensional plot
of shadow cast in Fig. 6, showing that the radius of shadow and photon ring
decreases with increase of parameter $\omega$, while the shadow and photon
rings luminosities increase. Comparing Fig. 5 for the static case with Fig. 6
for the infalling model shows that the shadow region for the infalling
accretion flow is darker than that of the static case which is caused by the
Doppler effect of the infalling matter. Note that the observed shadow size
depend only on the space-time geometry, while the luminosities of both the
shadow and photon rings are also affected by the accretion flow property.

Figure 6: The shadow and photon rings for KS black hole with radially
infalling gas with different values of $\omega$ and $M=1$.

##  5 Conclusions

In this paper, we studied the optical appearance of the KS black holes
surrounded by spherical accretion flow. First, we obtained the size of the
event horizon radius, the black hole shadow and photon sphere for different
values of Hořava parameter $\omega$, and found that the larger the $\omega$
parameter is, the smaller the radius of the event horizon, photon sphere and
shadow will be. We also obtain the constraint on the parameter $\omega$ using
the observed shadow diameters measured by the EHT, which are $0<\omega<0.8695$
for Sgr A*, and $0<\omega<0.4504$ for M87*. Then, we study the observed
intensity and the luminosity of KS black hole shadows and photon rings by
assuming that the black hole is surrounded by static/infalling spherical
accretion flow. For both scenarios, we found that the luminosities of the
shadow and photon rings of KS black holes increase with the increase of Hořava
parameter. This is because by increasing $\omega$, the impact parameter
decreases which means that the photons gain more kinetic energy and thus not
easily captured by KS black hole which leads to increase the luminosity.
Therefore, it can be argued that this observational appearance may offer a way
to distinguish a KS black hole in Hořava-Lifshitz gravity from a Schwarzschild
black hole in GR.

## Acknowledgments

The work of Mohaddese Heydari-Fard is supported by the Iran National Science
Foundation (INSF) and the Research Council of Shahid Beheshti University under
research project No. 4016024.

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# Experimental observation of inter-orbital coupling

Diego Guzmán-Silva, Gabriel Cáceres-Aravena, and Rodrigo A. Vicencio
Departamento de Física and Millenium Institute for Research in Optics - MIRO,
Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Chile

(September 3, 2024)

###### Abstract

Inter-orbital coupling refers to the possibility of exciting orbital states by
otherwise orthogonal non-interacting modes, a forbidden process in photonic
lattices due to an intrinsic propagation constant detuning. In this work,
using a femtosecond laser writing technique, we experimentally demonstrate
that fundamental and excited orbital states can couple each other when located
at different spatial positions. We perform a full characterization of an
asymmetric double-well like potential and implement a scan method to
effectively map the dynamics along the propagation coordinate. Our fundamental
observation constitutes also a direct solution for a spatial mode converter
device, which could be located in any position inside a photonic glass chip.
By taking advantage of the phase structure of higher-order photonic modes and
the effective negative coupling generated, we propose a trimer configuration
as a phase beam splitter ($\pi$-BS), which could be of great relevance for
multiplexing and interference-based photonic concatenated operations.

The way in which atoms assemble and interact with each other determines their
potential to form molecules and matter r1 . These interactions are
characterized by different atom states called orbitals. In order to understand
and explore otherwise occulted properties, diverse areas of research have
proposed the creation of artificial atoms in different physical contexts r2 ;
r3 ; r4 ; r5 ; r6 ; r7 ; r8 ; r9 . This includes numerical and experimental
research suggesting new materials and the possibility of observing new exotic
interactions, as solutions for transport and localization of energy, key
scientific goals in science. However, although theoretical researchers have
assumed the basic and fundamental concept of inter-orbital interactions as
obvious and trivial r10 ; r11 , no systematic experimental evidence is found
in literature. In this Article, we demonstrate, both numerically and
experimentally, that orthogonal orbital states, located at different two-
dimensional (2D) photonic atoms, couple each other by an evanescent
interaction. We show that symmetric S-like wave functions could couple to
P-like orbitals and also to higher order states. We use a femtosecond (fs)
laser technique r12  to fabricate elliptical 2D waveguides having different
propagation constants, which is analogous to different orbital energies. We
construct double-well photonic asymmetric potentials r13 , such that the on-
site energies of orthogonal orbitals can be finely tuned and coupling between
them becomes possible. Our results offer a new way of studying lattice
dynamics where, historically, S-like orbitals have been assumed mostly. An
inter-orbital coupling gives the opportunity of tuning the sign of hopping
between atoms, increasing the available tools for researchers of different
areas and, also, amplifying the possibilities to discover new phenomena r14 ;
r15 .

The paraxial wave equation in optics and the Schrödinger equation in quantum
mechanics have exactly the same mathematical form and, therefore, their
mathematical solutions must be equivalent r2 . Theory tells us that a
potential-well structure will always have a bound state r1 ; r16 ,
independently of its size (width and depth). However, if we want to excite and
observe higher-orbital states, we must satisfy a specific condition that, for
one-dimensional (1D) structures, reads as

$8mV_{o}a^{2}\geqslant h^{2}\hskip 14.22636pt\text{and}\hskip
14.22636pt8n_{o}\Delta n\ w^{2}\geqslant\lambda^{2}\ ,$ (1)

for quantum mechanics and optics, respectively. Here $m$ is the particle mass,
$V_{o}$ and $a$ are the potential depth and width, respectively, and $h$ the
Planck constant; while $n_{o}$ corresponds to the bulk refractive index,
$\Delta n$ is the refractive index contrast, $w$ is the waveguide width, and
$\lambda$ the wavelength. We notice that both relations are equivalent and
depend on phenomenologically similar parameters: if the size of
atoms/waveguides is larger than some lower bound, higher orbitals are allowed
to exist. Specifically in optics, we notice that a larger index contrast
and/or a wider waveguide facilitates the generation of a larger number of
eigenstates and waveguides could become multimode depending on the excitation
wavelength r16 ; r17 .

Resonance is a key concept in physics and many interactions can be well
understood by finding conditions for matching the eigen-frequencies of a given
system. Of course, complex resonance problems including different types of
interactions r18  are neither trivial nor simple to be described, but the
fundamental idea continues relying on matching or tuning a set of system
parameters. Similar to what happens in any physical problem having waves and
restrictions, eigenmodes are classified by their symmetry, where the ground
state r10  is always symmetric with respect to the center of the
atom/waveguide. This state is the so-called TE00 mode in optics r19  or ground
state in quantum mechanics r1 , and has a Gaussian-like profile and larger
propagation constant $\beta_{S}$ (lower energy). A second state corresponds to
a wavefunction which has a node at the center and, as a consequence, a
different phase at different lobes. In optics this state is called TE01 or
TE10 depending on its orientation r19 , while in atom physics is usually
called P-state, having also some variations depending on spatial orientation
r20 . For simplicity we use the Hydrogen atom-like nomenclature: S and P
states for TE00 and TE01/TE10 modes, respectively.

Figure 1: a) Propagation constant tuning concept. $C_{s}$ and $C_{sp}$
correspond to coupling constants in a discrete approach, and define an
interaction dynamical scale proportional to $1/C_{s,sp}$. (b) Numerically
computed normalized maximum power at waveguide 2 versus refractive index
contrast $\Delta\bar{n}$, for two waveguides separated a distance $d=20\ \mu
m$. Insets show numerical intensity profiles at indicated regions. White,
light gray, and gray shaded areas indicate one, two, or three modes supported
at waveguide 2, respectively. A sketch of a vertically oriented dimer
configuration is included.

In order to explicitly show inter-orbital coupling, we follow a simple
theoretical idea suggested for cold atoms systems r13 , as sketched in Fig.
1(a) considering a photonic dimer configuration. Single and multimode
waveguides have, in principle, different propagation constants for different
orbital states where, for example, $\beta_{P}<\beta_{S}$ (in general,
propagation constants are always lower for higher-order modes). Therefore, by
modifying the local structure of a given waveguide, we can tune its properties
and find conditions for an optimal orbital interaction, such that orthogonal
states located at different sites have a closer - if not equal - propagation
constant. This process would imply an effective dynamical transformation from
a S-S interaction into a S-P one, what is the main goal of the present work.
We investigate this theoretical concept by, first, numerically integrating a
paraxial wave equation considering a coupler (dimer) configuration, which
consists of two vertically oriented elliptical waveguides separated by a given
distance $d$ (see sketch in Fig. 1(b) and Supplemental Material r20p5  for
details). We define a relative contrast $\Delta\bar{n}\equiv\Delta
n_{2}-\Delta n_{1}$, where $\Delta n_{1}$ and $\Delta n_{2}$ correspond to
refractive index contrasts at waveguides 1 and 2, respectively. When both
waveguides are equal, a perfect evanescent coupling occurs between S states r2
; r12 ; r21 . In Fig. 1(b) we show that for $\Delta\bar{n}=0$ almost $100\%$
of power is effectively transferred to waveguide 2, as expected considering a
coupled-mode (discrete) approach r2 , which is governed for coupling constant
$C_{s}$ (see Supplemental Material r20p5 ). By increasing the relative
contrast $\Delta\bar{n}$, an effective detuning is produced in between the S
modes at different waveguides. This diminishes the effective interaction
between S states at different positions and the transferred power is
drastically reduced as Fig. 1(b) shows. We observe a minimum transference for
a contrast $\Delta\bar{n}\approx 0.25\times 10^{-3}$. This occurs at the
parameter region where the second waveguide starts supporting two modes, as
condition (1) suggests. A detuning in propagation constants produces a
tendency to localization, due to the creation of a non-symmetric dimer system
with corresponding non-symmetric eigenstates. An on-site detuning reduces the
effective coupling interaction and the energy tends to remain trapped at the
input site. This is phenomenologically similar to a nonlinear dimer r22 ; r23
, where the nonlinear response of the system effectively decouples the
waveguides due to an effective refractive index change.

By further increasing the refractive index contrast at site 2, in a region
where waveguide 2 already supports two modes, we find that power transfer
increases abruptly. We notice that this enhanced transfer is now occurring due
to an interaction between S and P states, as inset profiles show in Fig. 1(b).
This implies that the effective inter-orbital interaction is switched on, with
a coupling constant $C_{sp}$ now governing the dynamics. We find a new peak at
$\Delta\bar{n}\approx 0.53\times 10^{-3}$, where the S power at the first
waveguide is almost completely transferred to a P mode at waveguide 2. This
sharp peak is an indication that an exact condition for a perfect energy
transfer is hard to be achieved numerically and, even more, experimentally
(peak at $\Delta\bar{n}=0$ is sharp too, but easier to calibrate as both
waveguides are equal). Before and after this peak, the transferred energy at
site 2 decreases abruptly due to a detuning between S and P propagation
constants, as described in Supplemental Material r20p5  for a non symmetric
dimer model. By further increasing contrast $\Delta\bar{n}$, we observe that
the transferred power reduces to a minimum again, this time at
$\Delta\bar{n}\approx 1.0\times 10^{-3}$, in a region where the lower
waveguide starts supporting a third state as expected from condition (1). By
further increasing the relative index contrast, we find a new resonant peak at
$\Delta\bar{n}\approx 1.5\times 10^{-3}$. Now, the enhanced transfer of energy
at site 2 is due to an inter-orbital coupling between an S state and a third
mode, that we simply call “Tripole” or “D” state (TE20 or TE02 in optics r19
). This mode possesses three lobes and two nodes, with the corresponding phase
structure and a vertically oriented profile, due to the waveguide ellipticity.
Again, the enhanced peak is quite pronounced and the condition for perfect
conversion is hard to be achieved numerically, and even more experimentally.
So, in principle, this concept could be applied for the excitation - and
controlled generation - of any excited state, as soon as the propagation
constants at different sites match (see r20p5  for more details).

Figure 2: (a) Microscope image after white light illumination of dimer
examples fabricated using a fixed (variable) writing power for upper (bottom)
waveguides. (b) Experimental setup for characterizing waveguides and dimers. P
(lower, orange) waveguides have a shorter propagation length $\Delta z$. (b1)
Photograph of the experimental setup. (c) $z$-scan showing inter-orbital
coupling oscillation at different propagation lengths $\Delta z$ (indicated
below each sub-figure). (d) Maximum normalized transferred power at waveguide
2 versus fs average laser power difference ($\Delta P$). Every data point was
taken at a different distance

. Insets show experimental output intensity profiles at parameters indicated
by arrows.

We study optical waveguides as a simil of 2D photonic atoms by using a
femtosecond-laser writing technique r12  (see Supplemental Material r20p5  for
fabrication details). We fabricate several photonic dimer configurations as
the examples presented in Fig. 2(a), where bottom waveguides become multimode
as the writing power increases (see arrow). In order to characterize the
asymmetric dimers, we define the following procedure: (i) single-mode
waveguides are fabricated using a fixed writing power $P_{1}=78$ mW along the
whole sample ($50$ mm long); (ii) we set a writing power ($P_{2}$) for
waveguide 2 and fabricate 8 dimers for this power, having 8 different final
lengths ($\Delta z$) in the interval $\\{6,20\\}$ mm, with a step of 2 mm, as
sketched in Fig. 2(b). From a discrete model, we expect a cosine-like dynamics
along the propagation coordinate, which strongly depends on the coupling
constant and effective detuning. As a consequence, the maximum transferred
power is obtained at different $z$-values and the implementation of a $z$-scan
configuration is mandatory in our experiment. We characterize these dimers by
focusing a horizontally polarized HeNe laser beam at the input single mode
waveguide, as described in Fig. 2(b). In Fig. 2(c) we show an example of the
oscillatory (cosine-like) dynamics for $\Delta P\approx 23$ mW. We clearly
observe how light from the S (upper) waveguide starts to, first, weakly excite
a P state at the bottom waveguide, while for a larger propagation distance the
transferred power increases, with a maximum transference at $\Delta z=14$ mm.
Fig. 2(c) directly and quite clearly shows the effective transformation from a
S mode into a P state. This observation demonstrates experimentally the
theoretical concept described in Fig. 1 and gives a strong experimental
support for the existence of inter-orbital coupling in any physical system
having orbital-like states. We clearly observe a dynamical oscillation along
$z$ as a proof of a periodic energy transfer mechanism between neighboring
coupled waveguides, which is also a strong support for the study of tight-
binding (discrete) models including inter-orbital interactions.

In order to construct a complete parameter space, we analyze several intensity
profiles (as the examples shown in Fig. 2(d)-insets) for 23 different writing
powers $P_{2}$. Then, we look for the maximum transferred power at waveguide
2, which naturally occurs at different distances r20p5 . We integrate the
intensity at upper and lower halves of each image and obtain a normalized
value. Then, we look for the maximum transference for the 8 different
propagation lengths, as a result of the $z$-scan configuration. Fig. 2(d)
shows our compiled experimental results for maximum transferred power at
waveguide 2 versus the fs average laser power difference, which is defined as
$\Delta P\equiv P_{2}-P_{1}$. For $\Delta P=0$, we clearly observe that two
identical single-mode waveguides have an excellent transference of energy,
with almost $100\%$ of efficiency. Then, by continue increasing the writing
power $P_{2}$, we observe how the transferred power decreases abruptly, what
is in perfect agreement with direct numerical simulations [see Fig. 1(b)], and
with a discrete theoretical approach r20p5 .

As our main goal is the observation of inter-orbital coupling, we continue
increasing $P_{2}$ and, theferore, $\Delta\bar{n}$ (in our experiment,
$\Delta\bar{n}\sim\Delta P$ r20p5 ). For $\Delta P\approx 7$ mW, we observe
that the transferred power starts to increase weakly [see Fig.2(d)], however
this increment is in the order of the experimental error. We observe a
transition region with less than $10\%$ of transfer, for a large $\Delta P$
range. However, for $\Delta P\geqslant 19$ mW we observe a clear increasing
tendency and the excitation of a weak P mode at the second waveguide, as a
first indication of inter-orbital interaction. While continuing to increase
the writing power $P_{2}$, we observe an abrupt increment of energy at the
bottom waveguide, with a clear peak at $\Delta P=23$ mW, with more than $95\%$
of transference. This peak constitutes a concrete experimental proof for the
excitation of an orthogonal state at a neighbour waveguide, based on the
theoretical concepts described in the previous section. This observation is a
result of transforming a given single-mode waveguide into a multimode one, at
a given wavelength. We are indeed observing experimentally a propagation
constant tuning process, which allows switching on the effective interaction
between otherwise non-interacting orthogonal states. The experimental
intensity image at peak [inset profile in Fig. 2(d)] shows a perfect generated
dipole at the bottom waveguide. This simple observation constitutes two main
outcomes of our work: the demonstration of inter-orbital coupling and a
photonic mode converter. The first one is quite a fundamental result: it gives
support to different theoretical ideas r10 ; r13 ; r14  coming from diverse
research areas r2 ; r3 ; r4 ; r5 ; r6 ; r7 ; r8 ; r9  and it suggests the
possibility of studying new lattice configurations considering hybridized
interactions r24 ; r25 ; r26 ; r27 ; r28 . Without the demonstrated tuning
mechanism, orthogonal states simply do not interact on a lattice and
hybridized physics is simply not possible on a linear regime. The second
outcome of our observation is the generation of a simple and concrete method
for exciting higher-order spatial states inside a photonic chip, with perfect
spatial controllability that can be a key of success for concatenated photonic
operations r29 ; r30 ; r31 ; r32 ; r33 ; r34 .

Coupled-mode theory r2  and tight-binding approximation r1  show that modes at
different sites/atoms evanescently interact by a coupling constant $C$. In
Fig. 3(a) we show an experimental characterization of coupling coefficients
for different separation distances and for different orbital states (see
examples of SP dimers at Fig. 3(a)-inset and r20p5  for more details). We
observe a clear exponentially decaying tendency r12  for all coupling
coefficients. As profiles in Fig. 2(c) show, the S mode wavefunction, although
being broad, is very well trapped at the waveguide region, while the P mode,
in general, occupies a larger vertical area. Therefore, we expect that
$|C_{s}|<|C_{sp}|<|C_{p}|$, as our experimental data shows in Fig. 3(a) for
any distance $d$ (in general, $C_{s}>0$ and $C_{p}<0$, while $C_{sp}$ sign
depends on orbital orientation r15 ; r17 ; r27 ; r28 ). This is the first time
that this tendency and dependence are clearly and directly shown
experimentally for S and P states, giving a strong support for theoretical
studies considering different and more complex spatial configurations. It is
worth mentioning that the tendency shown in Fig. 3(a) strongly depends on
waveguide orientation, which in our case is vertical (see Fig. 1(b)-inset).

Figure 3: (a) Absolute coupling constant $|C|$ versus distance for S-S (red
dots), S-P (black dots), and P-P (purple dots) interactions, with lines
representing exponential fits. Inset shows a white light microscope image for
S-P dimers having different separation distances $d$. (b) SP-dimer + SPS-
trimer configuration to demonstrate a phase beam splitter. (c) White light
microscope image for a SP-dimer and a SPS-trimer. Inset shows the output
profile for the SP dimer only. (d1)–(d3), Output images, for $\Delta
z_{2}=2,4,6$ mm, respectively, after laser excitation of an S waveguide at the
input facet. Top and bottom images correspond to intensity and phase profiles,
respectively.

As an application of the inter-orbital coupling, we experimentally show the
different phases induced due to hybridized interactions. As it has been
suggested theoretically r13 ; r28 , an inter-orbital interaction produces the
appearance of negative coupling constants, which could be of great impact on
the study of topology and effective magnetic fields on lattices r15 ; r35 .
Therefore, our method looks as a promising and simple way of generating
synthetic magnetic fields r36  on lattices, which could have an important
impact when considering topological properties r37  as well. In order to show
an experimental evidence for this effect, we fabricate a setup consisting of a
SP-dimer plus a SPS-trimer configuration, as sketched in Fig. 3(b). We focus a
HeNe laser beam at the single-mode (white) waveguide and generate a perfect P
state at the multimode (orange) waveguide, after a coupling distance of
$\Delta z_{1}=14$ mm (Fig.3(c)-left shows an SP-dimer). Afterwards, the P
state propagates freely along the $z$ direction for about $30$ mm. Then it
acts as an input excitation for a SPS-trimer configuration (see Fig.
3(c)-right) and interacts symmetrically with both neighbouring single-mode
waveguides. As expected, inter-orbital coupling occurs again and new S modes
are generated back in waveguides 1 and 3, as shown in Figs. 3(d)-top.
Interestingly, by taking advantage of the phase structure of the P state we
are able to induce a $\pi$ phase shift in between the two generated S states.
Figs. 3(d)-bottom show different interferograms (see Supplemental Material
r20p5  for details) with a clear $\pi$ phase shift in between the right and
left parts of the output profiles. This clearly indicates that the coupling
interaction to the right and to the left single-mode waveguides are
effectively opposite in sign, which is a direct proof for a negative coupling
constant r15 . This system is also an example for a phase $\pi$ beam splitter
($\pi$-BS), which could be quite useful for interferometric quantum optics r33
; r39 ; r40 , considering concatenated operations r29 . The output profile
also coincides with the flat-band mode of a rhombic lattice r41 , which is an
important subject of research nowadays in photonic lattices.

In conclusion, we have demonstrated numerically and experimentally an inter-
orbital coupling interaction between S and P states on a photonic platform. We
first corroborated a resonance picture by numerically integrating a paraxial
wave equation and clearly observed the coupling between different orbitals, as
a consequence of asymmetrizing a double-well like potential photonic structure
r13 . This process was implemented with high precision using a fs laser
writing technique, allowing the excitation of higher order states and
experimentally probing negative coupling interactions. Our method also offers
a new technique for exciting higher-order spatial states inside an optical
chip, which could have a great impact on multiplexing applications r42 ; r43 .
The possibility of locating the mode conversion system in any arbitrary
position is a direct advantage compared to lithographic-like techniques r39 ;
r44 ; r45 ; r46 ; r47 ; r48 . Our observation constitutes a fundamental
validation of inter-orbital coupling, which could be an important new
ingredient for further studies in lattice science; but, also, a key tool for
concatenated or multiplexing photonic operations in the classical and quantum
regimes.

Authors acknowledge stimulating discussions with Magnus Johansson, Luis Foa
and Bastian Real. This work was supported in part by Millennium Science
Initiative Program ICN17-012 and FONDECYT Grants 1191205 and 3190601.

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