# Deflection angle, quasinormal modes and optical properties of a de Sitter
black hole in $\bm{f(\mathcal{T},\mathcal{B})}$ gravity

Nashiba Parbin  0000-0002-3570-0117<EMAIL_ADDRESS>Department of
Physics, Dibrugarh University, Dibrugarh 786004, Assam, India  Dhruba Jyoti
Gogoi  0000-0002-4776-8506<EMAIL_ADDRESS>Department of Physics,
Dibrugarh University, Dibrugarh 786004, Assam, India  Jyatsnasree Bora
0000-0001-9751-5614<EMAIL_ADDRESS>Department of Physics,
Dibrugarh University, Dibrugarh 786004, Assam, India  Umananda Dev Goswami
0000-0003-0012-7549<EMAIL_ADDRESS>Department of Physics, Dibrugarh
University, Dibrugarh 786004, Assam, India

###### Abstract

The current study aims to examine the impact of the boundary term on the
bending angle of light for a static spherically symmetric black hole in the
modified gravity described by the $f(\mathcal{T},\mathcal{B})$ function. To
accomplish this objective, we employ the Ishihara et al. method, which enables
us to compute the deflection angle of light for a receiver and source situated
at finite distances from a lens object in a non-asymptotically flat spacetime.
This method considers the receiver’s viewpoint, and the resulting deflection
angle diverges as the distance from the lens object increases, owing to the
non-asymptotically flat spacetime. Nevertheless, the divergence can be
regulated by the boundary term parameter $c_{0}$. For lower values of the
parameter $c_{0}$, the divergence can be minimized within the finite range of
the observer and source. Furthermore, we calculate the quasinormal modes of
massless scalar perturbations in the black hole’s background using the
asymptotic iteration method (AIM) and Padé averaged sixth-order Wentzel-
Kramers-Brillouin (WKB) approximation method. Our findings indicate that the
real quasinormal modes and damping rates are significantly impacted by the
model parameter $c_{0}$. Subsequently, we investigate two optical
characteristics of the black hole, namely the shadow and the emission rate.
Our results show that with an increase in the boundary term parameter $c_{0}$,
both the shadow’s size and the evaporation rate decrease.

Modified gravity; Deflection angle; Quasinormal modes, Black hole shadow and
emission rate

##  I Introduction

The revolutionary idea of the theory of general relativity (GR) first struck
in the great mind of Albert Einstein, which he published in $1915$ einstein .
After the time of publication of this theory, it has gained utmost importance
in describing many gravitational phenomena. Among different predictions of GR,
black holes and gravitational waves (GWs) are two most significant ones.
However, it has shown some limitations while explaining the current
accelerated expansion of the universe reiss ; perlmutter  as well as the
observed rotational dynamics of the galaxies rubin ; young1 ; harko ; gergely
; parbin . GR needs some exotic stuff called dark energy and dark matter
bertone ; swart ; tim ; frenk ; strigari ; will2014  in order to overcome its
limitations. Thus, in order to represent the observed facts of the present
universe, it can be conjectured that either the universe is filled with
mysterious dark matter and dark energy, or a plausible modified theory of
gravity (MTG) should be used such that it can explain the correct outcomes
from observations. At present there are lots of proposed MTGs that are
introduced to resolve unexplained issues of the universe Clifton ; wagoner ;
ronald ; capozziello ; clifton ; oikonomou ; odintsov ; sergei ; gogoi1 ;
nashiba ; gogoi7 ; gogoi_cosmo .

The first experimental verification of GR was observed in $1919$. It was the
observation of the gravitational bending of light, during a solar eclipse
eddington  as per the prediction of GR. Moreover, from the interpretation of
gravitational bending of light in the framework of geometrical optics of a
lens, a key idea, the gravitational lensing has emerged schneider ; trimble ;
renn ; valls . It was in $1979$, when the first instance of gravitationally
lensed object, the double quasar Q$0957+561$A,B was identified for the first
time young . In the modern observational cosmology for exploring the
exoplanets and for measuring the distribution of dark matter and dark energy,
an important parameter is the angle of gravitational bending of light. For
different black holes, the gravitational lensing around them has been
investigated in a plethora of articles. For the case of Schwarzschild black
hole such investigation was carried out in the Ref. virbhadra . For black
holes in presence of the cosmological constant the studies of such lensing are
reported in Ref.s zhao ; rindler . Again, the gravitational lensing by the
naked singularity and horizonless ultracompact objects are studied in the
Ref.s ellis ; shaikh .

In 2008, an alternative method for the derivation of deflection angle of light
in a spherically symmetric black hole, introduced by Gibbons and Werner, by
implementing the Gauss-Bonnet theorem (GBT) carmo ; bonnet  was reported in
the Ref. gibbons . This method is found to be good to calculate the exact form
of the deflection angle in the weak field limit for Schwarzschild black hole
spacetime gibbons . Quite a few studies show that this approach can be used to
derive the deflection angle for different black hole solutions banerjee ;
jusufi ; kimet ; sakalli ; Kjusufi ; izzet ; ovgun1 ; ovgun2 ; ovgun3 ; anisur
; eslam ; panah . To obtain the deflection angle by a Kerr black hole the
Gibbons-Werner method was then extended using the Kerr-Randers optical
geometry werner ; saavedra . This method was further extended in $2016$ by
Ishihara et al. to consider finite distances between the source and the
observer for static as well as stationary black hole solutions ishihara . For
stationary black hole spacetimes, extension of this method is reported in the
Ref.s ono ; kumar ; ghosh ; jamil ; crisnejo . For the case of asymptotically
non-flat spacetimes such study can be found in the Ref.s asada ; takizawa ;
carvalho . Using different MTGs such studies on gravitational deflection angle
using GBT are reported in carvalho ; jamil ; mubasher . Using GBT the effect
of dark matter on deflection angle is also reported earlier in a few articles
pantig1 ; pantig2 ; javed ; ovgun4 .

In our work, we intend to explore the impact of the modification of spacetime
curvature on the gravitational lensing of a static spherically symmetric black
hole spacetime. For this purpose, we consider the black hole solution in the
$f(\mathcal{T},\mathcal{B})$ modified gravity theory 2015_Bahamonde . The
$f(\mathcal{T},\mathcal{B})$ modified gravity has been investigated in various
aspects 2017_Bahamonde ; 2018_Bahamonde ; 2020_cappozielo ; 2020_farrugia ;
2022_Bahamonde ; camci , but for the case of gravitational lensing, we study
for the first time the effects of $f(\mathcal{T},\mathcal{B})$ gravity on the
deflection angle of light for a static spherically symmetric black hole using
the GBT. Here we shall implement the Ishihara method for the asymptotically
non-flat spacetime to obtain the deflection angle.

In this work we also study the quasinormal modes rayimbaev ; ghasemi  of the
considered black hole spacetime. In general these are the modes of emission of
GWs from compact and massive perturbed objects in the universe and are
represented by some complex numbers 1970_Vishveshwara ; 1971_press ;
1975_Chandrasekhar ; chandrasekhar . The real part of these complex numbers is
related to the emission frequency, while the imaginary part is related to the
damping of quasinormal modes of GWs. In the last few years a number of authors
have investigated the properties of GWs and also quasinormal modes of black
holes in different MTGs 2008_ca ; gogoi1 ; 2017_liang ; 2021_oliveira ;
2021_gogoi ; 2018_graca ; 2022_gogoi ; gogoi2 ; gogoi4 ; gogoi5 ; gogoi6 .

Furthermore, we analyze the effect that the modification of curvature has on
the shadow cast by the black hole in $f(\mathcal{T},\mathcal{B})$ gravity. The
shadow is a characteristic feature of a black hole and is cast as a result of
the strong gravitational field of the black hole. The shape and size of the
shadow depend directly on the mass and angular momentum of the black hole.
Studies on the shadow cast by a black hole have been of current interest to
the scientific community ovgun5 ; wei ; cunha ; wang ; dastan ; jamil ; kumar
; 1s ; shnew02 ; shnew01 ; 2s ; 3s ; 4s ; 5s ; 6s ; 7s ; 8s ; 9s ; 10s ; 11s ;
12s ; 13s ; 14s ; 15s ; 16s ; 17s ; 18s ; 19s ; 102-1 ; 102-2 ; 102-3 ; 102-4
; 102-5 ; 102-6 ; 102-7 ; 102-8 ; 102-9 ; 102-10 ; 102-11 ; 102-12 ; 102-13 ;
102-14 ; 102-15 , more so after the successful release of the black hole
images. The studies of the shadow cast by black holes surrounded by exotic
dark matter have also been carried out in various literature haroon ; salucci
; hou ; konoplya . Also, the analysis of the shadow cast by a black hole can
aid in characterizing the various gravity theories 3s . Finally, we discuss
and analyze the behaviour of the emission rate of particles around the black
hole. It is another observable phenomena which has also gained interest and
has been studied in literature Wei2013 ; mashhoon ; page ; crispino ; higuchi
; oliviera ; carlos .

Our paper is organized as follows. In Sec. II, we briefly review the field
equations related to the $f(\mathcal{T},\mathcal{B})$ modified gravity theory.
In Sec. III, we derive the deflection angle in the spacetime of the static
black hole in $f(\mathcal{T},\mathcal{B})$ gravity applying Ishihara et al.
method for asymptotically non-flat spacetime. The quasinormal modes of the
black hole are studied in Sec. IV. In Sec. V, we have studied the optical
properties, namely, the shadow and emission rate of the black hole. Finally,
in Sec. VI, we summarize and conclude the results of our work. Throughout our
work, we have imposed the sign convention ($-,+,+,+$).

##  II $f(\mathcal{T},\mathcal{B})$ gravity and a static spherically symmetric
black hole solution

An alternative geometrical formulation of GR can be made through the
teleparallel equivalent of general relativity (TEGR). In other words the TEGR
is an analogous theory of GR as this theory asserts that GR can be formulated
from the tetrad fields as well as from the torsion tensor 2013_Aldrovandi ;
2013_Maluf . A generalization of the TEGR is the $f(\mathcal{T})$ gravity, a
function of the torsion scalar $\mathcal{T}$, which is obtained by the
contraction of the torsion tensor ${\mathcal{T}^{a}}_{\mu\nu}$ defined as
2015_Bahamonde ; 2013_Maluf

${\mathcal{T}^{a}}_{\mu\nu}=\partial_{\mu}{h^{a}}_{\nu}-\partial_{\nu}{h^{a}}_{\mu},$
(1)

where ${h^{a}}_{\mu}$ are the tetrad fields, related to the metric
$g_{\mu\nu}$ via the Minkowski metric $\eta_{ab}$ as

$g_{\mu\nu}={h^{a}}_{\mu}{h^{b}}_{\nu}\eta_{ab}.$ (2)

A similar relationship between the inverse metric $g^{\mu\nu}$ and the inverse
tetrads ${h_{a}}^{\mu}$ can be obtained. The conditions satisfied by the
tetrads and the inverse tetrads are

${h_{b}}^{\mu}{h^{a}}_{\mu}=\delta^{a}_{b},\;\;\;\;{h_{a}}^{\mu}{h^{a}}_{\nu}=\delta^{\mu}_{\nu}.$
(3)

A more generalization to the $f(\mathcal{T})$ gravity is done by introducing a
boundary term $\mathcal{B}$ related to the divergence of the torsion vector
$\mathcal{T}_{\mu}={\mathcal{T}^{\alpha}}_{\alpha\mu}$ as
$\mathcal{B}=\frac{2}{h}\,\partial_{\mu}(h\,\mathcal{T}^{\mu})$, where
$h=det({h^{a}}_{\mu})=\sqrt{-g}$ is the determinant of the tetrads
${h^{a}}_{\mu}$ with $g$ being determinant of the metric $g_{\mu\nu}$. This
theory was introduced in 2015_Bahamonde . The action of the
$f(\mathcal{T},\mathcal{B})$ gravity is

$S_{f(\mathcal{T},\mathcal{B})}=\int
d^{4}x\,h\\!\left[\frac{1}{2\kappa^{2}}f(\mathcal{T},\mathcal{B})+\mathcal{L}_{m}\right],$
(4)

with $\kappa^{2}=8\pi G$ and $\mathcal{L}_{m}$ being the Lagrangian density of
matter. The field equations of this theory can be obtained as 2015_Bahamonde ;
2022_Bahamonde

$\displaystyle\delta^{\lambda}_{\nu}\accentset{\scriptstyle\circ}{\square}f_{\mathcal{B}}-\accentset{\scriptstyle\circ}{\nabla}^{\lambda}\accentset{\scriptstyle\circ}{\nabla}_{\nu}f_{\mathcal{B}}+\frac{1}{2}\mathcal{B}f_{\mathcal{B}}\delta^{\lambda}_{\nu}+\Big{[}(\partial_{\mu}f_{\mathcal{B}})+(\partial_{\mu}f_{\mathcal{T}})\Big{]}{S_{\nu}}^{\mu\lambda}+h^{-1}{h^{a}}_{\nu}\partial_{\mu}\\!\left(h{S_{a}}^{\mu\lambda}\right)\\!f_{\mathcal{T}}$
$\displaystyle\quad\quad\quad\quad\quad\quad-
f_{\mathcal{T}}{\mathcal{T}^{\sigma}}_{\mu\nu}{S_{\sigma}}^{\lambda\mu}-\frac{1}{2}f\delta^{\lambda}_{\nu}=\kappa^{2}T_{\nu}^{\lambda}.$
(5)

In this expression the circle over the terms is used to denote the quantities
which are determined using the Levi-Civita connection and the subscripts below
the functional $f$ denote the respective derivatives. The term
$T_{\nu}^{\lambda}$ represents the energy-momentum tensor and the term
${S_{\nu}}^{\mu\lambda}$ is known as the superpotential tensor 2022_Bahamonde
defined as

${S_{\nu}}^{\mu\lambda}={K^{\mu\lambda}}_{\nu}-\delta^{\mu}_{\nu}{\mathcal{T}_{\sigma}}^{\sigma\lambda}+\delta^{\lambda}_{\nu}{\mathcal{T}_{\sigma}}^{\,\sigma\mu}=-{S_{\nu}}^{\lambda\mu},$
(6)

where ${{K^{\nu}}}_{\mu\lambda}$ is the contortion tensor as given by

${K^{\nu}}_{\mu\lambda}={\Gamma^{\nu}}_{\mu\lambda}-{\accentset{\scriptstyle\circ}{{\Gamma}}^{\nu}}_{\mu\lambda}=\dfrac{1}{2}\left({{\mathcal{T}_{\mu}}^{\nu}}_{\lambda}+{{\mathcal{T}_{\lambda}}^{\nu}}_{\mu}-{\mathcal{T}^{\nu}}_{\mu\lambda}\right).$
(7)

In Ref. 2022_Bahamonde  different exact and spherically symmetric perturbative
solutions of field Eqs. (5) have been obtained. The black hole solution that
we consider in our work is obtained by using the complex tetrad found in the
Weitzenböck gauge (${\omega^{A}}_{B\mu}=0$) krssak . This complex tetrad has
been uniquely derived in 2022_Bahamonde , and is given by

$h^{A}_{(2)\mu}=\begin{pmatrix}0&iB(r)&0&0\\\
iA(r)\sin\vartheta\cos\phi&0&-\chi
r\sin\phi&-r\chi\sin\vartheta\cos\vartheta\cos\phi\\\
iA(r)\sin\vartheta\cos\phi&0&\chi
r\cos\phi&-r\chi\sin\vartheta\cos\vartheta\sin\phi\\\
iA(r)\cos\vartheta&0&0&\chi r\sin^{2}\vartheta\end{pmatrix},\hskip
8.53581pt(\chi=\pm 1),$ (8)

where, $A(r)$ and $B(r)$ are the metric elements for a spherically symmetric
metric of the form

$ds^{2}=-A(r)\,dt^{2}+B(r)dr^{2}+r^{2}\,d\Omega^{2},$ (9)

where $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\\!\theta\,d\phi^{2}$.

Following the approach described in 2022_Bahamonde  and using the complex
tetrad (8) in the field equations (5) of $f(\mathcal{T},\mathcal{B})$ gravity
theory, the symmetric field equations for the complex tetrad are obtained as

$\begin{split}\kappa^{2}\rho&=-\frac{1}{2}f+\frac{2f_{\mathcal{T}}(rBA^{\prime}+A(B-rB^{\prime}))}{r^{2}AB^{3}}+\frac{f_{\mathcal{B}}(r(B(rA^{\prime\prime}+4A^{\prime})-rA^{\prime}B^{\prime})+2A(B-rB^{\prime}))}{r^{2}AB^{3}}\\\
&+\frac{B^{\prime}f_{\mathcal{B}}^{\prime}}{B^{3}}-\frac{f_{\mathcal{B}}^{\prime\prime}}{B^{2}}+\frac{2f_{\mathcal{T}}^{\prime}}{rB^{2}},\end{split}$
(10)
$\begin{split}\kappa^{2}p_{r}&=\frac{1}{2}f+\frac{(rA^{\prime}+2A)f_{\mathcal{B}}^{\prime}}{rAB^{2}}-\frac{2f_{\mathcal{T}}(2rA^{\prime}+A)}{r^{2}AB^{2}}+\frac{f_{\mathcal{B}}(r(rA^{\prime}B^{\prime}-B(rA^{\prime\prime}+4A^{\prime}))-2A(B-rB^{\prime}))}{r^{2}AB^{3}},\end{split}$
(11)
$\begin{split}\kappa^{2}p_{l}&=\frac{1}{2}f-\frac{(rA^{\prime}+A)f_{\mathcal{T}}^{\prime}}{rAB^{2}}+\frac{f_{\mathcal{B}}(r(rA^{\prime}B^{\prime}-B(rA^{\prime\prime}+4A^{\prime}))-2A(B-rB^{\prime}))}{r^{2}AB^{3}}\\\
&+\frac{f_{\mathcal{T}}(r(rA^{\prime}B^{\prime}-B(rA^{\prime\prime}+3A^{\prime}))-A(B-rB^{\prime}+B^{3}))}{r^{2}AB^{3}}-\frac{B^{\prime}f_{\mathcal{B}}^{\prime}}{B^{3}}+\frac{f_{\mathcal{B}}^{\prime\prime}}{B^{2}},\end{split}$
(12)

where, the primes represent derivation with respect to the radial coordinate
$r$, and $\rho$, $p_{r}$ and $p_{l}$ are the energy density, radial pressure
and lateral pressure of the fluid respectively.

Again, as reported in 2022_Bahamonde , for a functional form of the type
$f(\mathcal{T},\mathcal{B})=\kappa_{1}\mathcal{T}+F(\mathcal{B})$, Eqs. (10),
(11) and (12) were utilized to obtain

$-\frac{\kappa_{1}(r(B(rA^{\prime\prime}+A^{\prime})-rA^{\prime}B^{\prime})+A(rB^{\prime}+B^{3}-B))}{r^{2}AB^{3}}=0.$
(13)

Solving this equation led to the derivation of an exact solution which behaves
as Schwarzschild-de Sitter black hole solution. This black hole solution is of
our interest as we wish to study the phenomenon of gravitational lensing by a
black hole and its quasinormal modes in $f(\mathcal{T},\mathcal{B})$ gravity
theory. The Schwarzschild-de Sitter black hole solution is obtained by
considering $\kappa_{1}=1/2$ and relating the metric elements as
$A(r)=1/B(r)=f(r)$ and is given by

$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\,d\Omega^{2},$ (14)

where, the metric co-efficient is obtained as

$f(r)=1-\frac{2M}{r}-(\Lambda+c_{0}M)\,r^{2}.$ (15)

The functional form of the $f(\mathcal{T},\mathcal{B})$ gravity model
considered in this solution is 2022_Bahamonde

$f(\mathcal{T},\mathcal{B})=\frac{\mathcal{T}}{2}-\dfrac{8\,c_{0}}{3\sqrt{\mathcal{B}+18(Mc_{0}+\Lambda)}}-3\Lambda.$
(16)

This functional form of $f(\mathcal{T},\mathcal{B})$ is motivated from the
fact that it leads to the unique Schwarzschild-de Sitter solution (14). It can
be obtained by choosing $\kappa_{1}=1/2$ in the generalized functional form
$f(\mathcal{T,B})=\kappa_{1}\mathcal{T}+F(\mathcal{B})$, mentioned earlier,
and deducing the remaining part of this relation, i.e. $F(\mathcal{B})$, by
using Eq. (13) following the Ref. 2022_Bahamonde . The solution for the term
$F(\mathcal{B})$ is made in such a way that the model presented in Eq. (16)
leads to the Schwarzschild-de Sitter solution (14) (see 2022_Bahamonde  for
details). Here $c_{0}$ is a parameter of the model carrying the effect of the
boundary term $\mathcal{B}$. It is a dimensional parameter having the
dimension of Length${}^{\,-\,3}$. Thus the solution is found to have an
effective cosmological constant due to the contribution from the boundary term
as given by $\Lambda_{eff}=\Lambda+c_{0}M$. One can estimate an observational
limit on $c_{0}$ from the shadow of black holes. For example, considering the
black hole metric (14), we can have an expression of the shadow of an
asymptotically flat black hole for a static observer at large distance as
given by (details are discussed in the Section V(A))

$R_{s}=\frac{3M}{\sqrt{\frac{1}{3}-9M^{2}\left(c_{0}M+\Lambda\right)}}.$ (17)

Following Ref. sh01  and considering M$87^{*}$ black hole shadow, it is
possible to have the following cases,

$c_{0}\leq 1.11\times 10^{-49}-2.15\times
10^{-21}\Lambda\;\;\;\text{kpc}^{-3},$ (18)

and

$c_{0}\leq 1.59\times 10^{-49}-2.15\times
10^{-21}\Lambda\;\;\;\text{kpc}^{-3}.$ (19)

Although the tetrad used for the derivation of the exact black hole solution
is complex, however the torsion scalar $\mathcal{T}$ and the boundary term
$\mathcal{B}$ are real, and eventually, for the black hole solution (14), the
torsion scalar and the boundary term are obtained as

$\mathcal{T}=\frac{4}{r^{2}}+(\Lambda+c_{0}M)\left(\frac{16M}{r}-12\right),$
(20)
$\mathcal{B}=\frac{4}{r^{2}}+(\Lambda+c_{0}M)\left(\frac{40M}{r}-36\right).$
(21)

One important point to be noted here is that the black hole solution (14) is
not asymptotically flat because of the presence of $r^{2}$ term in the last
part of the metric function (15). In this work we shall use this black hole
solution to study the gravitational deflection of light and the quasinormal
modes as discussed in the following sections.

##  III Gravitational deflection of light

We follow the Ishihara et al. approach ishihara  to obtain the deflection
angle of light in the weak field limit of the non-asymptotically flat black
hole spacetime given by the solution (14). Here, as shown in Fig. 1, the black
hole works as a lens ($L$), which is at a finite distance from the source
($S$) and the receiver ($R$). In the equatorial plane ($\theta=\pi/2$), the
deflection angle can be expressed as ishihara ; ono

$\hat{\Theta}=\Psi_{R}-\Psi_{S}+\phi_{RS},$ (22)

where $\Psi_{R}$ and $\Psi_{S}$ are the angles of light that are measured with
respect to the lens at the positions of the receiver and the source
respectively. $\phi_{RS}=\phi_{R}-\phi_{S}$ is the separation angle between
the receiver and the source. Here, $\phi_{R}$ and $\phi_{S}$ are the
longitudes of the receiver and the source respectively.

Figure 1: Schematic figure for the quadrilateral
$\stackrel{{\scriptstyle\infty}}{{R}}\\!\square\\!\stackrel{{\scriptstyle\infty}}{{S}}$
embedded in a curved space takizawa .

Since the light rays follow the null geodesic for which $ds^{2}=0$, the metric
Eq. (14) can be rewritten for this geodesic as

$dt^{2}=\gamma_{ij}dx^{i}dx^{j}=\frac{1}{f(r)^{2}}\,dr^{2}+\frac{r^{2}}{f(r)}\,d\Omega^{2},$
(23)

where $\gamma_{ij}$ is usually known as the optical metric, which specifies a
3D Riemannian space. We will denote this space by $\mathcal{M}^{(3)}$, where a
ray of light is considered as a spatial curve. By using the optical metric
$\gamma_{ij}$, we can define the angles $\Psi_{R}$ and $\Psi_{S}$. The non-
vanishing components of this metric are

$\gamma_{rr}=\frac{1}{f(r)^{2}},\;\;\gamma_{\phi\phi}=\frac{r^{2}}{f(r)}.$
(24)

An important parameter in the study of gravitational deflection angle of light
is the impact parameter of light in the black hole spacetime. This is usually
defined as the ratio of the angular momentum ($L$) to the energy
($\mathcal{E}$) of photons, which are the constant of motion in the equatorial
plane of spacetime. For the spacetime of black hole (14) these two constants
of motion are $\mathcal{E}=f(r)\,\dot{t}$ and $L=r^{2}\,\dot{\phi}$, where the
over dot denotes the derivative with respect to the affine parameter $\lambda$
along the path of the light ray. Thus the impact parameter of a light ray is

$\xi\equiv\frac{L}{\mathcal{E}}=\frac{r^{2}}{f(r)}\frac{d\phi}{dt}.$ (25)

The unit radial vector from the center of the lens can be obtained as
$e_{rad}=(f(r),0)$, and the unit angular vector along the angular direction
can be found as $e_{ang}=(0,f(r)/r)$. Again, the components of the unit
tangent vector $\bm{K}\equiv d\bm{x}/dt$ along the light ray are obtained as
ishihara

$(K^{r},K^{\phi})=\frac{\xi f(r)}{r^{2}}\left(\frac{dr}{d\phi},1\right).$ (26)

Here $dr/d\phi$ gives the orbital path variation of the rays of light known as
the orbit equation, which can be expressed as

$\left(\frac{dr}{d\phi}\right)^{\\!2}=-\,r^{2}f(r)+\frac{r^{4}}{\xi^{2}}.$
(27)

Now, if $\Psi$ denotes the angle between the radial component of the tangent
vector and the radial vector, i.e. the angle of the light ray which is
measured from the radial direction, then we can write,

$\cos\Psi=\frac{\xi}{r^{2}}\frac{dr}{d\phi}.$ (28)

It gives,

$\sin\Psi=\frac{\xi\sqrt{f(r)}}{r}.$ (29)

Moreover, in a general form Eq. (27) can be expressed as

$\left(\frac{du}{d\phi}\right)^{2}=F(u),$ (30)

where we have considered a new variable $u=1/r$ and hence the function
$F(u)=-\,u^{2}\,f(u)+1/\xi^{2}$.

Figure 2: Schematic figure for Gauss-Bonnet theorem ishihara . The inner angle
is $\varepsilon_{a}$ and the jump angle is $\theta_{a}$ ($a=1,2,...,N$).

As mentioned earlier, we will use the GBT to calculate the deflection angle
$\hat{\Theta}$. The GBT has many formulations. The simplest one states that
the total Gaussian curvature of an embedded triangle can be expressed in terms
of the total geodesic curvature of the boundary and the jump angles at the
corners. Thus, mathematically this simplest version of the GBT can be
expressed as carmo

$\int\int_{T}\mathcal{K}dS+\sum^{N}_{a\,=\,1}\int_{\partial
T_{a}}\kappa_{g}dl+\sum^{N}_{a\,=\,1}\theta_{a}=2\pi,$ (31)

where $T$ is a two-dimensional orientable surface shown in Fig. 2. The
boundaries of the surface $\partial T_{a}$ ($a=1,2,...,N$) are differentiable
curves. $\theta_{a}$ are the jump angles between the curves. $\mathcal{K}$ is
the Gaussian curvature of the orientable surface $T$, $\kappa_{g}$ is the
geodesic curvature of $\partial T_{a}$, $dS$ is the infinitesimal area element
of the surface and $dl$ is the infinitesimal line element along the boundary.
Also, $dl>0$ for prograde motion of photons and for retrograde motion $dl<0$.
The sign of $dl$ is chosen to be consistent with the orientation of the
surface $T$.

At this stage, we should note that the quadrilateral
$\stackrel{{\scriptstyle\infty}}{{R}}\\!\square\\!\stackrel{{\scriptstyle\infty}}{{S}}$
shown in Fig. 1 is embedded in a curved space $\mathcal{M}^{(3)}$ and consists
of a spatial curve for a light ray from the source to the receiver, two
outgoing radial lines from $R$ and from $S$, and a circular arc segment
$C_{r}$ with the coordinate radius $r_{C}$ ($r_{C}\rightarrow\infty$). As
clear from the figure, within the asymptotically flat spacetime,
$\kappa_{g}\rightarrow 1/r_{C}$ and $dl\rightarrow r_{C}\,d\phi$ as
$r_{C}\rightarrow\infty$ gibbons . Hence, the deflection angle can be defined
in the domain
$\stackrel{{\scriptstyle\infty}}{{R}}\\!\square\\!\stackrel{{\scriptstyle\infty}}{{S}}$
as

$\hat{\Theta}=\Psi_{R}-\Psi_{S}+\phi_{RS}=-\int\int_{\stackrel{{\scriptstyle\infty}}{{R}}\square\stackrel{{\scriptstyle\infty}}{{S}}}\mathcal{K}\,dS$
(32)

The separation angle $\phi_{RS}$ for our system can be obtained by integrating
Eq. (30) as given by

$\phi_{RS}=2\int_{0}^{u_{0}}\frac{du}{\sqrt{F(u)}},$ (33)

where $u_{0}$ is the inverse of the distance of closest approach. As in the
Ishihara et al. method, if we consider that the source and the receiver are at
finite distances from each other, then deflection angle can be written as

$\hat{\Theta}=\Psi_{R}-\Psi_{S}+\int_{u_{R}}^{u_{0}}\frac{du}{\sqrt{F(u)}}+\int_{u_{S}}^{u_{0}}\frac{du}{\sqrt{F(u)}}.$
(34)

Again, for the metric (14) using Eq. (29), we obtain,

$\begin{split}\hskip 14.22636pt\Psi_{R}-\Psi_{S}&=\arcsin(\xi
u_{R})+\arcsin(\xi
u_{s})-\pi+\frac{\xi(\Lambda+c_{0}M)}{2}\left[\frac{u_{R}^{-1}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{u_{S}^{-1}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]+\xi
M\left[\frac{u_{R}}{\sqrt{1-\xi^{2}u_{R}^{2}}}\right.\\\
&\left.+\,\frac{u_{S}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]\\!+\frac{\xi
M\Lambda}{2}\left[\frac{1-2\xi^{2}u_{R}^{2}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{1-2\xi^{2}u_{S}^{2}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\frac{\xi
Mc_{0}\Lambda}{4}\left[\frac{u_{R}^{-3}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{u_{S}^{-3}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]\\\
&-\frac{\xi^{3}Mc_{0}\Lambda}{2}\left[\frac{u_{R}^{-1}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{u_{S}^{-1}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\mathcal{O}(M^{2},M\Lambda^{2},\Lambda^{2}).\end{split}$
(35)

Here, it can be seen that the expansion of $\Psi_{R}-\Psi_{S}$ becomes
divergent at $u_{R}\rightarrow 0$ and $u_{S}\rightarrow 0$ due to the fact
that our spacetime is non-asymptotically flat. Hence, this series Eq. (35)
must be used only within a certain limit of finite radius of convergence.
Moreover, for the metric (14) the function $F(u)$ can be expressed as

$F(u)=\frac{1}{\xi^{2}}-u^{2}+2Mu^{3}+(\Lambda+c_{0}M).$ (36)

Hence, the angle $\phi_{RS}$ for the metric (14) is obtained as

$\begin{split}\phi_{RS}&=\pi-\arcsin(\xi u_{R})-\arcsin(\xi
u_{S})+\frac{M}{\xi}\left[\frac{2-\xi^{2}u_{R}^{2}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{2-\xi^{2}u_{S}^{2}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]\\\\[2.0pt]
&+\frac{\xi^{3}(\Lambda+c_{0}M)}{2}\left[\frac{u_{R}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{u_{S}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]+\frac{\Lambda\xi
M}{2}\left[\frac{2-3\xi^{2}u_{R}^{2}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{2-3\xi^{2}u_{S}^{2}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]\\\\[2.0pt]
&-\frac{\Lambda\xi^{5}c_{0}M}{4}\left[\frac{3u_{R}-2\xi^{2}u_{R}^{3}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{3u_{S}-2\xi^{2}u_{S}^{3}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\mathcal{O}(M^{2},M\Lambda^{2},\Lambda^{2}).\end{split}$
(37)

Using Eqs. (35) and (37), we finally get the deflection angle of light for our
considered black hole as

$\begin{split}\hat{\Theta}&=\frac{\xi(\Lambda+c_{0}M)}{2}\left[\frac{1+\xi^{2}u_{R}^{2}}{u_{R}\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{1+\xi^{2}u_{S}^{2}}{u_{S}\sqrt{1-\xi^{2}u_{S}^{2}}}\right]+\xi
M\left[\frac{u_{R}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{u_{S}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]\\\
&\;\;\;+\frac{\xi
M\Lambda}{2}\left[\frac{3-5\xi^{2}u_{R}^{2}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{3-5\xi^{2}u_{S}^{2}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\frac{\xi
c_{0}M\Lambda}{4}\left[\frac{u_{R}^{-3}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{u_{S}^{-3}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]\\\
&\;\;\;-\frac{\xi^{3}c_{0}M\Lambda}{2}\left[\frac{u_{R}^{-1}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{u_{S}^{-1}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\frac{M}{\xi}\left[\frac{2-\xi^{2}u_{R}^{2}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{2-\xi^{2}u_{S}^{2}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right]\\\
&\;\;\;-\frac{\xi^{5}c_{0}M\Lambda}{4}\left[\frac{3u_{R}-2\xi^{2}u_{R}^{3}}{(1-\xi^{2}u_{R}^{2})^{3/2}}+\frac{3u_{S}-2\xi^{2}u_{S}^{3}}{(1-\xi^{2}u_{S}^{2})^{3/2}}\right]+\mathcal{O}(M^{2},M\Lambda^{2},\Lambda^{2}).\end{split}$
(38)

By virtue of Eq. (35) few terms in the above expression may diverge in the far
distance limit, $u_{R}\rightarrow 0$, $u_{S}\rightarrow 0$. As mentioned
earlier, this is due to the reason that the spacetime we have considered here
is non-asymptotically flat, similar to the Kottler spacetime kottler  used by
Ishiahara et al. ishihara . As discussed in ishihara , we can state that this
divergence in the deflection angle in the far distance limit does not matter
as the limit $u_{R}\rightarrow 0$, $u_{S}\rightarrow 0$ is not applicable for
astronomical observations. Also, the effect of the boundary term contribution
$\Lambda_{eff}=\Lambda+c_{0}M$ on the deflection angle can be seen in Eq.
(38). It is observed that the deflection angle will increase with an increase
in the effective cosmological constant term. Thus, the boundary term coming
from $f(\mathcal{T},\mathcal{B})$ gravity has significant effect on the
deflection angle. Further, from Eq. (38) for $\Lambda=c_{0}=0$, we can arrive
at

$\hat{\Theta}\simeq\frac{M}{\xi}\left[\frac{2-\xi^{2}u_{R}^{2}}{\sqrt{1-\xi^{2}u_{R}^{2}}}+\frac{2-\xi^{2}u_{S}^{2}}{\sqrt{1-\xi^{2}u_{S}^{2}}}\right],$
(39)

which at the far distance limit ($u_{R}\rightarrow 0$, $u_{S}\rightarrow 0$),
reduces to the deflection angle in the Schwarzschild case,

$\hat{\Theta}\simeq\frac{4M}{\xi}.$ (40)

Figure 3: Deflection angle as a function of impact parameter for the case of
M$87^{*}$ black hole. In the first panel, we have considered $r_{s}=1$ Mpc
with three positive values of $c_{0}=0.01,0.001,0.0001$ in unit of kpc-3 which
are shown by the solid curves, and three negative values of
$c_{0}=-0.0001,-0.001,-0.01$ in the same unit which are represented by the
dashed curves. In the second and the last panel, we have chosen $r_{s}=1$ Mpc,
$10$ kpc, $1$ kpc with $c_{0}=0.0001$ kpc-3 and $c_{0}=-0.001$ kpc-3
respectively.

Now, as an example we will explore the deflection angle for M$87^{*}$, the
central black hole candidate of M$87$ eht ; akiyama  in the light of the above
analysis. For this purpose we consider different values of the distance from
the source $r_{s}$ and also different values of the model parameter $c_{0}$.
The mass of this black hole is $\sim 6.5\times 10^{9}M_{\odot}\sim 10^{13}$ m.
The distance of the receiver from M$87^{*}$ is its distance from the earth,
which is $\sim 16$ Mpc $\sim 10^{23}$ m, and the recent cosmological data
suggest the cosmological constant to be equal to $\Lambda=10^{-52}$ m-2 planck
. In Fig. 3, we plot the deflection angle as a function of the impact
parameter, and compare our results with that of the Schwarzschild case. In the
first panel of this figure, we have chosen a particular value of $r_{s}=1$
Mpc. In this plot the solid curves depict the behaviour of the deflection
angle for three positive values of the model parameter
$c_{0}=0.01,0.001,0.0001$ and the dashed curves show the same for three
negative values of the model parameter $c_{0}=-0.0001,-0.001,-0.01$. All
values of $c_{0}$ in this and rest of the calculations are used in the unit of
kpc-3. It can be seen that for $c_{0}=0.01$, the deflection angle decreases
upto a certain value of the impact parameter and then abruptly increases. As
the value of $c_{0}$ decreases, the deflection angle decreases with the impact
parameter and increases slowly after a certain point. For $c_{0}=0.0001$, the
curve almost overlaps with the Schwarzschild case upto a certain value of the
impact parameter and then slowly increases. For the negative values of the
model parameter, it is seen that the deflection angle decreases and becomes
negative for a certain value of the impact parameter. For $c_{0}=-0.0001$, the
curve almost overlaps with the Schwarzschild case and then slowly becomes
negative. This negative deflection angle signifies the behaviour of light
around the black hole considered in our study. It can be said from our results
that at certain impact parameter values, for $c_{0}<0$ cases, the photons get
repelled leading to negative deflection of light. This can be comprehended as
the repulsive gravitational effect of the teleparallel gravity theory
bahamonde_rev ; hanafy . This type of negative deflection angle has been found
in various studies drgt ; nakashi ; kitamura ; hagiwara ; izumi ; kar . In the
second and the last panel, we plot the deflection angle as a function of the
impact parameter for $c_{0}=0.0001$ and $c_{0}=-0.001$ respectively, with
three values of the distance from the source $r_{s}=1$ Mpc, $10$ kpc, $1$ kpc.
It is seen that if the source (say, a cluster of galaxies) is at a distance of
$1$ kpc, the deflection angle decreases till a certain point and then abruptly
diverges for both the cases of $c_{0}=0.0001$ and $c_{0}=-0.001$. However, for
$c_{0}=0.0001$ the divergent behaviour is seen at a higher deflection angle
than for $c_{0}=-0.001$. If we consider a galaxy cluster further away, say at
$10$ kpc, there is a change in the way the deflection angle diverges for
$c_{0}=0.0001$. Upto a certain value of the impact parameter, the deflection
angle decreases in a way similar to the Schwarzschild case, but slowly becomes
divergent after a particular value of the impact parameter. On the other hand,
for $c_{0}=-0.001$, the deflection angle decreases and eventually becomes
negative after a certain value of the impact parameter. For both the values of
$c_{0}$, such similarities are also seen if we consider a cluster of galaxies
at a distance of $1$ Mpc. Thus, the divergent behaviour of the deflection
angle is observed at low impact parameter value when the galaxy cluster is
nearer to the lens.

##  IV Quasinormal modes

In this section, we will address the massless scalar perturbation in the
spacetime of the black hole. We will assume that the test field exerts
negligible influence on the black hole spacetime. To determine the quasinormal
modes, we will derive Schrödinger-like wave equations taking into account the
corresponding conservation relations of the concerned spacetime, which should
be of Klein-Gordon type for the case of a scalar field. Two different methods,
viz. the asymptotic iteration method (AIM) and the Padé averaged 6th order WKB
approximation method, will be used to calculate the quasinormal modes. In this
regard considering only the axial perturbations, we can express the perturbed
metric as presented in lopez2020 :

$ds^{2}=-\,|g_{tt}|\,dt^{2}+r^{2}\sin^{2}\\!\theta\,(d\phi-p_{1}dt-p_{2}dr-
p_{3}d\theta)^{2}+g_{rr}\,dr^{2}+r^{2}d\theta^{2},$ (41)

where $p_{1}$, $p_{2}$ and $p_{3}$ define the perturbation introduced to the
black hole spacetime and are functions of $t$, $r$ and $\theta$. The metric
functions $g_{tt}$ and $g_{rr}$ represent the zeroth order terms and hence are
only functions of $r$.

###  IV.1 Scalar Perturbation

We consider a massless scalar field near the previously established black
hole. As it is considered that the effect of scalar field on the black hole
spacetime is minimal, the perturbed metric Eq. (41) in this case can be
expressed as

$ds^{2}=-\,|g_{tt}|\,dt^{2}+g_{rr}\,dr^{2}+r^{2}d\Omega^{2}.$ (42)

Now, for this case, it is feasible to write the Klein-Gordon equation in
curved spacetime as

$\square\Phi=\dfrac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi)=0.$
(43)

With the help of this Eq. (43) the quasinormal modes associated with the
scalar perturbation can be described. For this purpose we decompose the scalar
field $\Phi$ as follows:

$\Phi(t,r,\theta,\phi)=\dfrac{1}{r}\sum_{l,m}\psi_{l}(t,r)Y_{lm}(\theta,\phi).$
(44)

In this equation, $Y_{lm}$ is the spherical harmonics with $l$ and $m$ are the
usual indices associated with it, $\psi_{l}(t,r)$ is the radial time-dependent
wave function. Using Eqs. (43) and (44) one can obtain the radial wave
equation as

$\partial^{2}_{r_{*}}\psi_{l}(r_{*})+\omega^{2}\psi_{l}(r_{*})=V(r)\psi_{l}(r_{*}).$
(45)

Here, in this expression $r_{*}$ is defined as

$\dfrac{dr_{*}}{dr}=\sqrt{g_{rr}\,|g_{tt}^{-1}|}$ (46)

and is known as the tortoise coordinate. The term $V(r)$ represents the
effective potential, whose explicit form is

$V(r)=|g_{tt}|\left(\dfrac{l(l+1)}{r^{2}}+\dfrac{1}{r\sqrt{|g_{tt}|g_{rr}}}\dfrac{d}{dr}\sqrt{|g_{tt}|g_{rr}^{-1}}\right),$
(47)

here the term $l$ represents the multipole moment of the black hole’s
quasinormal modes. In the present work, we will compute the quasinormal modes
of the scalar perturbation of the black hole specified by the metric (14)
using this potential expression.

###  IV.2 The asymptotic iteration method

The AIM is an influential mathematical tool employed to solve differential
equations numerically, especially to those that are intractable by analytical
means. A critical area where AIM is particularly useful to apply is the
quasinormal modes of black holes and other systems featuring a potential
barrier AIM1 ; AIM2 ; AIM3 ; AIM4 . Quasinormal modes refer to the
characteristic oscillations exhibited by a system after a disturbance and are
instrumental in analyzing the stability and attributes of black holes. AIM
utilizes a systematic iteration approach that facilitates the derivation of
precise approximations to the quasinormal modes by converting the initial
differential equation into a series of simpler equations that are readily
solvable. The technique has proved successful in various physical systems and
remains an active area of exploration.

With the previous definition $u=1/r$ and following Ref. AIM1 , we obtain the
master wave equation for our case as given by

$\frac{d^{2}\psi}{du^{2}}+\frac{\mathcal{Z}^{\prime}}{\mathcal{Z}}\frac{d\psi}{du}+\left[\frac{\omega^{2}-\mathcal{Z}\left(-\frac{2\left(c_{0}M+\Lambda\right)}{u^{2}}+l(l+1)+2Mu\right)}{\mathcal{Z}^{2}}\right]\\!\psi=0,$
(48)

where the parameter $\mathcal{Z}$ is explicitly given by

$\mathcal{Z}=-M\left(c_{0}+2u^{3}\right)-\Lambda+u^{2}.$ (49)

Now, one needs to scale out the divergent characteristics of quasinormal modes
at the cosmological horizon, which can be done by defining the wave function
$\psi(u)$ as

$\psi(u)=e^{i\omega r_{*}}\,\mathcal{G}(u),$ (50)

which gives us the privilege to have Eq. (48) as

$\mathcal{Z}\mathcal{G}^{\prime\prime}+(\mathcal{Z}^{\prime}-2i\omega)\mathcal{G}^{\prime}-\left[-\frac{2\left(c_{0}M+\Lambda\right)}{u^{2}}+l(l+1)+2Mu\right]\mathcal{G}=0.$
(51)

Again, the correct quasinormal condition at the black hole horizon $u_{1}$
results,

$\mathcal{G}(r_{*})=(u-u_{1})^{-\frac{i\omega}{\kappa_{1}}}\chi(r_{*}),$ (52)

where $\kappa_{1}$ is given by

$\kappa_{1}=\left.\frac{1}{2}\frac{df}{dr}\right|_{r\to
r_{1}}\\!\\!\\!\\!=Mu_{1}^{2}-\frac{c_{0}M+\Lambda}{u_{1}}\,.$ (53)

In the above expressions, $u_{1}=1/r_{1}$, where $r_{1}$ is the event horizon
radius of the black hole. With the new function $\chi(r_{*})$, the Eq. (51)
takes the conventional format to be used in the iterations of AIM for the
differential equation as

$\displaystyle\chi^{\prime\prime}$ $\displaystyle=$
$\displaystyle\lambda_{0}(u)\chi^{\prime}+s_{0}(u)\chi\,,$ (54)

where the parameters $\lambda_{0}$ and $s_{0}$ are defined as

$\displaystyle\lambda_{0}(u)$
$\displaystyle=-\frac{1}{\mathcal{Z}}\left[\mathcal{Z}^{\prime}-\frac{2i\omega}{\kappa_{1}(u-u_{1})}-2i\omega\right],$
(55) $\displaystyle s_{0}(u)$
$\displaystyle=\frac{1}{\mathcal{Z}}\left[l(l+1)+2Mu-\frac{2\left(c_{0}M+\Lambda\right)}{u^{2}}\
+\frac{i\omega}{\kappa_{1}(u-u_{1})^{2}}\,\Big{(}\frac{i\omega}{\kappa_{1}}+1\Big{)}+(\mathcal{Z}^{\prime}-2i\omega)\,\frac{i\omega}{\kappa_{1}(u-u_{1})}\right].$
(56)

We shall use this differential equation and follow the Ref. AIM1  to calculate
the scalar quasinormal modes for the case of our black hole.

###  IV.3 The Padé averaged WKB approximation method

Besides the AIM, in this study we use the Padé averaged sixth order WKB
approximation method to calculate the quasinormal modes of black hole defined
by the metric (14), as mentioned earlier. In this sixth order WKB method, the
expression of oscillation frequency $\omega$ of GWs can be given by

$\omega=\sqrt{-\,i\left[(n+1/2)+\sum_{k=2}^{6}\bar{\Lambda}_{k}\right]\sqrt{-2V_{0}^{\prime\prime}}+V_{0}}\,,$
(57)

where $n=0,1,2\ldots$, $V_{0}=V(r)|_{r\,=\,r_{max}}$ and
$V_{0}^{\prime\prime}=\dfrac{d^{2}V}{dr^{2}}|_{r\,=\,r_{max}}$. Here $r_{max}$
is the position at which the potential $V(r)$ has its maximum value.
$\bar{\Lambda}_{k}$ are the correction terms and the explicit forms of these
correction terms, as well as the Padé averaging recipe, can be found in Ref.s
Schutz ; Will_wkb ; Konoplya_wkb ; Maty_wkb .

Table 1: Quasinormal modes from the black hole defined by the metric (14) for different values of the multipole moment $l$ with the overtone number $n=0$ obtained by using the AIM (with $91$ iterations) and the $6$th order Padé averaged WKB approximation method. In this calculation we choose the model parameters $c_{0}=-0.01$ and $\Lambda=0.002$. Here we use $M=G=c=\hbar=1$ unit system. $l$ | AIM | Padé averaged WKB | $\Delta_{rms}$ | $\Delta_{6}$ | $\Delta_{m}$
---|---|---|---|---|---
$1$ | $0.3271473-0.1000942i$ | $0.329269-0.102651i$ | $0.00743811$ | $0.000107719$ | $0.97115\%$
$2$ | $0.5377497-0.1048303i$ | $0.538571-0.104694i$ | $0.00234137$ | $0.0000262313$ | $0.15196\%$
$3$ | $0.7481478-0.1053878i$ | $0.747713-0.105344i$ | $0.000294378$ | $5.808299\times 10^{-6}$ | $0.05784\%$
$4$ | $0.9592543-0.1056549i$ | $0.959321-0.105657i$ | $8.032384\times 10^{-6}$ | $2.580773\times 10^{-6}$ | $0.00691\%$
$5$ | $1.1707337-0.1057988i$ | $1.171000-0.105793i$ | $0.000233917$ | $1.280549\times 10^{-6}$ | $0.02266\%$

In Table 1, we list the quasinormal modes for different values of the
multipole moment $l$ with overtone number $n=0$. In the second column, the
quasinormal modes obtained from the AIM with $91$ iterations are listed and in
the third column, the quasinormal modes obtained from the 6th-order Padé
averaged WKB approximation method are shown. In this table, $\Delta_{rms}$
represents the rms error associated with the Padé averaged $6$th order WKB
approximation method and $\Delta_{6}$ provides a measurement of the error from
two nearby approximation orders defined as

$\Delta_{6}=\dfrac{|\omega_{7}-\omega_{5}|}{2},$ (58)

where $\omega_{5}$ and $\omega_{7}$ respectively represent the quasinormal
modes calculated by using the Padé averaged $5$th order and $7$th order WKB
approximation methods. In the last column, $\Delta_{m}$ represents the
percentage deviation of quasinormal modes calculated by using WKB method from
those calculated by using AIM. One can see that with an increase in the
multipole moment $l$, the error associated with the quasinormal modes
decreases. It is also seen that the AIM and the Padé averaged 6th-order WKB
approximation method provide very close results, that is the quasinormal modes
from both methods are in good agreement with each other. Moreover, the
agreement between these two methods becomes far better with the increasing $l$
values. It is to be noted that as a characteristic of the WKB approximation
method, it fails to provide significant results when the overtone number $n$
is greater than the multipole moment $l$ 2021_gogoi ; 2022_gogoi ; gogoi5 .
Hence, for smaller values of $l$, it seems that AIM provides more accurate
results than those obtained from the WKB method. One may further note that the
quasinormal frequencies shown in Table 1 are in geometric units with $M=1$. To
convert them to physical units, we can use the following conversion formula
Ferrari :

$f=\dfrac{32.26}{\eta}\,(M\omega_{R})\;\text{kHz},$ (59)

where $\eta=M/M_{\odot}$. As mentioned above, in the WKB approximation method,
the errors decrease with an increase in $l-n$ and for $n>l$ the method fails
to provide actual quasinormal frequencies with a reasonable accuracy
konoplya_new . Hence in the rest of the study, we shall consider $n=0$ and a
higher value of $l$ for a better accuracy of the results of quasinormal
frequency calculations.

Figure 4: Variation of real (on the left panel) and imaginary (on the right
panel) scalar quasinormal mode frequencies with the model parameter $c_{0}$
associated with the black hole metric (14) obtained by using $n=0,l=4$ and
$\Lambda=0.002$. Solid circles in both plots denote quasinormal modes for
asymptotically flat Schwarzschild black hole. Parameters are expressed in mass
units i.e. $M=1$.

In Fig. 4, we plot the real quasinormal frequencies on the left panel and the
imaginary quasinormal frequencies on the right panel with respect to the model
parameter $c_{0}$. For both the plots, we use the mass of the black hole
$M=1$, overtone number $n=0$, and multipole moment $l=4$. One can see that
with an increase in the value of $c_{0}$, the quasinormal frequency decreases
non-linearly and approaches zero towards $c_{0}=0.04$. However, beyond
$c_{0}=0.04$, oscillation frequencies of quasinormal modes or ringdown GWs
start to increase very slowly. So, it seems that the positive values of the
model parameter $c_{0}$ permit ringdown GWs of very large wavelengths, which
will be difficult to detect experimentally. Similarly, the decay rate or the
damping rate of the quasinormal modes also decreases non-linearly with an
increase in the model parameter $c_{0}$. Near $c_{0}=0.04$, the decay rate
also becomes very close to zero. However, beyond this point, the decay rate
increases drastically up to $c_{0}=0.1$, representing highly damped GWs.

###  IV.4 Time domain profiles

In the previous two subsections, we numerically calculated the quasinormal
modes and studied their behaviour with respect to the model parameter $c_{0}$.
In this subsection, we shall deal with the time domain profiles of the scalar
perturbation of the black spacetime. To obtain the time evolution profiles, we
shall implement the time domain integration formalism gundlach . For this
purpose, we define the wavefunction and potential as
$\psi(r_{*},t)=\psi(i\Delta r_{*},j\Delta t)=\psi_{i,j}$ and
$V(r(r_{*}))=V(r_{*},t)=V_{i,j}$. With these definitions we can express radial
wave Eq. (45) as

$\dfrac{\psi_{i+1,j}-2\psi_{i,j}+\psi_{i-1,j}}{\Delta
r_{*}^{2}}-\dfrac{\psi_{i,j+1}-2\psi_{i,j}+\psi_{i,j-1}}{\Delta
t^{2}}-V_{i}\psi_{i,j}=0.$ (60)

Now, we set the initial conditions
$\psi(r_{*},t)=\exp\left[-\,\dfrac{(r_{*}-k_{1})^{2}}{2\sigma^{2}}\right]$ and
$\psi(r_{*},t)|_{t<0}=0$, where $k_{1}$ and $\sigma$ are the median and width
of the initial wave-packet, and then calculate the time evolution of the
scalar field as

$\psi_{i,j+1}=-\,\psi_{i,j-1}+\left(\dfrac{\Delta t}{\Delta
r_{*}}\right)^{\\!2}\\!(\psi_{i+1,j+\psi_{i-1,j}})+\left(2-2\left(\dfrac{\Delta
t}{\Delta r_{*}}\right)^{\\!2}-V_{i}\Delta t^{2}\right)\psi_{i,j}.$ (61)

Using the above iteration scheme and choosing a fixed value of $\frac{\Delta
t}{\Delta r_{*}}$, one can easily obtain the profile of $\psi$ with respect to
time $t$. However, one should keep $\frac{\Delta t}{\Delta r_{*}}<1$ so that
the Von Neumann stability condition is satisfied during the numerical
procedure.

Figure 5: Time domain profiles with $n=0$ and $\Lambda=0.002$ for the massless
scalar perturbation. On the left panel, we use $l=2$ and on the right panel,
$c_{0}=0.005$ is used.

Fig. 5 shows the time evolution profiles of the massless scalar field
perturbation in the black hole spacetime (14). On the left panel, the time
domain profiles are shown for different values of the model parameter $c_{0}$.
One can see that with an increase in the value of $c_{0}$ from $0.001$ to
$0.005$, the oscillation frequencies decrease slowly. On the right panel, we
show the variation of the time domain profiles with different values of the
multipole moment $l$. It is clear from the variation of the time domain
profiles that with an increase in the value of the multipole moment $l$, both
the oscillation frequency and the damping rate increase gradually. These
results agree well with the numerically calculated quasinormal modes in the
previous subsections.

##  V Optical behaviour of the black hole

###  V.1 Shadow

The shadow of a black hole is the dark area of spacetime that is surrounded by
the event horizon of the black hole. Black holes are known to have an intense
gravitational pull. The pull is so intense that once it passes the event
horizon, even light cannot escape the gravitational attraction eslam ; kumar .
As a result, the area of the black hole creates a shadowy dark zone on the
background of the nearby matter or light. The dimension and shape of this
shadow can provide vital insights into the characteristics of black holes and
the nature of gravity. Recent astronomical and technical developments have
made it possible to photograph black hole shadows, significantly advancing our
understanding of these mysterious objects wei .

For the case of static and spherically symmetric spacetime metric, the
Lagrangian of the form:

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\,g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$
(62)

can be written as shnew02 ; ronit23

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\left[-f(r)\,\dot{t}^{2}+\frac{1}{f(r)}\,\dot{r}^{2}+r^{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right)\right].$
(63)

Here, the derivative with respect to the proper time $\tau$ is indicated by
the dot over the variables. The corresponding Euler-Lagrange equation is

$\frac{d}{d\tau}\\!\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)-\frac{\partial\mathcal{L}}{\partial
x^{\mu}}=0.$ (64)

For the present case of the study, choosing the equatorial plane, i.e.
$\theta=\pi/2$, the conserved energy $\mathcal{E}$ and angular momentum $L$
can be obtained using killing vectors $\partial/\partial\tau$ and
$\partial/\partial\phi$ as ronit23

$\mathcal{E}=f(r)\,\dot{t},\quad L=r^{2}\dot{\phi}.$ (65)

The geodesic equation for the case of photon results in the relation,

$-f(r)\,\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}\,+r^{2}\dot{\phi}^{2}=0.$ (66)

In this Eq. (66), using the conserved quantities i.e., $\mathcal{E}$ and $L$
one can obtain the orbital equation of photon as given by shnew01

$\left(\frac{dr}{d\phi}\right)^{\\!2}=V_{eff},$ (67)

where we define the right hand side of Eq. (67) as an effective potential
$V_{eff}$, given by

$V_{eff}=r^{4}\left[\frac{\mathcal{E}^{2}}{L^{2}}-\frac{f(r)}{r^{2}}\right].$
(68)

Furthermore, by writing Eq. (67) in the form of a radial equation, one can
have

$V_{r}(r)=\dfrac{1}{\xi^{2}}-\dot{r}^{2}/L^{2},$ (69)

where the impact parameter $\xi$ is given as $\xi=L/\mathcal{E}$ and
$V_{r}(r)$ is the reduced potential having the form:

$V_{r}(r)=\frac{f(r)}{r^{2}}.$ (70)

The study of this potential’s behaviour with respect to the radial distance
$r$ would be the most practical way to comprehend the nature of the photon
sphere around the black hole spacetime we have taken into consideration. This
potential governs the radial motion of photons in the black hole spacetime. In
Fig. 6, we show the behaviour of this potential with respect to $r$. One can
see that the behaviour of the potential is not similar for positive and
negative values of the model parameter $c_{0}$. For negative values of the
parameter $c_{0}$, it is seen that the potential decreases slowly after
reaching a peak point. For smaller values of the parameter $c_{0}$, the
potential increases to its peak value very swiftly and the peak of the
potential increases gradually with a decreasing value of $c_{0}$. On the other
hand, for positive values of the parameter $c_{0}$, the peaks are distinct,
reach to them comparatively slowly and the potential decreases drastically
after reaching the peak with increasing values of $r$. In this case also, the
peak value of the potential increases for lower values of the parameter
$c_{0}$. Further, positive $c_{0}$ gives a lower peak value and negative
$c_{0}$ gives a higher peak value than that of the Schwarzschild case.

Figure 6: Variation of the reduced potential $V_{r}(r)$ with respect to $r$.
In these plots $M=1$ and $\Lambda=0.02$ are used.

Considering the turning point of the trajectory, given by $r=r_{ph}$, which
is, in fact, the radius of the photon sphere or the light ring surrounding the
black hole, one can determine the shadow of the black hole. At this turning
point, the following conditions must be satisfied 18 ; synge ; Luminet:1979nyg
:

$\left.\frac{dr}{d\phi}\right|_{r_{ph}}\\!\\!\\!\\!\\!\\!\\!=0\;\;\text{or}\;\;\left.V_{eff}\right|_{r_{ph}}\\!\\!\\!\\!=0,\;\;\text{and}\;\;\;\left.\frac{d^{2}r}{d\phi^{2}}\right|_{r_{ph}}\\!\\!\\!\\!\\!\\!\\!=0\;\;\text{or}\;\;\left.V_{eff}^{\prime}\right|_{r_{ph}}\\!\\!\\!\\!=0.$
(71)

Using the first condition, the impact parameter $\xi$ at the turning point can
be obtained as

$\frac{1}{\xi_{crit}^{2}}=\frac{f(r_{ph})}{r_{ph}^{2}}.$ (72)

The radius of the photon sphere $r_{ph}$ can be determined using the second
aforementioned condition and solving the equation:

$\left.\frac{d}{dr}\,\mathcal{A}(r)\right|_{r_{ph}}\\!\\!\\!\\!\\!\\!\\!=0.$
(73)

This equation can be explicitly written as

$\frac{f^{\prime}(r_{ph})}{f(r_{ph})}-\frac{h^{\prime}(r_{ph})}{h(r_{ph})}=0,$
(74)

where $\mathcal{A}(r)=h(r)/f(r)$ with $h(r)=r^{2}$. Thus from Eqs. (72) and
(74) it is clear that the critical impact parameter is
$\xi_{crit}=3\sqrt{3}M/\sqrt{-27c_{0}M^{3}-27\Lambda M^{2}+1}$ and the photon
sphere is located at $r_{ph}=3M$.

Now, to obtain the expression for shadow of the black hole, we rewrite Eq.
(67) with Eq. (72) in terms of the function $\mathcal{A}(r)$ as

$\left(\frac{dr}{d\phi}\right)^{\\!2}=h(r)f(r)\left(\frac{\mathcal{A}(r)}{\mathcal{A}(r_{ph})}-1\right).$
(75)

Using this Eq. (75), the shadow radius can be determined. To this end, if one
considers the angle between the light rays from a static observer at $r_{0}$
and the radial direction of the photon sphere as $\alpha$, then this angle can
be calculated as Perlick:2021aok ; 18

$\cot\alpha=\frac{1}{\sqrt{f(r)h(r)}}\left.\frac{dr}{d\phi}\right|_{r\,=\,r_{0}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$
(76)

Together with Eq. (75), above equation can be written as

$\cot^{2}\\!\alpha=\frac{\mathcal{A}(r_{0})}{\mathcal{A}(r_{ph})}-1.$ (77)

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

$\sin^{2}\\!\alpha=\frac{\mathcal{A}(r_{ph})}{\mathcal{A}(r_{0})}.$ (78)

Substitution of the actual form of $\mathcal{A}(r_{ph})$ from Eq. (72) and
$\mathcal{A}(r_{0})=r_{0}^{2}/f(r_{0})$ the black hole’s shadow radius for a
static observer at $r_{0}$ is estimated as 15s

$R_{s}=r_{0}\sin\alpha=\sqrt{\frac{r_{ph}^{2}f(r_{0})}{f\left(r_{ph}\right)}}.$
(79)

In case of an asymptotically flat black hole and for a static observer at
large distance, i.e. at $r_{0}\rightarrow\infty$, $f(r_{0})\rightarrow 1$. So
for such an observer the shadow radius $R_{s}$ of this type of black hole
becomes,

$R_{s}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}.$ (80)

The apparent form of the shadow of a black hole can be determined via the
stereographic projection of the shadow from the black hole’s plane to the
observer’s image plane with coordinates $(X,Y)$. These coordinates are defined
as shnew03 ; ronit23

$\displaystyle X$
$\displaystyle=\lim_{r_{0}\rightarrow\infty}\left(-\,r_{0}^{2}\sin\theta_{0}\left.\frac{d\phi}{dr}\right|_{r_{0}}\right),$
(81) $\displaystyle Y$
$\displaystyle=\lim_{r_{0}\rightarrow\infty}\left(r_{0}^{2}\left.\frac{d\theta}{dr}\right|_{(r_{0},\theta_{0})}\right),$
(82)

where $\theta_{0}$ represents the angular position of the observer with
respect to the plane of the black hole. Depending on the value of $c_{0}$, the
black hole spacetime can be de-Sitter (dS) or Anti de-Sitter (AdS) for a fixed
value of $\Lambda$. It is to be noted that dS solution gives rise to two
horizons viz., event horizon and cosmological horizon, whereas AdS has only
event horizon similar to the case of the Schwarzschild black hole. In general,
one may note that physical observers can be present between the event horizon
and cosmological horizon with different physically allowed observer distance
i.e. $r_{1}<r_{0}<r_{\Lambda}$, where $r_{1}$ and $r_{\Lambda}$ are the event
and cosmological horizons respectively. Hence a variation in $r_{0}$ can have
impacts on the appearance of the black hole shadow Perlick:2021aok . On the
other hand, for the asymptotically flat black holes with a static observer at
large distance scenario as mentioned above, the apparent form of the shadow
remains independent of the observer position. Therefore, in this
investigation, we use the shadow radius expression (79) to obtain the
stereographic projections of the black hole shadow at some finite physical
observer distance $r_{0}$.

Figure 7: Stereographic projections of the shadow of the black hole with
$\Lambda=0.002$. On the left panel, we use $c_{0}=0.001$ and $r_{0}=10$ and on
the right panel, $M=1$ and $r_{0}=15$. The black dotted circle in both cases
denote Schwarzschild black hole shadow radius at respective finite observer
distances.

Figure 8: Stereographic projections of the shadow of the black hole with
$\Lambda=0.002$ and $M=1$. On the left panel, we use $c_{0}=0.001$ (dS case)
and on the right panel, $c_{0}=-0.003$ (AdS case). The black dotted circle in
each plot denotes the Schwarzschild black hole shadow radius at infinite
observer distance (i.e. at $r_{0}\rightarrow\infty$).

For the black hole considered in this study, the stereographic projections of
the black hole shadow at finite observer distances are shown in Fig. 7. On the
left panel, we show the behaviour of the black hole shadow radius for
different values of the black hole mass $M$. In this case, we observe that the
shadow radius increases with an increase in the value of black hole mass $M$
at finite observer distance $r_{0}=10$. On the right panel, we show the
dependency of the black hole shadow with the model parameter $c_{0}$. With an
increase in the value of the parameter $c_{0}$, we observe a decrease in the
size of the black hole shadow at finite observer distance $r_{0}=15$. That is,
the size of the black hole shadow is bigger for the AdS ones than that of the
dS cases. Finally, in Fig. 8, we show the dependency of the shadow radius with
the observer distance $r_{0}$ for both dS and AdS black holes considering
$\Lambda=0.002$ and $M=1$. For the dS case (left panel) we choose the model
parameter $c_{0}=0.001$ for which the black hole can have two horizons at
$r=2.02491$ (event horizon) and $r=17.1606$ (cosmological horizon). The
physical observer distance should lie within this range. We consider
$r_{0}=10$ to $16$ to obtain the stereographic projections of the black hole
shadow. For comparison, we have also plotted the Schwarzschild black hole
shadow at infinite observer distance. It is clear from the figure that the
observer distance plays a crucial role in depicting the black hole shadow
radius. In this dS case, we see that with an increase in the observer distance
$r_{0}$, the shadow radius decreases and they are smaller than the
Schwarzschild black hole shadow. In the case of AdS black holes we take the
parameter $c_{0}=-0.003$ (right panel). It is seen that with an increase in
the distance $r_{0}$ of the observer the shadow size increases and is
comparable to that of the Schwarzschild black one, in contrast to the case of
dS black holes.

###  V.2 Emission rate

By examining the black hole shadow, it is possible to investigate the emission
of particles in the vicinity of the black hole. It has been demonstrated that
for an observer located far away, the black hole shadow is indicative of its
high-energy absorption cross-section Wei2013 . Generally speaking, in the case
of a spherically symmetric black hole, the absorption cross-section exhibits
oscillatory behaviour around a constant limiting value $\sigma_{lim}$ at
extremely high energies. This limiting value $\sigma_{lim}$ corresponds to the
geometrical cross-section of the photon sphere around the black hole. Since
the shadow provides a means of visually detecting a black hole, it is roughly
equivalent to the area of the photon sphere ($\sigma_{lim}\approx\pi
R_{s}^{2}$). Thus the energy emission rate can be calculated by using the
equation Wei2013 ,

$\frac{d^{2}E(\omega_{p})}{dt\,d\omega_{p}}=\frac{2\pi^{3}\omega_{p}^{3}R_{s}^{2}}{e^{\omega_{p}/T}-1},$
(83)

in which $\omega_{p}$ is the emission frequency, and $T$ is Hawking
temperature given by

$T=\dfrac{\hbar f^{\prime}(r_{h})}{4\pi},$ (84)

where $r_{h}$ is the horizon radius of the black hole.

Figure 9: Emission rate of the black hole with respect to emission frequency
$\omega_{p}$ for $M=1$, $\Lambda=0.002$ and observer distance $r_{0}=5$.

Using the expression of Hawking temperature (84) in Eq. (83), we show the
variation of the emission rate with respect to $\omega_{p}$ for different
values of $c_{0}$ in Fig. 9. One can see that with an increase in the value of
the parameter $c_{0}$, the emission rate of the black hole decreases. Hence
for smaller values of the parameter $c_{0}$, the black hole will evaporate
more rapidly. Moreover, it is to be noted that for $c_{0}<-0.01$ the emission
rate is greater than that of the Schwarzschild black hole, whereas it is less
than Schwarzschild one for the positive values of $c_{0}$. For the
$c_{0}=-0.01$ case, the rate is very close to the Schwarzschild black hole.
Here we consider the observer distance $r_{0}=5$, the cosmological constant
$\Lambda=0.002$ and the black hole mass $M=1$. Thus in this figure the cases
for $c_{0}\leq-0.01$ correspond to the AdS black holes situations, whereas
other values of $c_{0}$ correspond to the dS black holes.

##  VI Conclusion

In this work, we study a recently introduced static black hole solution in an
extension of modified teleparallel gravity i.e., $f(\mathcal{T},\mathcal{B})$
modified gravity, which includes the function of the torsion scalar
$\mathcal{T}$ and a related boundary term $B$. In this modified gravity
theory, we compute the deflection angle of light by a non-asymptotically flat
black hole. Then we study the quasinormal modes associated with the axial
scalar perturbation in the background of the black hole.

Gibbons and Werner first introduced an alternative way to calculate the
gravitational bending angle using the Gauss-Bonnet theorem. They first
evaluated the deflection angle for a Schwarzschild black hole. Since then
their work has been extended in various ways for different kinds of black
holes. Few researchers have computed the deflection angle from the stationary
black holes. Few have again considered finite distances between source and
receiver, and derived the deflection angle by the static as well as stationary
black holes. Some recent studies also considered black holes in modified
gravity theories to obtain the deflection angle of light. In this work, we
implemented the Ishihara et al. method to evaluate the deflection angle of
light from the receiver point of view. This method does not depend on the
asymptotic flatness. Hence, we have applied this method in the non-
asymptotically flat black hole in $f(\mathcal{T},\mathcal{B})$ gravity. We
have computed the bending angle considering the source and the receiver at a
finite distance. It is found that when the source and the receiver are
considered near to the lens object, the deflection angle becomes divergent.
The boundary term coming from $f(\mathcal{T},\mathcal{B})$ gravity has a
significant effect on the deflection angle. In the near future, we wish to
extend our work to obtain the deflection angle of massive particles using
Ishihara et al. method in the black hole as well as in wormhole backgrounds in
different MTGs.

Quasinormal modes are some complex numbers related to the emission of GWs from
the compact objects in the universe. The real part of these modes is related
to the emission frequency and the imaginary part is related to the damping. In
this work, we try to find a Schrödinger type equation containing an effective
potential, which allows us to compute the quasinormal modes associated with
the massless scalar perturbations in the background of the considered black
hole in the $f(\mathcal{T},\mathcal{B})$ gravity. For this purpose, we use the
AIM and Padé averaged sixth-order WKB approximation method. We obtained the
quasinormal modes for different values of the multipole moment $l$ with the
overtone number $n=0$. It has been observed that for smaller values of $l$,
the error associated with the quasinormal frequencies calculated with WKB
method is higher in magnitude. With an increase in the value of $l$, the
errors associated with the frequencies decrease significantly. The AIM stands
in agreement with the results obtained by using the WKB method, confirming the
validity of the results. Our study shows that the parameter $c_{0}$ has a
significant impact on the quasinormal modes of the black hole. Moreover, the
time domain profile analysis of evolution of massless scalar field
perturbation in the black hole spacetime also confirms all these outcomes of
quasinormal modes’ calculations. However, for the observational constraints on
the model from the quasinormal modes, we might need to wait for the LISA
2022_gogoi . The reason behind this is that assuming the nearest black holes,
such as Sgr A∗ or M $87^{*}$ to be almost static, their respective quasinormal
modes are beyond the sensitivity range of current and near future ground-based
detectors 2022_gogoi . Nonetheless, the upcoming LISA space-based detector
exhibits a remarkable increase in sensitivity and will be proficient in
detecting quasinormal modes with a greater precision.

We also study the optical properties of the black hole viz., the shadow and
the emission rate. It is seen from our investigation that the shadow of the
black hole decreases with an increase in the value of the parameter $c_{0}$.
Similarly, an increase in the value of $c_{0}$ decreases the emission rate or
evaporation rate of the black hole, hence increasing its stability.

Our investigation demonstrates that the boundary-derived parameter $c_{0}$
exerts considerable influence on the spacetime of a black hole, impacting
various observable characteristics of it. We demonstrate that $c_{0}$ has
effects on the gravitational deflection of light, ringdown GWs or quasinormal
modes, the black hole shadow and the emission rate, potentially providing
observable evidence for $f(\mathcal{T},\mathcal{B})$ gravity. Constraining the
model with available observational data will provide more useful insights into
the $f(\mathcal{T},\mathcal{B})$ gravity which we keep as a future prospect of
our study.

## Acknowledgments

UDG is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for the Visiting Associateship of the institute.

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