# Noncommutative black hole in de Rham-Gabadadze-Tolley like massive gravity

Piyali Bhar  0000-0001-9747-1009<EMAIL_ADDRESS>Department of
Mathematics, Government General Degree College Singur, Hooghly, West Bengal
712 409, India  Dhruba Jyoti Gogoi  0000-0002-4776-8506<EMAIL_ADDRESS>Department of Physics, Moran College, Moranhat, Charaideo 785670, Assam, India
Theoretical Physics Division, Centre for Atmospheric Studies, Dibrugarh
University, Dibrugarh  Supakchai Ponglertsakul<EMAIL_ADDRESS>Strong
Gravity Group, Department of Physics, Faculty of Science, Silpakorn
University, Nakhon Pathom 73000, Thailand

###### Abstract

We examine the behavior of non-commutative Schwarzschild black holes in the
context of massive gravity. According to the investigation, corresponding to a
minimal mass, the black hole can have two horizons, one horizon, or no horizon
at all. Our results imply the existence of a stable black hole remnant, whose
mass can be uniquely calculated in terms of the non-commutative parameter
$\theta$ and the graviton mass m. Thermodynamic features such as heat capacity
and Hawking temperature are studied. We also examine a scalar linear
perturbation on the black hole. Quasinormal frequencies are computed via
Wentzel–Kramers–Brillouin (WKB) method with Padé improvement. All quasinormal
frequencies considered in this work have a negative imaginary part. In the
eikonal limit, we investigate the angular velocity and the Lyapunov exponent
as a function of $M/\sqrt{\theta}$. Additionally, we explore the black hole’s
shadow across various model parameters. Our findings indicate that non-
commutativity leads to a reduction in the black hole’s shadow, with this
effect exhibiting a nonlinear relationship. Furthermore, we observe that the
inclusion of a massive graviton in the theory results in an increase in the
black hole’s shadow radius, particularly at greater observer distances.

##  I Introduction

The phenomenology of the dRGT massive gravity model is very fascinating. The
dRGT black hole (BH) and black string (BS) solutions differ from the standard
general relativity solution in the presence of the heavy graviton because the
graviton mass plays a role in this situation. Similar terms can also be found
in Bergshoeff, Hohm, and Townsend (BHT) massive gravity Oliva et al. (2009)
and the minimal theory of massive gravity De Felice et al. (2018). Recently
black hole (BH), black string (BS), and, more recently, rotating black string
solutions have been found and examined in the context of the dRGT massive
gravity.

For a generic choice of the parameters of dRGT massive gravity, Ghosh et al.
Ghosh et al. (2016) provided a precise spherical black hole solution and also
discussed the black hole’s thermodynamics and phase structure (for the charged
case) in the grand canonical and canonical ensembles. Tannukij et al. Tannukij
et al. (2017) proposed a cylindrically symmetric black string solution (both
charged and uncharged) to the nonlinear ghost-free massive gravity discovered
by de Rham, Gabadadze, and Tolley (dRGT). According to the author, this “dRGT
black string” is considered as a generalization of Lemos’s black string
solution. Furthermore, since the graviton mass contributes to both the global
monopole term and the cosmological-constant term, the dRGT black string
solution encompasses additional classes of black string solutions, such as the
monopole-black string ones. A rotating black string solution in de Rham-
Gabadadze-Tolley (dRGT) massive gravity is discussed in Ghosh et al. (2020).
This solution has two more variables that characterize the structure of
graviton mass, making it a kind of generalized variant of the rotating anti-de
Sitter (AdS)/dS black string solution. Refs. Burikham et al. (2017);
Ponglertsakul et al. (2018); Boonserm et al. (2018); Nieuwenhuizen (2011);
Boonserm et al. (2019); Panpanich et al. (2019); Wuthicharn et al. (2022)
contain some other intriguing works on black holes under massive gravity.

Since non-commutative spacetime in gravity theories can be considered as an
alternative to quantum gravity, it has gained attention as an active research
topic in recent years Nicolini (2009); Snyder (1947). Several studies have
been carried out about non-commutative gravity. Specifically, the impact of
non-commutativity on black hole physics has drawn significant interest, mainly
due to the increased abundance of the final stage of the noncommutative black
hole. It is familiar that the non-commutativity can be applied to General
Relativity by changing the matter source Nicolini et al. (2006), eliminating
pointlike structures in favour of smeared objects in flat spacetime Smailagic
and Spallucci (2003a, b). Consequently, mass density is changed to introduce
non-commutativity by substituting a Gaussian distribution for the Dirac delta
function Nicolini et al. (2006). The Schwarzschild black hole, inspired by
four-dimensional non-commutative geometry, was first postulated by Nicolini
Nicolini et al. (2006). This black hole solution was then extended to include
the scenario of charged Ansoldi et al. (2007). In 2010, this study was further
extended to the general case of charged rotating non-commutative black holes
by Modesto and Nicolini Modesto and Nicolini (2010). Hamil et al. Hamil and
Lütfüoğlu (2024) investigated the shadow images in the presence of plasma and
used Hawking temperature, entropy, and specific heat functions to study the
thermodynamics of non-commutative Schwarzschild black holes embedded in
quintessence matter. They also explored phase transition and stability
features and the quasinormal modes in WKB and Mashhoon approximations.
Finally, they thoroughly discussed the impacts of quintessence matter and non-
commutative parameter on the black hole spacetime. Heidari et al. Heidari et
al. (2024) computed the thermodynamic parameters of the system and examined
the gravitational signatures of a non-commutative stable black hole, comparing
their findings with those found in recent research. They used the WKB
approximation and the Pöschl–Teller fitting method to investigate the
quasinormal modes of massless scalar perturbations. Additionally, they looked
at the geodesics of massive and massless particles, showing how the non-
commutative parameter $\theta$ greatly affects event horizons and light
pathways. Additionally, they computed the shadows, demonstrating that larger
values of $\theta$ correspond to larger shadow radii. Yan and colleagues Yan
et al. (2023) computed the quasinormal modes (QNMs) of a charged non-
commutative black hole using three different methods: scalar, electromagnetic,
and gravitational fields. They discussed the effects of charge $Q$ and the
non-commutative parameter $\theta$ on QNMs in various domains. After that,
scientists used the event horizon telescope’s constraints on the shadow radii
of M87* and Sgr A* to compute the black hole’s shadow radius and provided the
valid range of $\theta$ and Q. Saleem and Aslam Saleem and Aslam (2023)
examined the observable behavior and shadow of the non-commutative (NC)
charged Kiselev black hole, which was surrounded by various profiles of
accretions. The authors selected particular values for the model parameters to
produce the BH shadow profile, and they concluded that changes in each
parameter directly affected the BH size and light trajectories. In ref
Iftikhar (2023), Iftikhar investigated the shadow cast by a non-commutative
rotating Hayward black hole. Based on the study, the author demonstrates that
the spin, non-commutative parameter, and parameter $g$ of the mentioned black
hole determine both the apparent shape and size of the shadow. Both the size
of the shadow and the non-commutative parameter decrease with $g$.
Additionally, the author discovered that at large values of $g$ and spin, the
shape of the shadow deviates from a complete circle. Zeng et al.Zeng et al.
(2023) examined the light rings and observational characteristics of the BH
shadow encircled by various accretion flow models in their system of
consideration. They next looked at how model factors affected the space-time
structure and observational display of BHs. Investigation on wormholes under
non-commutative geometry were studied in Tayde et al. (2024); Farooq and
Zubair (2023); Pradhan et al. (2023); Chalavadi et al. (2023); Debnath et al.
(2023); Tayde et al. (2023); Kavya et al. (2023); Rahaman et al. (2012); Bhar
and Rahaman (2014)

The evolution of a linear matter wave around black hole is generally governed
by quasi-normal modes (QNMs). QNMs are associated to complex frequency (quasi-
normal frequency) where real part describes emission frequency and imaginary
part represents the decaying time scale. Remarkably, quasinormal frequency can
be uniquely determined by black hole’s mass and angular momentum which allows
one to obtain various properties of black holes by studying QNMs of black
holes. The QNMs study can be traced back to 1970 where the first QNMs of
Schwarzschild black hole was investigated Vishveshwara (1970). Since then, an
astronomical amount of studies devoted to black hole’s QNMs have been explored
(see for example, Press (1971); Kokkotas and Schmidt (1999); Konoplya and
Zhidenko (2011); Gogoi et al. (2024, 2023a); Konoplya (2002, 2003); Konoplya
and Stuchlík (2017); Konoplya et al. (2019); Sekhmani and Gogoi (2023); Gogoi
et al. (2023b); Gogoi and Ponglertsakul (2024); Okyay and Övgün (2022); Pantig
et al. (2022a); Pedrotti and Vagnozzi (2024)). In addition, charged black
holes in the dRGT massive gravity are superradiantly unstable Burikham et al.
(2017). In contrast, no instability modes are found in the case of neutral
black string in the dRGT massive gravity Ponglertsakul et al. (2018). On the
other hand, within noncommutative framework, various QNMs of black holes are
explored e.g. the Schwarzschild black hole Giri (2007); Liang (2018a, b);
Campos et al. (2022); Zhao et al. (2023), the Reissner-Nordström black hole
Ćirić et al. (2018) and the BTZ black hole Gupta et al. (2017, 2015).

The study of black holes reveals their event horizon as a crucial boundary
beyond which no particles, even light, can escape due to intense gravitational
forces Synge (1966). Matter falling into a black hole, known as accretion,
spirals inward, emitting radiation across various frequencies, including
detectable radio waves, forming a bright background with a dark region called
the black hole shadow Luminet (1979). Pioneered by Falcke et al. Falcke et al.
(2000), recent advancements, notably by the Event Horizon Telescope, have
allowed the imaging of black hole shadows in galaxies like Messier 87* (M 87*)
and Sagittarius A* (Sgr. A*) Akiyama et al. (2019, 2022). These observations
offer insights into photon sphere distortion caused by the non-spherical
horizons of rotating black holes Övgün et al. (2018); Kocherlakota et al.
(2021); Belhaj et al. (2021, 2020); Belhaj and Sekhmani (2022); Gogoi et al.
(2023c); Övgün and Sakallı (2020); Övgün et al. (2020); Kuang and Övgün
(2022); Kumaran and Övgün (2022); Mustafa et al. (2022); Cimdiker et al.
(2021); Okyay and Övgün (2022); Atamurotov et al. (2023); Pantig et al.
(2023); Abdikamalov et al. (2019); Abdujabbarov et al. (2016); Atamurotov and
Ahmedov (2015); Papnoi et al. (2014); Abdujabbarov et al. (2013); Atamurotov
et al. (2013); Cunha and Herdeiro (2018); Gralla et al. (2019). These
advancements have led to a comprehensive study of shadows in various modified
gravities, including comparisons with M87* and Sgr. A* observations, providing
valuable geometric data Pantig et al. (2022a); Lobos and Pantig (2022); Gogoi
et al. (2023c); Uniyal et al. (2023); Panotopoulos et al. (2021); Panotopoulos
and Rincon (2022); Khodadi and Lambiase (2022); Khodadi et al. (2021); Zhao et
al. (2023); Khodadi et al. (2020); Khodadi and Saridakis (2021); Vagnozzi et
al. (2023); Khodadi (2022). Furthermore, the study of black hole shadows has
revealed insights into thermodynamic phase transitions, with the shadow radius
serving as a key tool for studying thermodynamic black hole systems Zhang and
Guo (2020). Additionally, a significant relationship has been established
between black hole shadows and quasi-normal modes, particularly in the eikonal
limit, demonstrating that the real parts of quasi-normal modes are connected
to the shadow radii of black holes Jusufi (2020a, b, 2021).

In this work, we investigate thermodynamics properties, perturbation and
optical properties of noncommutative black hole in dRGT massive gravity.
Hence, this paper is organized as follows. Section II, we discuss the action
and modified field equation within the framework of the dRGT massive gravity.
Noncommutative black hole is introduced in Section III. Then, horizon,
asymptotic structure and curvature scalars of the black hole solution are
discussed. Later, thermodynamics properties such as temperature and heat
capacity are explored. In Section IV, we calculate quasinormal frequencies of
noncommutative black holes and investigate the eikonal limit. Optical
appearance i.e., black hole’s shadow is studied in Section V. Finally, we
summarize the results and discuss the novel finding in the conclusions
section.

##  II action and field equation in massive gravity

To examine the black hole model within the framework of de Ramadanze–Tolley
(dRGT) massive gravity, we take into consideration the following action:

$\mathcal{I}=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[\mathcal{R}+m^{2}\sum_{i}^{4}c_{i}\mathcal{U}_{i}(g,f)\right]+L_{\text{m}},$
(1)

where $L_{\text{m}}$ is the matter Lagrangian, $g$ is the determinant of the
metric tensor $g_{\mu\nu}$. $f$ and $g$ are fixed symmetric tensors and metric
tensors, respectively, and $\mathcal{U}_{i}$ is the self-interacting potential
of the graviton with mass $m$, $c_{i}$’s are constant. The potential
$\mathcal{U}_{i}$ in this case is defined as,

$\displaystyle\mathcal{U}_{1}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right],\;\;\;\;\;\;\;\mathcal{U}_{2}=\left[\mathcal{K}\right]^{2}-\left[\mathcal{K}^{2}\right],$
$\displaystyle\mathcal{U}_{3}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right]^{3}-3\left[\mathcal{K}\right]\left[\mathcal{K}^{2}\right]+2\left[\mathcal{K}^{3}\right],$
$\displaystyle\mathcal{U}_{4}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right]^{4}-6\left[\mathcal{K}^{2}\right]\left[\mathcal{K}\right]^{2}+8\left[\mathcal{K}^{3}\right]\left[\mathcal{K}\right]+3\left[\mathcal{K}^{2}\right]^{2}-6\left[\mathcal{K}^{4}\right].$

$\left[\mathcal{K}\right]$ represents the trace of the metric
$\mathcal{K}^{\mu}_{\nu}=\delta^{\mu}_{\nu}-\sqrt{g^{\mu\sigma}f_{ab}\partial_{\sigma}\phi^{a}\partial_{\nu}\phi^{b}}$.
The $f_{ab}$ is fiducial or reference metric to be defined later. The four
Stuckelberg fields take the form $\phi^{a}=x^{\mu}\delta^{a}_{\mu}$.

By varying the action (1) for the metric tensor $g^{\mu}_{\nu}$, the equation
of motion for this gravity can be obtained as follows:

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+m^{2}\chi_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu},$ (2)

the expression for $\chi_{\mu\nu}$ is represented by,

$\displaystyle\chi_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{c_{1}}{2}\left(\mathcal{U}_{1}g_{\mu\nu}-\mathcal{K}_{\mu\nu}\right)-\frac{c_{2}}{2}\left(\mathcal{U}_{2}g_{\mu\nu}-2\mathcal{U}_{1}\mathcal{K}_{\mu\nu}+2\mathcal{K}_{\mu\nu}^{2}\right)$
(3)
$\displaystyle-\frac{c_{3}}{2}(\mathcal{U}_{3}g_{\mu\nu}-3\mathcal{U}_{2}\mathcal{K}_{\mu\nu}+6\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{2}-6\mathcal{K}_{\mu\nu}^{3})$
$\displaystyle-\frac{c_{4}}{2}\left(\mathcal{U}_{4}g_{\mu\nu}-4\mathcal{U}_{3}\mathcal{K}_{\mu\nu}+12\mathcal{U}_{2}\mathcal{K}_{\mu\nu}^{2}-24\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{3}+24\mathcal{K}_{\mu\nu}^{4}\right),$

and $T_{\mu\nu}$ represents the energy momentum tensor.
In ($3+1$) dimensions, let us express the line element that describes the
interior spacetime of a spherically symmetric distribution of matter as
follows:

$ds^{2}=-h(r)dt^{2}+[h(r)]^{-1}dr^{2}+r^{2}(d\Theta^{2}+\sin^{2}\Theta
d\phi^{2}),$ (4)

where $h(r)=e^{2\phi(r)}$ and $[h(r)]^{-1}=e^{2\lambda(r)}$, and $\phi$ and
$\lambda$ depend on ‘r’, i.e., purely radial. Generalizing and using the
ansatz in Cai et al. (2015), we employ the reference metric that follows:

$f_{\mu\nu}=\text{diag}(0,0,C^{2},C^{2}\sin^{2}\Theta),$ (5)

$C$ being a positive constant. With the reference metric (5), we have Cai et
al. (2015),

$\mathcal{U}_{1}=\frac{2C}{r},\text{ \ \
}\mathcal{U}_{2}=\frac{2C^{2}}{r^{2}},\,\mathcal{U}_{3}=0,\,\mathcal{U}_{4}=0.$

We assume that the matter distribution is anisotropic, and therefore we select
the most general energy-momentum tensor in the form:

$T_{\mu}^{\nu}=(\rho+p_{t})u^{\nu}u_{\mu}+p_{t}g_{\mu}^{\nu}+(p_{r}-p_{t})v^{\nu}v_{\mu},$
(6)

The symbols $v^{\nu}$ are the spacelike vector and $u_{\mu}$ represent fluid
four velocities. $\rho(r),\,p_{r}(r)$, and $p_{t}(r)$ stand for the energy
density, radial pressure, and tangential pressure, respectively. The field
equations (assuming $G=c=1$) in de Rham-Gabadadze-Tolley like massive gravity,
are given as Li et al. (2023):

$\displaystyle 8\pi r^{2}\rho$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(2r\lambda^{\prime}-1)+1+m^{2}C(c_{2}C+c_{1}r),$ (7)
$\displaystyle 8\pi r^{2}p_{r}$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(2r\phi^{\prime}+1)-1-m^{2}C(c_{2}C+c_{1}r),$ (8) $\displaystyle
8\pi rp_{t}$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(\phi^{\prime}+r\phi^{\prime
2}-\lambda^{\prime}-r\lambda^{\prime}\phi^{\prime}+r\phi^{\prime\prime})-\frac{m^{2}c_{1}C}{2}.$
(9)

Here the derivative with respect to $r$ is indicated by the prime. The
conservation law for the energy–momentum tensor is obtained by using
$T_{\mu;\nu}^{\nu}=0$ as,

$\displaystyle\frac{dp_{r}}{dr}=-(\rho+p_{r})\frac{d\phi}{dr}+\frac{2}{r}(p_{t}-p_{r})=0.$
(10)

The goal of the upcoming section is to derive the black hole model by solving
the field equations in the presence of non-commutative geometry.

##  III Black hole model

In ($3+1$) dimensions, when a static, spherically symmetric, noncommutative
distributed, particle-like gravitational source is taken into consideration,
one finds,

$\displaystyle\rho=\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}},$
(11)

Here, $M$ denotes the source’s total mass. The matter distribution in models
inspired by noncommutative geometry is smeared on a scale that is directly
related to the noncommutativity parameter, $\theta$. The assertion is that the
uncertainty principle between coordinates prevents the matter from being fully
localized in the presence of noncommutativity. According to phenomenological
findings, noncommutativity is not apparent at presently accessible energies,
i.e., $\theta<10^{-16}$ cm.

In addition, to obtain a black hole solution, one requires
$g_{tt}=(g_{rr})^{-1}$ that implies, $p_{r}=-\rho$. Using the expression of
$\rho$ given in (11), we solve the equation (7) for $e^{-2\lambda}$ which is
obtained as,

$\displaystyle e^{-2\lambda}$ $\displaystyle=$ $\displaystyle
1+Cm^{2}\left(Cc_{2}+\frac{c_{1}r}{2}\right)+\frac{2M}{\sqrt{\theta\pi}}e^{-\frac{r^{2}}{4\theta}}-\frac{2M}{r}\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right),$
(12)

where $\text{erf}(r)$ is the error function. Using field equations (7)-(9), we
get,

$\displaystyle e^{2\phi}=e^{-2\lambda}$ $\displaystyle=$ $\displaystyle
1+Cm^{2}\left(Cc_{2}+\frac{c_{1}r}{2}\right)+\frac{2M}{\sqrt{\theta\pi}}e^{-\frac{r^{2}}{4\theta}}-\frac{2M}{r}\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right).$
(13)

This metric is clearly not an asymptotically flat spacetime. In the absence of
graviton mass $m\rightarrow 0$, one can readily be obtained as,

$\displaystyle e^{2\phi}=e^{-2\lambda}$ $\displaystyle=$ $\displaystyle
1+\frac{2M}{\sqrt{\theta\pi}}e^{-\frac{r^{2}}{4\theta}}-\frac{2M}{r}\text{erf}\left(\frac{r}{2\sqrt{\theta}}\right),$
(14)

which is the expression of the non-commutative Schwarzschild black hole
proposed by Nicolini Nicolini (2005), and, for
$\frac{r}{\sqrt{\theta}}\rightarrow\infty$, and $m\rightarrow 0$, the standard
Schwarzschild metric $e^{-2\lambda}=e^{2\phi}=1-\frac{2M}{r}$ can be
constructed from (12).
The radial and transverse pressures can be obtained as,

$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{M}{(4\pi\theta)^{\frac{3}{2}}}e^{-\frac{r^{2}}{4\theta}},$
(15) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{M(r^{2}-4\theta)}{32\pi^{\frac{3}{2}}\theta^{\frac{5}{2}}}e^{-\frac{r^{2}}{4\theta}}$
(16)

Analyzing the geometrical and thermodynamical behavior of the solution is
worthwhile to comprehend the remaining surprise characteristics fully.

###  III.1 Horizons of the black hole

Let us begin our investigation with the black hole’s horizon equation. The
horizon ($r_{H}$) of the black hole occurs where $g_{rr}(r_{H})=0$. That
gives,

$\displaystyle
1+Cm^{2}\left(Cc_{2}+\frac{c_{1}r_{H}}{2}\right)+\frac{2M}{\sqrt{\theta\pi}}e^{-\frac{r_{H}^{2}}{4\theta}}-\frac{2M}{r_{H}}\text{erf}\left(\frac{r_{H}}{2\sqrt{\theta}}\right)=0$
(17)

From Eq. (17), we are unable to find a closed-form solution for $r_{H}$.
Plotting $g_{tt}$, on the other hand, allows one to mathematically determine
the presence of horizon(s) and their radius by identifying intersections with
the $\frac{r}{\sqrt{\theta}}$-axis. As seen in Fig. 1, horizons and their
radii are present at the locations where $g_{tt}(\frac{r}{\sqrt{\theta}})$
crosses the $\frac{r}{\sqrt{\theta}}$-axis. Noncommutativity introduces new
behavior concerning the regular Schwarzschild black hole, as Fig. 1
illustrates. Several possibilities exist, rather than just one event horizon.
The point at which the function $g_{tt}(\frac{r}{\sqrt{\theta}})$ touches with
the $\frac{r}{\sqrt{\theta}}$-axis indicates the existence of a minimum mass
$M_{0}=1.90\sqrt{\theta}$, below which there is no event horizon. In the
borderline situation of $M=M_{0}$, we discover one single degenerate horizon
at $r_{H}\sim 3.05\sqrt{\theta}$, corresponding to the extremal black hole,
while the line element exhibits two separate horizons for $M>M_{0}$.
Additionally, the figure also indicates that for $M<M_{0}$ no black hole
exists. Additionally, Fig. 1 shows that as the mass of the black hole
increases, so will the distance between the horizons.

Figure 1: $g_{tt}(\frac{r}{\sqrt{\theta}})$ is shown versus
$\frac{r}{\sqrt{\theta}}$. The curves are drawn for various values of
$\frac{M}{\sqrt{\theta}}$. For $M=1.4\sqrt{\theta}$ no horizon exists. For
$M=1.9\sqrt{\theta}$ one degenerate horizon exists at $r=3.05\sqrt{\theta}$
and for $M=2.4\sqrt{\theta}$, two horizon exists. For drawing the plots we
have taken $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$, and $\theta=0.01$.

Intriguingly, the black hole (17) is finite as $r\to 0.$ i.e.,

$\displaystyle e^{2\phi}$
$\displaystyle\sim\left(1+C^{2}c_{2}m^{2}\right)+O(r),$ (18)

However, the Ricci scalar $R$ and the Kretchmann scalar $K$ are found to be

$\displaystyle R$ $\displaystyle\sim
O(r^{-2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K\sim
O(r^{-4}).$ (19)

as $r\to 0.$ which indicate curvature singularity of the black hole solution.
In Fig. 2, the Ricci scalar and the Kretchmann scalar are displayed. It is
apparent that these diverge at $r=0$. Moreover, at the location of event
horizon as illustrated in Fig. 1, these curvature scalars are finite.
Therefore despite the black hole solution being regular at $r=0$ there exists
curvature singularity. Remarkably, in the $M=1.4\sqrt{\theta}$ case, the
solution can be interpreted as a naked singularity.

Figure 2: Scalar curvature quantities plot against $\frac{r}{\sqrt{\theta}}$
for $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$, and $\theta=0.01$. Left: the
Ricci scalar. Right: the Kretchmann scalar. Note that, a subplot displays
behavior of the curvature scalar at small $\frac{r}{\sqrt{\theta}}.$

In addition, the energy density $(\rho)$, radial and transverse pressure
($p_{r},p_{t}$) are plotted as a function of $\frac{r}{\sqrt{\theta}}$ in Fig.
3. All of them are finite from within the centre of black hole toward the
exterior region. With increasing $\frac{M}{\sqrt{\theta}}$, the energy density
becomes larger while both pressures ($p_{r},p_{t}$) are smaller. Moreover as
one moves away from the black hole’s centre, $\rho$ decreases and $p_{r}$
increases with $\frac{r}{\sqrt{\theta}}$. The transverse pressure $p_{t}$ is
increasing before reaching its maximum at $r=2.8284\sqrt{\theta}$. A similar
type of behavior of the transverse pressure was obtained by Roy Chowdhury et
al. Chowdhury et al. (2021) when describing the model of a noncommutative
black hole in the Finslerian spacetime and by Rahaman et al Rahaman et al.
(2013) when studying BTZ black hole in presence of non-commutative geometry.
It can also be noticed that the strong energy conditions $\rho+p_{r}+2p_{t}$
do not hold for $r<2.8284\sqrt{\theta}$ This suggests that the classical
description of energy and matter breaks down at the region where quantum
phenomena dominate despite nonlinear gravity.

Figure 3: Energy density (left), radial pressure (centre), and transverse
pressure (right) plot against $\frac{r}{\sqrt{\theta}}$ for $\theta=0.01$. The
other parameters are taken as, $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$,
and $\theta=0.01$.

###  III.2 Temperature, total mass and the heat capacity of the black hole

The Hawking temperature of the black hole can be calculated as,

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left(\frac{dg_{tt}}{dr}\right)_{r=r_{H}},$ (20)
$\displaystyle=$
$\displaystyle\frac{\frac{r_{H}^{3}\left(Cm^{2}\left(2c_{2}C+c_{1}r_{H}\right)+2\right)}{\theta\left(r_{H}-\sqrt{\pi}\sqrt{\theta}e^{\frac{r_{H}^{2}}{4\theta}}\text{erf}\left(\frac{r_{H}}{2\sqrt{\theta}}\right)\right)}+4Cm^{2}\left(c_{2}C+c_{1}r_{H}\right)+4}{16\pi
r_{H}}.$

The plot of the Hawking temperature is shown in Fig. 4 for our present
noncommutative black hole. The figure indicates a minimum horizon radius
$r_{0}$ since $T_{H}$ cannot be negative in general, and this radius occurs
where $\frac{dg_{tt}}{dr}$ vanishes. One can note that at this radius $r_{0}$,
the Hawking temperature $T_{H}$ vanishes.

In the absence of massive gravity i.e., $m=0$ and considering the limit
$r/\sqrt{\theta}\gg 1$, the expression of $T_{H}$ $\eqref{temp}$ resemble
those of the Schwarzschild black hole

$\displaystyle T_{H}$ $\displaystyle=\frac{1}{4\pi r_{H}}.$ (21)

Figure 4:  The profile of Hawking temperature is shown against
$\frac{r_{H}}{\sqrt{\theta}}$. The values of the parameters are taken as,
$C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$, and $\theta=0.01$.

We can obtain the mass in terms of $r_{H}$ as follows:

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{r_{H}^{2}}{4\theta}}\sqrt{\pi}r_{H}(2+Cm^{2}(2Cc_{2}+c_{1}r))\sqrt{\theta}}{-4r_{H}+4e^{\frac{r_{H}^{2}}{4\theta}}\sqrt{\pi}\sqrt{\theta}erf(\frac{r_{H}}{2\sqrt{\theta}})}.$
(22)

The left panel of Fig. 5 displays the variation of the overall mass with
respect to $\frac{r_{H}}{\sqrt{\theta}}$. The figure shows that the minimal
mass $M_{0}$ validates the degenerate horizon found in Fig. 1.

Figure 5: (Left) the total mass $M$ is shown against
$\frac{r_{H}}{\sqrt{\theta}}$, (right) heat capacity is shown against
$\frac{r_{H}}{\sqrt{\theta}}$. For drawing the plots, the values of the
parameters are taken as, $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$, and
$\theta=0.01$.

The heat capacity of the model can be obtained as

$\displaystyle\mathcal{C}=\frac{\partial M}{\partial
r_{H}}\left(\frac{\partial T_{H}}{\partial r_{H}}\right)^{-1},$ (23)

We have shown the graphical representation of the heat capacity in Fig 5
(right panel). The heat capacity is positive for
$2.9\sqrt{\theta}<r_{H}<4.7\sqrt{\theta}$. Because of their negative heat
capacity, black holes are unstable for $r_{H}>4.7\sqrt{\theta}$. The heat
capacity approaches $0$ when $r_{H}$ approaches $2.9\sqrt{\theta}$.

##  IV Quasinormal modes

Here we consider as massive scalar perturbation on spacetime background (4). A
scalar field propagating on curved spacetime is given by the Klein-Gordon
equation

$\displaystyle\nabla_{a}\nabla^{a}\psi-\alpha^{2}\psi$ $\displaystyle=0,$ (24)

where $\alpha^{2}$ is scalar field’s mass squared. In spherical symmetric
spacetime, scalar field $\psi$ can be separated into the following form

$\displaystyle\psi$ $\displaystyle=e^{-i\omega
t}\frac{R(r)}{r}Y(\Theta,\phi),$ (25)

where $Y(\Theta,\phi)$ is spherical harmonic function. The scalar radial field
equation can be recast into the Schrödinger-like form as

$\displaystyle 0$
$\displaystyle=\frac{d^{2}R}{dr_{\ast}^{2}}+\left[\omega^{2}-V\right]R,$ (26)
$\displaystyle V$
$\displaystyle=e^{2\phi}\left(\alpha^{2}+\frac{\ell(\ell+1)}{r^{2}}+\frac{2e^{2\phi}\phi^{\prime}}{r}\right),$
(27)

where prime denotes derivative with respect to $r$ and $\ell(\ell+1)$ is the
eigenvalue of angular operator. The tortoise coordinate is defined such that

$\displaystyle\frac{dr_{\ast}}{dr}$ $\displaystyle=e^{-2\phi},$ (28)

It can be shown that in the massless case $\alpha=0$, the effective potential
is finite i.e., $V\to C^{2}c_{1}m^{4}/4$ as $r\to\infty$. In contrast, $V$, in
the massive case $\alpha\neq 0$, diverges as $r\to\infty$. In fact, the
effective potential is monotonically increasing with $r$. These are
demonstrated in Fig. 6. Clearly, the maximum of effective potential increases
with angular index $\ell$ as displayed in the left figure of Fig. 6. With the
chosen spacetime parameters i.e.,
$C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05$, $V\to 6.25\times 10^{-8}$ as
$r\to\infty$. It is possible to render the $C^{2}c_{1}m^{4}/4$ non-zero finite
value. Therefore, the $\alpha_{eff}\equiv C^{2}c_{1}m^{4}/4$ term effectively
behaves as a scalar field mass. Thus, in the massless case $\alpha=0$, the
appropriate boundary conditions leading to the quasinormal modes are $R\sim
e^{\pm i\omega r_{\ast}}$ as $r_{\ast}\to\pm\infty$. In contrast, $V$ diverges
asymptotically in the massive case as it is shown in the right figure of Fig.
6. As scalar mass increases, the effective potential diverges more rapidly.
The presence of mass term dramatically changes the potential’s asymptotic
structure. This requires special treatment when considering appropriate
boundary conditions of the quasinormal modes. For the sake of simplicity, we
shall particularly consider the massless case $\alpha=0.$

Figure 6: Left: effective potential in massless case $\alpha=0$ for various
value of $\ell$ Right: effective potential in massive case $\alpha\neq 0$ for
various value of $\alpha$ and fixed $\ell=2$. The other parameters are fixed
to $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05,\theta=0.1$ and
$M/\sqrt{\theta}=2.4.$

To compute the quasinormal frequencies, we implement Wentzel–Kramers–Brillouin
(WKB) approximation method. This method is introduced to a study of black hole
scattering problem Schutz and Will (1985). Later, WKB method is extended to
third Iyer and Will (1987) and sixth order Konoplya (2003). In Matyjasek and
Opala (2017), the WKB method is extended further to thirteenth order with the
Padé averaged. Several previous results can be reproduced with significant
accuracy when implementing the Padé averaged technique Matyjasek and Telecka
(2019). We note that, our computational calculation is based on the Wolfram’s
Mathematica code provided in Konoplya et al. (2019). In this work, we shall
apply the third order WKB with the Padé averaged to calculate the quasinormal
frequencies of scalar field around noncommutative black hole in dRGT massive
gravity. The WKB approximation works efficiently for $\ell>n$ when $n$ is
overtone number as it is pointed out in Panotopoulos and Rincón (2019);
Cardoso et al. (2004). In Table 1, we show example of quasinormal frequencies
of massless scalar perturbation when varying $\ell$ and $n$. Remark that, the
most fundamental mode ($\ell=n=0$) is also displayed but it might not be well-
approximated by the WKB method. It can be seen that as $\ell$ increases, the
real part of $\omega$ significantly increases while the imaginary part
slightly decreases (in magnitude). On the contrary, the real part marginally
decreases while the imaginary part notably increases (in magnitude) as the
overtone number is larger. We note that for this point onward, only the most
fundamental frequencies ($n=0$) are considered.

Table 1: Quasinormal frequencies of massless scalar perturbation on noncommutative black hole in dRGT massive graivty with $C=0.05,\,m^{2}c_{1}=0.01,\,m^{2}c_{2}=0.05,M/\sqrt{\theta}=2.4$, and $\theta=0.1$. The frequencies are computed via the $3^{rd}$ order WKB with the Padé averaged. $\ell$ | $n=0$ | $n=1$ | $n=2$ | $n=3$
---|---|---|---|---
$0$ | $0.108428-0.045389i$ | $-$ | $-$ | $-$
$1$ | $0.374373-0.040713i$ | $-$ | $-$ | $-$
$2$ | $0.630769-0.040309i$ | $0.627717-0.121194i$ | $-$ | $-$
$3$ | $0.885705-0.040209i$ | $0.883507-0.120758i$ | $0.879193-0.201681i$ | $-$
$4$ | $1.140155-0.040169i$ | $1.138441-0.120585i$ | $1.135050-0.201229i$ | $1.130058-0.282232i$

Figure 7: Quasinormal frequencies of noncommutative black hole in dRGT massive
gravity against $M/\sqrt{\theta}$ for $\ell=2,\theta=0.01$ and
$m^{2}c_{2}=0.05$ for various value of $\alpha_{eff}.$

Now we illustrate how $M/\sqrt{\theta}$ effect the quasinormal frequencies in
Fig. 7. In this plot, we pick four different values of effective scalar
field’s mass ($\alpha_{eff}$). This can be archived by adjusting $C$ and
$c_{1}=j/m^{2}$. For instance, $\alpha_{eff}=6.25\times 10^{-8}$ corresponds
to $C=0.05,j=0.01$, $\alpha_{eff}=2.5\times 10^{-3}$ for $C=1,j=0.1$,
$\alpha_{eff}=0.100806$ for $C=1.27,j=0.5$ and $\alpha_{eff}=0.5041$ for
$C=1.42,j=1$. It is worth-mentioning that, changing $C$ and $c_{1}$ leads to
different asymptotic structure of the black hole metric (13). From this
figure, we find that as the mass increases, the real part of $\omega$
decreases. An opposite trends can be seen from the imaginary part (the right
figure of Fig. 7). The $Im(\omega)$ becomes less negative as $M/\sqrt{\theta}$
increases. We also observe that the higher the $\alpha_{eff}$, the lower
$Re(\omega)$ and $Im(\omega)$. Remark that in each curve, the smallest
$M/\sqrt{\theta}$ indicates the black hole that is very close to its extremal
limit.

Figure 8: Quasinormal frequencies of noncommutative black hole in dRGT massive
gravity against $\ell$ for $\theta=0.01,m^{2}c_{2}=0.05$ and
$\alpha_{eff}=6.25\times 10^{-8}$ for three distinct values of
$M/\sqrt{\theta}.$

In Fig. 8, quasinormal frequencies as function of angular index $\ell$ are
displayed with three different values of $M/\sqrt{\theta}$. The angular index
$\ell$ affects the $Re(\omega)$ much more significantly than the $Im(\omega)$.
The real part increases monotonically with $\ell$. In contrast, the imaginary
part narrowly increases with $\ell$. For $M/\sqrt{\theta}=1.9$, the increasing
of the $Im(\omega)$ is noticeable at small $\ell$. However as
$M/\sqrt{\theta}$ increases, this behaviour becomes less obvious. Lastly, with
smaller $M/\sqrt{\theta}$, the rate at which the $Re(\omega)$ changes with
$\ell$ is higher than those of large $M/\sqrt{\theta}$.

Figure 9: Quasinormal frequencies of noncommutative black hole in dRGT massive
gravity against $\alpha_{eff}$ for $\theta=0.01,m^{2}c_{2}=0.05$ and $\ell=5$
for four distinct values of $M/\sqrt{\theta}.$

In addition, we show quasinormal frequencies as function of effective scalar’s
mass $\alpha_{eff}$ in Fig. 9. The real part of $\omega$ is larger as
$\alpha_{eff}$ increases. Remarkably, the imaginary part becomes more negative
as $\alpha_{eff}$ is larger. This is in opposition to many previous results
that increasing scalar field mass tends to slower the decay Simone and Will
(1992); Konoplya (2002); Konoplya and Zhidenko (2005). It is important to
emphasize that $\alpha_{eff}$ is not an actual scalar field mass but rather
play a similar role as $\alpha^{2}$ in the effective potential (27). Moreover,
the presence of $\alpha_{eff}$ directly relates to the spacetime parameters
$C$ and $c_{1}$. Therefore, non-zero value of $\alpha_{eff}$ inevitably
affects the spacetime’s geometry.

###  IV.1 The eikonal limit

The WKB approximation technique is considerably more accurate in the eikonal
limit $\ell\to\infty$. Under this limit, the effective potential (27) is
reduced to the following form

$\displaystyle V_{eik}$ $\displaystyle=e^{2\phi}\frac{\ell^{2}}{r^{2}}.$ (29)

With the simplified effective potential, it is possible to express quasinormal
frequency in an analytic form Cardoso et al. (2009)

$\displaystyle\omega_{eik}$
$\displaystyle=\Omega\ell-i\left(n+\frac{1}{2}\right)|\lambda_{L}|,$ (30)

where the coordinate angular velocity $\Omega$ at the unstable null orbit and
the Lyapunov exponent $\lambda$ are given by

$\displaystyle\Omega$
$\displaystyle=\left.\sqrt{\frac{e^{2\phi}}{r^{2}}}\right|_{r=r_{0}},$ (31)
$\displaystyle\lambda_{L}$
$\displaystyle=\frac{1}{\sqrt{2}}\left.\sqrt{\frac{r^{2}}{e^{2\phi}}\frac{d^{2}}{dr_{\ast}^{2}}\left(\frac{e^{2\phi}}{r^{2}}\right)}\right|_{r=r_{0}},$
(32)

where $\Omega$ and $|\lambda_{L}|$ are evaluated at $r=r_{0}$ i.e., the peak
of the potential. More interestingly, there is a connection between the real
part of eikonal quaisnormal modes and black hole shadow radius Jusufi (2020a,
b, 2021). It is suggests that Jusufi (2020a)

$\displaystyle Re(\omega_{eik})$ $\displaystyle=\lim_{\ell\gg
1}\frac{\ell}{R_{s}},$ (33)

where $R_{s}$ is the shadow radius. We will come back to this point when
discussing optical property in the following section.

Figure 10: Coordinate angular velocity $\Omega$ (left) and the Lyapunov
exponent $|\lambda|$ (right) plotted against $M/\sqrt{\theta}$ for
$\theta=0.1$ with four chosen values of $\alpha_{eff}.$

Now let’s explore the behaviour of coordinate angular velocity and the
Lyapunov exponent under influence of $M/\sqrt{\theta}$ in Fig. 10. Clearly,
both $\Omega$ and $|\lambda|$ is smaller as the black hole mass increases.
Moreover, the higher the effective scalar field’s mass, the larger angular
velocity and the Lyapunov exponent.

##  V Optical Property: Shadow

This segment investigates the enigmatic shadows produced by black
holes—shadowy regions in space-time covering the event horizon of these
astronomical occurrences. Black holes are distinguished by an incredibly
potent gravitational force, so formidable that light cannot escape once it
crosses the event horizon Eslam Panah et al. (2020); Konoplya (2019).
Consequently, a mysterious and darkened circle emerges around the periphery of
the black hole, standing out against the backdrop of adjacent matter or light.
The size and configuration of this obscure silhouette offer valuable insights
into the characteristics of black holes and the essence of gravity. Recent
progress in astronomy and technology now enables the acquisition of images
depicting these elusive black hole shadows, markedly advancing our
comprehension of these puzzling celestial entities Gogoi et al. (2023c, a);
Gogoi and Ponglertsakul (2024).

To obtain the shadow of the black hole, we begin with the standard Euler-
Lagrange equation as given below:

$\frac{d}{d\tau}\\!\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)-\frac{\partial\mathcal{L}}{\partial
x^{\mu}}=0,$ (34)

where the term $\mathcal{L}$ represents the Lagrangian of the system and it
can be given by,

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\,g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}.$
(35)

In this investigation, we have considered a static and spherically symmetric
spacetime metric in the presence of noncommutative geometry and hence the
Lagrangian can be further written as

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\left[-h(r)\,\dot{t}^{2}+\frac{1}{h(r)}\,\dot{r}^{2}+r^{2}\left(\dot{\Theta}^{2}+\sin^{2}\Theta\dot{\phi}^{2}\right)\right].$
(36)

One may note that the dot over the variables in the above expression stands
for derivative with respect to the proper time $\tau$.

When we select the equatorial plane, where the polar angle $\Theta$ is fixed
at $\pi/2$, we can determine the conserved energy $\mathcal{E}$ and angular
momentum $L$ of a particle using the killing vectors $\partial/\partial\tau$
and $\partial/\partial\phi$. This yields the expressions

$\mathcal{E}=h(r)\,\dot{t},\quad L=r^{2}\dot{\phi},$ (37)

where $h(r)$ is nothing but the metric function of the black hole previously
denoted by $e^{2\phi}$. It is a function of the radial coordinate $r$. In the
above expression, $\dot{t}$ and $\dot{\phi}$ are the time and azimuthal
components of the particle’s four-velocity.

For photons, which travel along null geodesics in curved spacetime, the
geodesic equation on the equatorial plane takes the form

$-h(r)\,\dot{t}^{2}+\frac{\dot{r}^{2}}{h(r)}\,+r^{2}\dot{\phi}^{2}=0.$ (38)

By combining Eq. (38) with the conserved quantities $\mathcal{E}$ and $L$, we
can derive the orbital equation for photons which is given by Gogoi et al.
(2023c); Gogoi and Ponglertsakul (2024):

$\left(\frac{dr}{d\phi}\right)^{2}=V_{eff},$ (39)

where the effective potential $V_{eff}$ has the following explicit form:

$V_{eff}=r^{4}\left[\frac{\mathcal{E}^{2}}{L^{2}}-\frac{h(r)}{r^{2}}\right].$
(40)

This effective potential helps us to obtain the photon sphere position around
the black hole. To obtain the same, we first try to reduce the potential
expression defined above. In a more convenient radial form, using a reduced
form of potential, Eq. (39) becomes

$V_{r}(r)=\dfrac{1}{\xi^{2}}-\dot{r}^{2}/L^{2},$ (41)

where $\xi$ is the impact parameter defined as $\xi=L/\mathcal{E}$. The
reduced form of the potential is denoted by $V_{r}(r)$ and it has the
following explicit form:

$V_{r}(r)=\frac{h(r)}{r^{2}}.$ (42)

To study the shadows of black holes, we need to focus on a specific point
along the trajectory, denoted as $r_{ph}$, which corresponds to the turning
point. This point marks the location of the light ring that encircles the
black hole and is also the radius of the photon sphere, a key element in
analyzing the black hole’s shadow. At this turning point, the effective
potential $V_{eff}$ as well as $\dot{r}$ vanishes, indicating the critical
impact parameter $\xi_{crit}$ Pantig et al. (2022b); Övgün et al. (2018);
Papnoi et al. (2014); Övgün et al. (2020):

$\left.V_{eff}\right|_{r_{ph}}=0,$ (43)

which can be expressed as

$\frac{1}{\xi_{crit}^{2}}=\frac{h(r_{ph})}{r_{ph}^{2}}.$ (44)

One may note that this same expression also can be obtained by using
$\dot{r}=0$ at $r=r_{ph}$ in the equation of reduced potential Eq. (41).

The radius of the photon sphere can be obtained from

$\left.V^{\prime}_{eff}\right|_{r_{ph}}=0,$ (45)

where the prime represents derivative w.r.t. $r$. This condition can be
further simplified by using Eq. (43). Similarly, one can also use the
expression of reduced potential to obtain the photon sphere radius by
calculating the extremum position. It shows that the radius of the photon
sphere $r_{ph}$ can be determined by finding the extremum of the function
$\mathcal{A}(r)$:

$\left.\frac{d}{dr}\,\mathcal{A}(r)\right|_{r_{ph}}=0,$ (46)

which, in turn, can be rewritten as

$\frac{h^{\prime}(r_{ph})}{h(r_{ph})}-\frac{\mathcal{Q}^{\prime}(r_{ph})}{\mathcal{Q}(r_{ph})}=0,$
(47)

where $\mathcal{A}(r)=\mathcal{Q}(r)/h(r)$ and $\mathcal{Q}(r)=r^{2}$. This
critical radius $r_{ph}$ and its corresponding impact parameter provide
crucial insights into the nature of black hole shadows.

To determine the shadow of a black hole, we can rewrite Eq. (39) with Eq. (44)
in terms of the function $\mathcal{A}(r)$ as

$\left(\frac{dr}{d\phi}\right)^{\\!2}=\mathcal{Q}(r)h(r)\left(\frac{\mathcal{A}(r)}{\mathcal{A}(r_{ph})}-1\right).$
(48)

This equation allows us to calculate the shadow radius of the black hole. When
considering a scenario where a static observer is positioned at a distance
$r_{0}$ from the black hole, we can find the angle $\alpha$ between the light
rays emitted by the observer and the radial direction of the photon sphere.
This angle is given by

$\cot\alpha=\frac{1}{\sqrt{h(r)\mathcal{Q}(r)}}\left.\frac{dr}{d\phi}\right|_{r\,=\,r_{0}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$
(49)

Combining this with Eq. (48), we get

$\cot^{2}\\!\alpha=\frac{\mathcal{A}(r_{0})}{\mathcal{A}(r_{ph})}-1,$ (50)

which can be rewritten using the relation
$\sin^{2}\\!\alpha=1/(1+\cot^{2}\\!\alpha)$ as

$\sin^{2}\\!\alpha=\frac{\mathcal{A}(r_{ph})}{\mathcal{A}(r_{0})}.$ (51)

The above equation helps us determine the angular size of the black hole’s
shadow as seen by an observer at a distance $r_{0}$ from the black hole.

By substituting the expression for $\mathcal{A}(r_{ph})$ from Eq. (44) and
$\mathcal{A}(r_{0})=r_{0}^{2}/h(r_{0})$, the black hole’s shadow radius for a
static observer at $r_{0}$ is estimated as Gogoi et al. (2023c):

$R_{s}=r_{0}\sin\alpha=\sqrt{\frac{r_{ph}^{2}h(r_{0})}{h\left(r_{ph}\right)}}.$
(52)

In the limit of asymptotically flat spacetime as $r_{0}\rightarrow\infty$,
which corresponds to a static observer at a large distance where
$h(r_{0})\rightarrow 1$, the shadow radius $R_{s}$ simplifies to:

$R_{s}=\frac{r_{ph}}{\sqrt{h(r_{ph})}}.$ (53)

This expression gives the shadow radius of the black hole as observed by a
distant static observer for asymptotically flat spacetime. Clearly, the right
hand side of the above equation is the inverse of angular velocity i.e.,
$1/\Omega$. Hence, it is in agreement with (33) discussed earlier. However, it
is important to mention that the black hole solution being investigated in
this work is not asymptotically flat and hence with a variation in $r_{0}$,
even at a very far distance away, can have impacts on the black hole shadow.
Due to this nature, we have thoroughly investigated the impacts of $r_{0}$ on
the black hole shadow for different sets of the model parameters.

Figure 11: Stereographic projection of black hole shadow using
$M=1,c_{1}=0.5,c_{2}=0.15,m=0.1$ and $\theta=0.2$. On the first panel, we have
used $r_{0}=20$ and on the second panel $r_{0}=50$. The black dotted circle
represents Schwarzschild’s black hole shadow.

Figure 12: Stereographic projection of black hole shadow using
$M=1,C=0.5,c_{1}=0.35,c_{2}=0.2$ and $\theta=0.2$. On the first panel, we have
used $r_{0}=20$ and on the second panel $r_{0}=50$. The black dotted circle
represents Schwarzschild’s black hole shadow.

Figure 13: Stereographic projection of black hole shadow using
$M=1,C=0.5,c_{2}=0.15,m=0.1$ and $\theta=0.2$. On the first panel, we have
used $r_{0}=20$ and on the second panel $r_{0}=50$. The black dotted circle
represents Schwarzschild’s black hole shadow.

Figure 14: Stereographic projection of black hole shadow using
$M=1,C=0.5,c_{1}=0.35,m=0.1$ and $\theta=0.2$. On the first panel, we have
used $r_{0}=20$ and on the second panel $r_{0}=50$. The black dotted circle
represents Schwarzschild’s black hole shadow.

Figure 15: Stereographic projection of black hole shadow using
$C=0.5,c_{1}=0.35,c_{2}=0.2,m=0.3$ and $\theta=0.1$. On the left panel, we
have used $r_{0}=20$ and on the right panel $M=1$. The black dotted circle
represents Schwarzschild’s black hole shadow.

Figure 16: Stereographic projection of black hole shadow using
$C=0.5,c_{1}=0.35,c_{2}=0.1,m=0.2$ and $M=1$. On the left panel, we have used
$r_{0}=20$ and on the right panel $r_{0}=50$. The black dotted circle
represents Schwarzschild’s black hole shadow.  Figure 17: Variation of black
hole shadow w.r.t $\theta$ at observer distance $r_{0}=20$ using
$C=0.5,c_{1}=0.35,c_{2}=0.1,m=0.2$ and $M=1$.

In Fig. 11, we have shown the variation of black hole shadow for different
values of model parameter $C$. On the left panel, we choose observer distance
$r_{0}=20$ and on the right panel, $r_{0}=50$. One may observe that with an
increase in the value of the model parameter $C$, the size of the black hole
shadow increases slowly. This variation is, however, very small. The black
dotted circle in these figures represent the shadow radius of a standard
Schwarzschild black hole. It is also seen that an increase in $r_{0}$
increases the shadow size of the black hole and at the same time, the impact
of the model parameter also becomes more significant at larger $r_{0}$ values.
So, the impacts of such a model parameter can be effectively constrained using
observational data. In Fig. 12, we have shown the impacts on the black hole
shadow by the parameter $m$. Similar to the scenario of $C$, we observe an
increase in the shadow radius with an increase in the model parameter $m$
value. However, the impacts of $m$ on the black hole shadow are more
significant. Similar to the previous case, at far distance away from the black
hole, the impacts of $m$ are more clearly visible. In Fig. 13, we have shown
the effects of parameter $c_{1}$ on the black hole shadow. On the left panel,
we have used the observer at $r_{0}=20$ and on the right panel, we have used
the observer at $r_{0}=50$. One may note that, at a far distance away from the
black hole, the impact of the model parameter $c_{1}$ is more significant. In
both cases, with an increase in the model parameter $c_{1}$, we observe an
increase in the shadow size of the black hole. From Fig. 14, one may observe
that $c_{2}$ has a very small impact on the black hole shadow. The impact is
also almost negligible with a variation of the observer’s distance. Unlike the
previous scenarios, here, an increase in $c_{2}$ decreases the black hole
shadow size slowly. In Fig. 15, impacts of black hole mass $M$ and observer
distance $r_{0}$ on the black hole shadow are shown. Mass $M$ has a usual
impact on the shadow of the black hole. With an increase in the observer
distance $r_{0}$, the shadow size of the black hole increases significantly.
Impacts of the noncommutative parameter $\theta$ on the black hole shadow are
shown in Figs. 16 and 17. From Fig. 16, one may note that the impact of
$\theta$ on the black hole shadow is very small and it is also almost
negligible with a variation of the observer’s distance from the black hole.
Since noncommutative parameter is an important part of our investigation, we
further plot Fig. 17 to get a clearer idea of the impact of $\theta$ on the
shadow radius of the black hole. As predicted earlier, the impact of $\theta$
on the black hole shadow radius is found to be very small. As expected
$\theta$ impacts the black hole shadow nonlinearly and with an increase in
$\theta$, the black hole shadow radius decreases as shown in Fig. 17.

##  VI Concluding Remarks

In the context of de Rham-Gabadadze-Tolley-like massive gravity, we have
examined the properties and dynamics of the noncommutative black hole with
Gaussian distribution in this article. Through our investigation, we were able
to determine the relationship between the graviton mass m and the metric
potential, temperature, entropy, heat capacity, and free energy. To explain
various physical features of our model, we take into consideration a
particular set of values for the model parameters. Fig. 1 shows that, for a
particular set of model parameters, the non-commutative black hole has a
minimum mass ($M_{r_{H}}$) below which it does not exist. For a given BH mass
$M=M_{r_{H}}$ (where $M_{r_{H}}$ is the critical mass), there is one event
horizon, and for $M>M_{r_{H}}$, there are two event horizons. Notably, we
found that despite the various values of $\frac{M}{\sqrt{\theta}}$, we
observed for the particular Gaussian form of density, the minimal horizon is
always located at the distance $r=2.9\sqrt{\theta}$. Our research indicates
that the temperature of the non-commutative black hole disappears as the
horizon radius approaches the minimal horizon. The weak energy condition is
always satisfied for our model but for $r>2.9\sqrt{\theta}$, it is noteworthy
that the strong energy condition breaks down.

A linear scalar perturbation on the noncommutative black hole in dRGT massive
gravity is explored. We find that the model parameters $C,m$ and $c_{1}$
combined effectively as scalar field’s mass ($\alpha_{eff}$). We have seen
that the real part of frequency decreases with $M/\sqrt{\theta}$. In contrast,
the real part of frequency increases with $\ell$ and $\alpha_{eff}$. In
addition, the imaginary part of frequency becomes less negative as
$M/\sqrt{\theta}$ is larger. We have also observed that increasing $\ell$
leads to slightly increasing $Im(\omega)$. Remarkably, the $Im(\omega)$ is
more negative with increasing in $\alpha_{eff}$. Additionally, we investigate
the QNMs in the eikonal limit where the angular velocity $\Omega$ and the
Lyapunov exponent $\lambda$ are shown to be decreasing function with
$M/\sqrt{\theta}$.

We have also investigated the shadow of the noncommutative black hole in this
investigation. We have seen that the shadow radius has a dependency on the
observer’s distance from the black hole due to the non-flat asymptotic
behaviour of the black hole spacetime. Hence, we investigated the shadow of
the black hole with respect to the model parameters for different observer
distances. The impacts of the model parameters $C$, $m$ and $c_{1}$ are more
significant when the observer distance is large. The impacts of the other
model parameters are almost independent of the observer distance. The impact
of the noncommutative parameter $\theta$ on the black hole shadow is very
small. We found that with an increase in the value of $\theta$, shadow radius
$R_{s}$ decreases nonlinearly.

## Acknowledgments

P.B. is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, Government of India, for providing visiting associateship. DJG
acknowledges the contribution of the COST Action CA21136 – “Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse)”. SP acknowledges funding support from the NSRF via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation [grant number B39G670016].

## Data Availability Statement

There are no new data associated with this article.

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