# Observational signature of Lorentz violation in Kalb-Ramond field model and
Bumblebee model: A comprehensive comparative study

Sohan Kumar Jha<EMAIL_ADDRESS>Chandernagore College, Chandernagore,
Hooghly, West Bengal, India

###### Abstract

Abstract

This article is devoted to the comparative study of the effects of the Lorentz
symmetry violation (LV) arising in Kalb-Ramond (KR) and Bumblebee (BM) field
models. We study optical appearance with accretion, Quasinormal modes,
ringdown waveforms, Hawking radiation, and weak gravitational lensing for both
black holes (BHs) and compare their observational impacts. The horizon radius,
photon radius, and the critical impact parameter for KR BHs decrease with the
LV parameter $\alpha$. In contrast, they remain independent of the BM
parameter $\beta$ and have values the same as those for Schwarzschild BH.
Their qualitative and quantitative variation with $\alpha$ are illustrated for
KR BH. Our study reveals that the observed intensity for static and infalling
accretions peaks at the critical impact parameter. We find that a KR BH is
brighter than a Schwarzschild or BM BH for static as well as infalling
accretion. The BM BH, on the other hand, is brighter than a Schwarzschild BH
when the accretion is static but becomes darker for an infalling accretion.
The shadow’s size, independent of the accretion type, is smaller for a KR BH.
Our investigation into quasinormal modes (QNMs) with the help of the $6$th
order Padé averaged WKB method and ringdown waveforms provide deeper insight
into the difference in observational imprints of LV parameters. It reveals
that GWs emitted by KR BHs have larger frequencies and decay faster than those
emitted by Schwarzschild or BM BHs for scalar and electromagnetic
perturbations. GWs emitted by BM BHs have lower frequency and decay slower
than those emitted by Schwarzschild BHs for scalar perturbation, whereas the
frequency is higher for electromagnetic perturbation. We then study the
greybody factor (GF) and power emitted for both BHs. The Hawking temperature
is higher for a KR BM and lower for a BM BH than a Schwarzschild BH. It also
reveals that the transmission probability decreases with $\alpha$ and $\beta$.
A comparison of GFs for KR and BM BHs reveals that the transmission
probability is higher for BM BH. We also study the effect of LV on the power
emitted in the form of Hawking radiation. Power received by an asymptotic
observer is larger for a KR BH. We finally try to differentiate between KR and
BM BHs based on their observational signatures in the weak gravitational
lensing. We obtain higher-order corrections in the deflection angle and
graphically illustrate the impact of $\alpha$ and $\beta$. We observe that a
light ray gets deflected most from its path when passing by a Schwarzschild
BH, and the deflection is least when it passes by a KR BH. Our study
conclusively shows that we can differentiate between KR and BM BHs based on
astrophysical observations.

##  I Introduction

The Lorentz symmetry is the central pillar of the two most successful
theories: general relativity and the standard model of particle physics.
Despite their astounding success in predicting various experimental
observations, it is widely believed that the Lorentz symmetry may be violated
at higher energies in various theories, such as string theory, non-commutative
field theory, and so on [1-8]. The Lorentz symmetry can either be broken
spontaneously or explicitly. When broken explicitly, the Lagrange density no
longer remains invariant under the Lorentz transformation, and we have
physical laws taking different forms in certain reference frames. When the
Lorentz symmetry is spontaneously broken, although the Lagrange density shows
Lorentz invariance, the Lorentz symmetry is broken in the ground state. The
standard model extension provides a general framework for incorporating LV
Kostelecky2004a . Bumblebee models within this extension are one of the
simplest theories where LV occurs spontaneously due to the non-zero vacuum
expectation value of a vector field called bumblebee field Kostelecky1989a ;
Kostelecky1989 ; Kostelecky1989b ; Bailey2006 ; Bluhm2008a . Casana et al., in
their article bm , reported a Schwarzschild -like solution. Various aspects of
the BH, such as lensing, accretion, etc., have extensively been studied in
[15-24]. Some other notable articles related to BHs in the bumblebee model are
[25-41]. In the KR model, LV is spontaneously broken due to a rank-two
antisymmetric tensor field known as the Kalb-Ramond field assumes non-zero
vacuum expectation value Altschul2010  . Various properties of the KR field
have been studied in [44-49]. Lessa et al. obtained an exact non-rotating
solution in the KR model Lessa2020 . For more details regarding different
aspects of Lessa solution and rotating KR BHs, please see [51-55]. A different
exact static and spherically symmetric solution in the KR field model is
obtained by Ke Yang et al. in kr . This solution will be the subject of our
study along with the solution given in bm .
BHs acquire mass (and possibly angular momentum) through the accretion process
by capturing the ambient matter surrounding BHs IG . The kinetic energy of
accreting matter increases at the cost of its gravitational energy, which
enhances the energy radiated by the central object. The accretion process is
believed to be the primary energy source of astronomical massive objects such
as quasars and active galactic nuclei. It helps us explain astrophysical
phenomena such as gamma-ray bursts and tidal disruption events. Exploring
differences in the impact LV in the BM model and LV in the KR model have on
the shadow with static and infalling accretion is our primary goal here.
Shadow of a BH encodes important information regarding the central object. It
directly manifests the strong gravitational field of BHs. Since one can find
the observational signature of LV in the shadow and accretion, it would be
instructive to study the shadow in the presence of accretion. Please see
articles [58-65] for more details.
The emergence of QNMs is a result of perturbation of BHs. These modes are
transient in nature. BH, on getting perturbed, emits gravitational waves
(GWs), and eventually, the perturbed state of the BH dies down. There are
three phases that a BH, on perturbation, experiences KONOR : initially, there
is an outburst of radiation depending on the initial condition, then we have
the ringdown phase, and the final one is the tail where the perturbation dies
down by a power or exponential law. QNMs are associated with the ringdown
phase. They are functions of the model parameters characterizing the central
object and the type of perturbation BH encounters. They do not depend on the
initial conditions, signifying that the BH continually interacts with the
field responsible for the perturbation. Thus, probing BH perturbation and QNMs
would be imperative to gauge the impact of LV and to differentiate between two
BHs that incorporate the LV effect on the basis of an observational signature.
The prospect of QNMs getting detected by LIGO, VIRGO, and LISA in the future
makes it more pertinent to study QNMs LIGO1 ; LIGO2 ; LIGO3 ; LIGO4 ; VIRGO .
A great deal of research has been carried out to understand QNMs for different
BH metrics [78-116].
Hawking radiation and the association of temperature with BH horizon are
fascinating subjects that have intrigued researchers for quite some time.
Hawking showed that a BH emits radiation by considering quantum consequences
HAWKING . It can be explained with the help of a pair production near the
horizon where one of the particles gets consumed by the BH, whereas the second
particle moves towards an asymptotic observer. It is the second particle that
constitutes the Hawking radiation [127-129]. It was later shown by Bekenstein
and Keif that to be consistent with thermodynamics, a temperature needs to be
associated with the BH horizon BEK ; KEIF . We can obtain Hawking temperature
using different techniques [132-134]. The greybody factor (GF) is a
significant quantity that provides the probability of Hawking radiation
getting detected by a distant observer. Out of various methods at our disposal
to bound GF[135-150], we employed the elegant technique illustrated in GB ;
GB1 ; GB2 . Here, Hawking radiation is also investigated to differentiate
between KE and BM BHs on the basis of LV imprints in observation. We also
explore the weak gravitational lensing (GL) due to its dependence on the
nature of the underlying spacetime. In gravitational lensing, a massive
central object acting as a lens deflects light rays from its path, where the
deflection angle is a function of model parameters. It has been extensively
studied to probe dark matter and energy and also for the determination of the
mass of galaxies [153-155]. GL can be studied in the strong field limit and
weak field limit. It is the latter we are interested in. The central pillar of
studies made in the weak GL is the seminal work by Gibbons and Werner GW .
This technique was subsequently extended to stationary spacetimes in WERNER
and to finite distances in ISHIHARA1 ; ISHIHARA2 . These studies form the
basis of further research in ONO1 ; ONO2 ; ONO3 . Higher order corrections in
the deflection angle were later obtained in CRISNEJO . Please refer to the
articles for studies in different spacetimes [165-175]. We will explore weak
GL to find differential observational signatures of LV arising in KR and BM
models.
We compose the article in the following manner. Sec. II briefly introduces KR
and BM metrics, and unstable circular orbits are studied. Sec. III is where
the optical appearance of BH with accretion is examined, and a qualitative
comparison is carried out to differentiate between KR and BM BHs. Sec. IV and
V are devoted to studying differences in the implication of LV on QNMs in KR
and BM models. Differential observational signatures of LV in KR and BM models
with respect to Hawking radiation are explored in Sec. VI. We try to
differentiate between KR and BM BHs based on the deflection angle in the weak
field limit in Sec. VII. We finally elaborate on our findings in Sec. VIII.

##  II Static and spherically symmetric BH solutions with Lorentz symmetry
violation

We introduce static and spherically symmetric solutions resulting from KR and
BM. In both models, the the non-zero vacuum expectation value of a field
breaks the Lorentz symmetry spontaneously : it is a vector field in BM and a
tensor field in KR. The static and spherically symmetric solution in the BM
model is bm

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+h(r)\left(d\theta^{2}+sin^{2}\theta
d\phi^{2}\right)$ (1) $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{1+\beta}{\frac{1}{1-\alpha}-\frac{2M}{r}}dr^{2}+r^{2}\left(d\theta^{2}+sin^{2}\theta
d\phi^{2}\right),$ (2)

where $f(r)=1-\frac{2M}{r}$, $g(r)=\frac{1-\frac{2M}{r}}{1+\beta}$, and
$h(r)=r^{2}$. The BH solution obtained from the KR model is kr

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+h(r)\left(d\theta^{2}+sin^{2}\theta
d\phi^{2}\right)$ (3) $\displaystyle=$
$\displaystyle-\left(\frac{1}{1-\alpha}-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\frac{1}{1-\alpha}-\frac{2M}{r}}+r^{2}\left(d\theta^{2}+sin^{2}\theta
d\phi^{2}\right),$ (4)

where $f(r)=g(r)=\frac{1}{1-\alpha}-\frac{2M}{r}$ and $h(r)=r^{2}$. The KR(BM)
model reduces to Schwarzschild solution in the limit $\alpha(\beta)\rightarrow
0$. The Kretschmann scalars for the KR BH ($K_{\alpha}$) and BM BH
($K_{\beta}$) are given by

$\displaystyle K_{\alpha}=\frac{48M^{2}}{r^{6}}-\frac{16\alpha
M}{(1-\alpha)r^{5}}+\frac{4\alpha^{2}}{(\alpha-1)^{2}r^{4}},$ $\displaystyle
K_{\beta}=\frac{4}{\left(1+\beta\right)^{2}}\left(\frac{12M^{2}}{r^{6}}+\frac{4\beta
M}{r^{5}}+\frac{\beta^{2}}{r^{4}}\right).$ (5)

Since the Kretschmann scalars for KR and BM BHs differ from that for a
Schwarzschild BH, there exists no coordinate transformation that can eliminate
LV from metrics [$2$] and [$4$]. A KR BH has an event horizon at
$r_{h}=2M(1-\alpha)$, which depends on the parameter $\alpha$, whereas the
horizon radius for a BM BH is $r_{h}=2M$, irrespective of the value of
$\beta$. To observe shadow, we must consider the motion of massless particles
in the background of the BH. Since the BH solutions under consideration are
spherically symmetric, we may consider null geodesics confined to the
equatorial plane. As such, the Lagrangian corresponding to the metric [$1$] is

$\mathscr{L}=\frac{1}{2}\left(-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{g(r)}+h(r)\dot{\phi}^{2}\right).$
(6)

Here, $\dot{}$ is the differentiation with respect to the affine parameter
$\lambda$. Owing to the static and spherically symmetric nature of the
spacetime, we have two conserved quantities: energy
$\mathcal{E}=-p_{t}=-\frac{\partial\mathscr{L}}{\partial\dot{t}}=f(r)\dot{t}$
and angular momentum
$\mathcal{L}=p_{\phi}=\frac{\partial\mathscr{L}}{\partial\dot{\phi}}=h(r)\dot{\phi}$.
With these conjoined with the fact that $\mathscr{L}=0$ for null geodesics, we
obtain

$\displaystyle\dot{r}^{2}+\left(-\mathcal{E}^{2}\frac{g(r)}{f(r)}+\mathcal{L}^{2}{g(r)}{h(r)}\right)=0$
(7) $\displaystyle\Rightarrow$ $\displaystyle\dot{r}^{2}+V_{eff}(r)=0,$

where $V_{eff}=-\mathcal{E}^{2}\frac{g(r)}{f(r)}+\mathcal{L}^{2}{g(r)}{h(r)}$
is the effective potential. Following are the conditions for an unstable
spherical orbit at radius $r_{p}$:

$V_{eff}(r_{p})=0,\quad\frac{\partial V_{eff}}{\partial
r}|_{r=r_{p}}=0,\quad\text{and}\quad\frac{\partial^{2}V_{eff}}{\partial
r^{2}}|_{r=r_{p}}<0.$ (8)

The above conditions yield $f(r)^{\prime}h(r)=h(r)^{\prime}f(r)$ evaluted at
$r=r_{p}$, where ′ implies differentiation with respect to $r$. For a KR BH,
the position of the unstable spherical orbit is $r_{p}=3M(1-\alpha)$, and the
spherical radius for a BM BH is $r_{p}=3M$. Thus, LV does not affect the
photon orbit in the case of a BM BH. However, due to the presence of LV, the
spherical radius decreases with $\alpha$. The critical impact parameter
associated with the spherical orbit is

$b_{p}=\sqrt{\frac{h(r)}{f(r)}}.$ (9)

For a KR BH, $b_{p}=3\sqrt{3}\left(1-\alpha\right)^{3/2}M$ and for a BM BH,
$b_{p}=3\sqrt{3}M$. Similar to the photon radius, increasing the KR parameter
$\alpha$ would decrease the impact parameter. Although LV leaves an imprint on
the impact parameter in KR, it is absent in BM. This is because the LV
parameter in BM appears in the radial component of the metric. In contrast,
the spherical radius and the impact parameter depend on the metric’s time and
the azimuthal components.

Figure 1: Variation of $r_{p}$ and $b_{p}$ against $\alpha$. Table 1: Various values of horizon radius, photon radius, and impact parameter for different values of $\alpha$. $\alpha$ | 0. | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
---|---|---|---|---|---|---|---|---|---|---
$r_{\text{eh}}$ | 2. | 1.8 | 1.6 | 1.4 | 1.2 | 1. | 0.8 | 0.6 | 0.4 | 0.2
$r_{p}$ | 3.0 | 2.7 | 2.4 | 2.1 | 1.8 | 1.5 | 1.2 | 0.9 | 0.6 | 0.3
$b_{p}$ | 5.19615 | 4.43655 | 3.71806 | 3.04319 | 2.41495 | 1.83712 | 1.31453 | 0.853815 | 0.464758 | 0.164317

The qualitative variation of the photon radius and the critical impact
parameter are shown in Fig. [1], and their quantitative nature of variation is
shown in Table [1]. The significant impact of KR parameter $\alpha$ on these
observables is evident from above. Next, we will explore LV’s impact on the
optical appearance of BH with different accretion models.

##  III Black hole shadow with accretion

Accretion is the process whereby massive astronomical objects gain mass by
capturing surrounding matter. It is believed to be the primary energy source
of many massive objects, such as quasars and active galactic nuclei. The
existence of an accretion disk around a BH significantly impacts its optical
appearance. Since the accretion and the BH shadow largely depend on the nature
of the underlying spacetime, it would be instructive to study the BH shadow
with accretion. Static and infalling accretion will be considered in our
study.

###  III.1 Shadow with static accretion

The static accretion disk is considered to be optically and geometrically
thin. The measured intensity by an asymptotic observer is mj97 ; bambi13

$I(\nu_{obs})=\int_{ray}g_{f}^{3}j(\nu_{em})dl_{prop},$ (10)

where $\nu_{obs}$ and $\nu_{em}$ are, respectively, the observed and emitted
frequencies, the redshift factor $g_{f}=\sqrt{f(r)}$, $j(\nu_{em})$ is the
emissivity per unit volume, and the proper differential length
$dl_{prop}=\sqrt{\frac{1}{g(r)}+h(r)\Big{(}\frac{d\phi}{dr}\Big{)}^{2}}dr$. We
need to obtain the expression for $\frac{d\phi}{dr}$.

Fig 2. Observed intensity and black hole shadows with static spherical
accretion for KR BH. The upper right is for $\alpha=0$, the lower left one is
for $\alpha=0.1$, and the lower right one is for $\alpha=0.2$.

To this end, we use the Eq. [7] conjoined with
$\dot{\phi}^{2}=\frac{\mathcal{L}^{2}}{h(r)^{2}}$. This yields

$dl_{prop}=\sqrt{\frac{h(r)}{g(r)(h(r)-b^{2}f(r))}}$ (11)

With the assumption that the emission is monochromatic, we take
$j(\nu_{em})\propto\frac{\delta(\nu_{em}-\nu_{t})}{r^{2}}$, $\nu_{t}$ being
the rest-frame frequency. Thus, the expression for the measured intensity
becomes

$I(\nu_{obs})=\int_{ray}\frac{f(r)^{3/2}}{r^{2}}\sqrt{\frac{h(r)}{g(r)(h(r)-b^{2}f(r))}}.$
(12)

Photons with an impact parameter equal to their critical value get trapped in
the unstable circular orbit. If they get perturbed from their circular path,
they may either move towards the BH and get swallowed or move away from it and
reach the distant observer. The latter photons are responsible for the bright
circular ring seen in Figs. [$2$, $3$]. Photons with impact parameter
$b<b_{p}$ come too close to the BH to escape its strong gravitational pull and
get consumed by it. Those photons with $b>b_{p}$ get deflected towards the
asymptotic observer.

Fig 3. Observed intensity and black hole shadows with static spherical
accretion BM BH. The upper right is for $\beta=0$, the lower left one is for
$\beta=0.1$, and the lower right one is for $\beta=0.2$.

It is evident from Figs. [$2$, $3$] that the intensity initially increases
with the impact parameter before reaching its peak value at $b_{p}$ and then
sharply decreases with $b$. The peak position remains unaltered, and the peak
value changes marginally with the BM parameter $\beta$. However, the position
of the peak shifts towards the left, and the maximum intensity increases
significantly with increasing KR parameter $\alpha$. It is because the
intensity reaches its maximum value at $b=b_{p}$, which does not change with
$\beta$ but decreases with $\alpha$. The region within the shadow is not
entirely black since a small fraction of photons can escape a BH. It is also
to be noted that KR and BM BHs are brighter than a Schwarzschild BH.

Fig 4. A comparison of observed intensity with static accretion. We have taken
$\alpha=\beta=0.2$.

A comparison of observed intensity between KR and BM BHs in Fig. [$4$] reveals
that the intensity will always be greater for a KR BH, and as a result, KR BHs
will be brighter than BM BHs. Now, we turn our attention towards infalling
accretion.

###  III.2 Shadow with infalling accretion

Accretion disks are believed not to be static but have radial velocity, i.e.,
infalling. Thus, studying shadow with infalling accretion would be closer to
reality. The equation for the intensity observed in the case of infalling
accretion is the same as that for static accretion with the redshift factor
modified to

$g_{f}=\frac{k_{\gamma}u_{obs}^{\gamma}}{k_{\zeta}u_{em}^{\zeta}},$ (13)

where $k^{\mu}$ represents photon four-velocity, four-velocity of the
assymptotic observer is $u_{obs}^{\mu}\equiv(1,0,0,0)$, and accretion four-
velocity is $u_{em}^{\mu}$. We know $k_{t}=1/b$ and $k_{\mu}k^{\mu}=0$. This
helps us obtain the relation between $k_{t}$ and $k_{r}$ as

$\frac{k_{r}}{k_{t}}=\pm\sqrt{\frac{g(r)f(r)h(r)}{h(r)-b^{2}f(r)}}.$ (14)

The +(-) sign is when photons move towards(away from) the BH. The four-
velocity of accreting matter is

$\displaystyle
u_{em}^{t}=\frac{1}{f(r)},~{}~{}~{}~{}~{}u_{em}^{\theta}=u_{em}^{\phi}=0,~{}~{}~{}~{}~{}u_{em}^{r}=-\sqrt{\frac{g(r)}{f(r)}-g(r)}.$
(15)

Fig 5. Observed intensity and black hole shadows with infalling spherical
accretion. The upper right is for $\alpha=0$, the lower left one is for
$\alpha=0.1$, and the lower right one is for $\alpha=0.2$.

Four-velocities of the photon, distant observer, and accreting matter lead to
the following expression of the redshift factor:

$g_{f}=\frac{f(r)\sqrt{h(r)}}{\sqrt{h(r)}+\sqrt{h(r)-b^{2}f(r)}\sqrt{1-f(r)}}.$
(16)

Also, the differential proper distance for the infalling case is

$dl_{prop}=k_{\mu}u_{em}^{\mu}d\zeta=\frac{k_{t}}{g_{f}|k_{r}|}dr.$ (17)

With the above expressions at our disposal, we are now in a position to delve
into studying the impact of LV on the optical appearance of KR and BM models
in Figs. [$5$, $6$]. Similar to the static accretion case, the intensity peaks
at the critical impact parameter, albeit the variation with the impact
parameter, is smooth compared to the static case. However, the peak value of
intensity is much smaller than the static accretion. Although similar to the
static accretion, the intensity increases significantly with the KR parameter
$\alpha$, the impact of BM parameter $\beta$ is opposite to that in the static
case, i.e., increasing $\beta$ decreases the intensity. This adverse impact of
$\beta$ on the intensity is because the radial velocity of photons increases
with $\beta$, resulting in a decrease in the proper length evident from Eq.
[17]. Owing to the Doppler effect, the central region within the photon ring,
much darker than the static case, is observed.

Fig 6. Observed intensity and black hole shadows with infalling spherical
accretion. The upper right is for $\beta=0$, the lower left one is for
$\beta=0.1$, and the lower right one is for $\beta=0.2$.

We compare observed intensities for KR and BM BHs. Similar to the static
accretion, the intensity for the KR BH is much greater than the BM BH,
rendering the BM BH darker than the KR BH or Schwarzschild BH.

Figure 2: A comparison of observed intensity with infalling accretion. We have
taken $\alpha=\beta=0.2$.

##  IV Quasinormal modes

Quasinormal modes of a BH are complex-valued functions of parameters
characterizing the spacetime. As such, it embeds information regarding the
central object, making it a possible avenue to probe any deviation from the
Schwarzschild BH. In this section, we explore the impact of LV on QNMs in KR
and BM models. Assuming a negligible impact of scalar or electromagnetic
perturbation on the background spacetime, we take into account the Klein-
Gordon equation for the massless scalar field

$\frac{1}{\sqrt{-g}}{\partial\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\chi)=0,$
(18)

and the Maxwell equation for the electromagnetic field

$\frac{1}{\sqrt{-g}}{\partial\nu}(F_{\rho\sigma}g^{\rho\mu}g^{\sigma\nu}\sqrt{-g})=0,$
(19)

where $F_{\rho\sigma}={\partial\rho}A^{\sigma}-{\partial\sigma}A^{\rho}$,
$A_{\nu}$ being the electromagnetic four-potential. We need to reduce Eqs.
[18, 19] to a Schr$\ddot{o}$dinger-like equation so that we can employ the
$6$th order Padé averaged WKB approach. This can be done with the tortoise
coordinate defined by $\text{d}r_{\ast}=\frac{\text{d}r}{f(r)g(r)}$, where
$r_{*}\rightarrow-\infty$ as $r\rightarrow r_{h}$ and $r_{*}\rightarrow\infty$
as $r\rightarrow\infty$. This helps us reduce Eqs. [18, 19] to the desired
form:

$-\frac{\text{d}^{2}\phi}{\text{d}{r^{2}_{*}}}+V(r)\phi=\omega^{2}\phi.$ (20)

The effective potential has the form

$V(r)=|g_{tt}|\left(\frac{\ell(1+\ell)}{r^{2}}+\frac{1-s^{2}}{r\sqrt{|g_{tt}|g_{rr}}}\frac{\text{d}}{\text{d}r}\sqrt{|g_{tt}|g_{rr}^{-1}}\right),$
(21)

where $\ell$ and $s$ are the angular momentum and the spin, respectively.
$s=0$ gives the effective potential for the scalar perturbation, and the
potential for the electromagnetic perturbation is given by $s=1$. With the
effective potential, we can obtain quasinormal frequencies with the help of
the $6th$ order WKB method by solving the equation schutz ; iyer ; iyer1 ;
konoplya1

$\frac{\text{i}(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{{}^{\prime\prime}}}}-\sum^{6}_{\text{i}=2}\Omega_{\text{i}}=n+\frac{1}{2},$
(22)

Figure 3: Variation of gravitational wave frequency with KR parameter $\alpha$
for different $\ell$.

Figure 4: Variation of gravitational wave frequency with BM parameter $\beta$
for different $\ell$.

Figure 5: Variation of decay rate with KR parameter $\alpha$ for different
$\ell$.

Figure 6: Variation of decay rate with BM parameter $\beta$ for different
$\ell$.

Figure 7: Comparison of gravitational wave frequency with $\ell=2$.

Figure 8: Comparison of decay rate with $\ell=2$.

where $V_{0}$ is the peak value of the potential obtained at $r_{*}=r_{0}$,
$V^{\prime\prime}_{0}=\frac{d^{2}V(r_{*})}{dr_{*}^{2}}|_{r_{*}=r_{0}}$, and
$\Omega_{\text{i}}$ are WKB corrections schutz ; iyer ; iyer1 ; konoplya1 .
The accuracy of the WKB method can be increased with Padé approximants method
jerzy  where a polynomial of order $k$ in terms of order parameter $\epsilon$
is defined as

$P_{k}(\epsilon)=V_{0}+\Omega_{2}\epsilon^{2}+\Omega_{4}\epsilon^{4}+\Omega_{6}\epsilon^{6}+\ldots-i(n+\frac{1}{2})\sqrt{-2V_{0}^{{}^{\prime\prime}}}\left(\epsilon+\Omega_{3}\epsilon^{3}+\Omega_{5}\epsilon^{5}\ldots\right),$
(23)

Putting $\epsilon=1$ in the above equation will lead us to an equation of the
form $\omega^{2}=P_{\tilde{n}/\tilde{m}}(1)$ where jerzy ; konoplya2

$P_{\tilde{n}/\tilde{m}}(\epsilon)=\frac{\mathcal{Q}_{0}+\mathcal{Q}_{1}\epsilon+\ldots+\mathcal{Q}_{\tilde{n}}\epsilon^{\tilde{n}}}{\mathcal{R}_{0}+\mathcal{R}_{1}\epsilon+\ldots+\mathcal{R}_{\tilde{m}}\epsilon^{\tilde{m}}},$
(24)

with $\tilde{n}+\tilde{m}=k$. To gauge the impact of LV on the frequency and
the decay rate of GW, we plot them against $\alpha$ and $\beta$ for scalar and
electromagnetic perturbations. We also tabulate numerical values of QNMs for
scalar perturbations in Table [2] and electromagnetic perturbations in Table
[3].

Table 2: Quasinormal frequencies for scalar perturbation with $n=0$. $\alpha(\beta)$ | $\ell$=1(KR) | $\ell$=1(BM) | $\ell$=2(KR) | $\ell$=2(BM) | $\ell$=3(KR) | $\ell$=3(BM)
---|---|---|---|---|---|---
0. | 0.292931 -0.097661 i | 0.292931 -0.097661 i | 0.483644 -0.0967591 i | 0.483644 -0.0967591 i | 0.675366 -0.0964997 i | 0.675366 -0.0964997 i
0.1 | 0.345699 -0.120733 i | 0.291094 -0.09301 i | 0.568024 -0.119524 i | 0.482541 -0.0922123 i | 0.792124 -0.119172 i | 0.674579 -0.0919857 i
0.2 | 0.416377 -0.153054 i | 0.289557 -0.0889634 i | 0.680133 -0.15138 i | 0.481621 -0.0882512 i | 0.946873 -0.150884 i | 0.673923 -0.088051 i
0.3 | 0.514751 -0.200311 i | 0.288251 -0.0854008 i | 0.834629 -0.197898 i | 0.480842 -0.08476 i | 1.15949 -0.197169 i | 0.673367 -0.0845815 i
0.4 | 0.658685 -0.273324 i | 0.287128 -0.0822331 i | 1.05789 -0.269675 i | 0.480173 -0.0816526 i | 1.46553 -0.26854 i | 0.67289 -0.0814921 i
0.5 | 0.884172 -0.39516 i | 0.286151 -0.0793927 i | 1.40185 -0.388943 i | 0.479592 -0.0788636 i | 1.93457 -0.387037 i | 0.672476 -0.0787183 i
Table 3: Quasinormal frequencies for electromagnetic perturbation with $n=0$. $\alpha(\beta)$ | $\ell$=1(KR) | $\ell$=1(BM) | $\ell$=2(KR) | $\ell$=2(BM) | $\ell$=3(KR) | $\ell$=3(BM)
---|---|---|---|---|---|---
0. | 0.248251 -0.0924846 i | 0.248251 -0.0924847 i | 0.457595 -0.0950048 i | 0.457595 -0.0950048 i | 0.656899 -0.0956163 i | 0.656899 -0.0956163 i
0.1 | 0.287643 -0.11364 i | 0.250425 -0.0885174 i | 0.534146 -0.117121 i | 0.45885 -0.0906899 i | 0.768098 -0.117961 i | 0.657786 -0.0912194 i
0.2 | 0.338589 -0.14296 i | 0.252237 -0.0850152 i | 0.634686 -0.147964 i | 0.459896 -0.0869137 i | 0.914633 -0.149162 i | 0.658526 -0.0873781 i
0.3 | 0.406394 -0.185243 i | 0.25377 -0.0818949 i | 0.771229 -0.192809 i | 0.460781 -0.0835727 i | 1.11449 -0.194602 i | 0.659152 -0.0839844 i
0.4 | 0.499903 -0.249383 i | 0.255084 -0.0790924 i | 0.964788 -0.261613 i | 0.461539 -0.0805895 i | 1.39942 -0.26447 i | 0.659689 -0.0809576 i
0.5 | 0.63478 -0.353454 i | 0.256223 -0.0765576 i | 1.25523 -0.375054 i | 0.462196 -0.0779043 i | 1.83038 -0.380019 i | 0.660154 -0.0782362 i

It is evident from Figs. [3, 5] that the KR parameter significantly impacts
the frequency as well as the decay rate of GW. The frequency and the decay
rate increase with $\alpha$ for both perturbations. The variation of real and
imaginary parts of QNMs with BM parameter $\beta$ is shown in Figs. [4, 6]. It
shows that although the decay rate decreases with $\beta$ for both the
perturbations, the impact of $\beta$ on frequency is not the same for both of
them. The frequency decreases with $\beta$ for scalar perturbation but
increases for electromagnetic perturbation, albeit the variation is marginal.
Next, we compare the frequency and the decay rate of KR and BM BHs in Figs.
[7, 8]. It conclusively shows that except at $\alpha=\beta=0$ where we have
Schwarzschild BH, the frequency and decay rate are always greater for a KR BH.
Numerical values given in Tables [2, 3] are in congruence with conclusions
drawn from Figs.[3-8].

##  V time domain profile

The effect of LV on QNMs can be visualized with a time domain profile. We will
be using the integration method described in gundlach1 , which is based on the
finite difference method with the error proportional to the step $\Delta$
used. We can obtain ringdown waveforms with the help of the following
equation:

$\phi_{m,n+1}=-\,\phi_{m,n-1}+\left(\dfrac{\Delta
t}{\Delta\tilde{r}}\right)^{2}(\phi_{m+1,n+\phi_{m-1,n}})+\left(2-2\left(\dfrac{\Delta
t}{\Delta\tilde{r}}\right)^{2}-V_{m}\Delta t^{2}\right)\phi_{m,n},$ (25)

with $\phi(\tilde{r},t)=\phi(m\Delta\tilde{r},n\Delta t)=\phi_{m,n}$,
$V(r(\tilde{r}))=V(\tilde{r},t)=V_{m,n}$. Choosing initial conditions
$\phi(r_{*},t)=\exp\left[-\dfrac{(r_{*}-\hat{r}_{*})^{2}}{2\sigma^{2}}\right]$
and $\phi(r_{*},t)|_{t<0}=0$ with $r_{*}=5$, $\hat{r}_{*}=0.4$ and imposing
Von Neumann’s stability condition on the ratio $\frac{\Delta
t}{\Delta\tilde{r}}$, we obtain time domain profiles of scalar and
electromagnetic perturbations.

Figure 9: Time domain profile of electromagnetic field for KR BH. The left one
is for $\ell=2$, and the right one is for $\alpha=0.3$.

Figure 10: Time domain profile of scalar field for KR BH. The left one is for
$\ell=2$, and the right one is for $\alpha=0.3$.

Figure 11: Time domain profile of electromagnetic field for BM BH. The left
one is for $\ell=2$, and the right one is for $\beta=0.3$.

Figure 12: Time domain profile of scalar field for BM BH. The left one is for
$\ell=2$, and the right one is for $\beta=0.3$.

Figure 13: Comparison of time domain profiles with $\alpha=\beta=0.2$ and
$\ell=2$.

Similar to the previous section, we observe from Figs. [9, 12] that the
frequency and the decay rate increases with $\alpha$ for both perurbations.
However, it is evident from Figs. [
remkr2, 12] that the decay rate increases with $\ell$ for electromagnetic
perturbation but decreases for scalar field. Figs. [11, 12] reinforce our
findings in the previous section regarding the effect of the BM parameter on
QNMs. A comparison of time-domain profiles between KR and BM BHs is shown in
Fig. [13]. It can be conclusively inferred from the figure and Table [2, 3]
that GWs originated from a KR BH due to scalar or electromagnetic perturbation
have larger frequency and decay faster than GWs originating from a BM BH or a
Schwarzschild BH.

##  VI Greybody factor and Hawking spectrum

We will explore LV’s possible impacts on the transmission coefficient or GF
and Hawking spectrum. BHs emit radiation HAWKING  and possess a temperature
BEK ; KEIF . The temperature associated with the BH is

$T_{H}=\frac{1}{4\pi\sqrt{-g_{tt}g_{rr}}}\frac{dg_{tt}}{dr}|_{r=r_{h}}.$ (26)

We have $g_{tt}=-f(r)$ and $g_{rr}=\frac{1}{g(r)}$. The Hawking temperature of
a KR BH is $T_{H}=\frac{1}{8\pi M\left(1-\alpha\right)^{2}}$ and for a BM BH
$T_{H}=\frac{1}{8\pi M\sqrt{1+\beta}}$. It is evident from these expressions
that LV in the KR model increases the Hawking temperature but impacts
adversely in the case of the BM model. The radiation observed by a local
observer differs from that observed by a distant one, owing to the redshift
factor. The radiation received by an asymptotic observer is described by
greybody distribution. There exists a number of methods to calculate GF qn32 ;
qn36 ; qn37 ; qn38 ; qn39 ; qn40 ; qn41 ; qn42 ; qn43 ; qn44 ; qn45 ; qn46 .
Visser and Boonserm gave an elegant method in GB ; GB1 ; GB2  to lower bound
GF as

$T\left(\omega\right)\geq\sec
h^{2}\left(\frac{1}{2w}\int_{-\infty}^{+\infty}Vdr_{\ast}\right).$ (27)

The above equation can also be written as

$T\left(\omega\right)\geq\sec
h^{2}\left(\frac{1}{2w}\int_{r_{h}}^{+\infty}V\frac{dr}{\sqrt{f(r)g(r)}}\right).$
(28)

With the help of the above equation, the lower bound of GF for a KR BH is
obtained as $T\left(\omega\right)=\sec
h^{2}\left[-\frac{2(\alpha-1)\ell(\ell+1)+s^{2}-1}{8(\alpha-1)^{2}M\omega}\right]$
and for a BM BH, the lower bound is $T\left(\omega\right)=\sec
h^{2}\left[\frac{2\sqrt{\beta+1}\ell(\ell+1)-\sqrt{\frac{1}{\beta+1}}\left(s^{2}-1\right)}{8M\omega}\right]$.
We want to probe the impact of LV on GF and compare its effect in KR and BM
models. The qualitative nature of variation of GF with KR parameter $\alpha$
is shown in Fig. [14], and for BM BH, it is shown in Fig. [15]. A comparison
between KR and BM BHs is illustrated in Fig. [16].

Figure 14: Greybody bounds for KR BH. The left one is for the electromagnetic
field, and the right one is for the scalar field. We have taken $\ell=2$.

Figure 15: Greybody bounds for BM BH. The left one is for the electromagnetic
field, and the right one is for the scalar field. We have taken $\ell=2$.

Figure 16: Comparison of Greybody bounds. The left one is for the
electromagnetic field, and the right one is for the scalar field. We have
taken $\ell=2$ and $\alpha=\beta=0.2$.

It can be inferred from Figs. [14, 15] that the transmission probability
decreases with $\alpha$ and $\beta$. However, the variation of GF with $\beta$
is smaller. Our comparison of GFs between two BHs shows that the probability
of thermal radiation reaching spatial infinity is less for a KR BH. Next, we
investigate the power emitted in the $\ell$th mode which is given by yg2017 ;
fg2016

$P_{\ell}\left(\omega\right)=\frac{A}{8\pi^{2}}T(\omega)\frac{\omega^{3}}{e^{\omega/T_{H}}-1}.$
(29)

Here, $A$ is the horizon area yg2017 .

Figure 17: Variation of power emitted by KR BH in the $\ell$th mode with
$\omega$ for different values of $\alpha$. The left one is for the
electromagnetic field, and the right one is for the scalar field. We have
taken $\ell=1$.

Figure 18: Variation of power emitted by BM BH in the $\ell$th mode with
$\omega$ for different values of $\beta$. The left one is for the
electromagnetic field, and the right one is for the scalar field. We have
taken $\ell=1$.

Figure 19: Comparison of power emitted in $\ell$th mode. The left one is for
the electromagnetic field, and the right one is for the scalar field. We have
taken $\ell=2$ and $\alpha=\beta=0.2$.

A qualitative study of the variation of power emitted with $\omega$ for
different values of LV parameters $\alpha$ and $\beta$ as shown in Figs. [17,
18] reveal that the power emitted by a KR BH increases with the LV parameter
and the position of the peak shifts towards the right. On the other hand, the
energy emitted decreases with the LV parameter for a BM BH, and the frequency
where the power peaks shifts towards the left. A comparison between KR and BM
BHs regarding the thermal power they emit reveals that an observer at spatial
infinity will receive much more thermal radiation from a KR BH than a BM BH.

##  VII weak gravitational lensing

BHs, acting as lenses, force even light rays to deviate from their straight-
line path. Due to its dependence on the underlying spacetime, gravitational
lensing becomes a potent tool to gauge the impact of LV parameters. We will
study weak gravitational lensing in the background of KR and BM BHs with the
help of the Gauss-Bonnet theorem proposed in GW . The deflection angle can be
obtained using the formula ISHIHARA1 ; CARMO

$\gamma_{D}=-\int\int_{{}_{R}^{\infty}\Box_{S}^{\infty}}KdS,$ (30)

where ${}_{O}^{\infty}\Box_{S}^{\infty}$ is the quadrilateral shown in Fig.
(20) and K is the Gaussian curvature. We consider null geodesics where
$ds^{2}=0$ resulting in

$dt^{2}=\zeta_{ij}dx^{i}dx^{j},$ (31)

where

$\zeta_{ij}dx^{i}dx^{j}=\frac{1}{f(r)g(r)}dr^{2}+\frac{h(r)}{f(r)}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right).$
(32)

Figure 20: Schematic diagram of the quadilateral
${}_{O}^{\infty}\Box_{S}^{\infty}$.

We can obtain the Gaussian curvature using the expression WERNER

$\displaystyle K$ $\displaystyle=$ $\displaystyle\frac{{}^{3}R_{r\phi
r\phi}}{\zeta},$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\zeta}}\left(\frac{\partial}{\partial\phi}\left(\frac{\sqrt{\zeta}}{\zeta_{rr}}{}^{(3)}\Gamma^{\phi}_{rr}\right)-\frac{\partial}{\partial
r}\left(\frac{\sqrt{\zeta}}{\zeta_{rr}}{}^{(3)}\Gamma^{\phi}_{r\phi}\right)\right)$
, (33)

where $\zeta=\frac{h(r)}{f(r)^{2}g(r)}$, $\zeta_{rr}=\frac{1}{f(r)g(r)}$,
${}^{(3)}\Gamma^{\phi}_{rr}=0$, and
${}^{(3)}\Gamma^{\phi}_{r\phi}=\frac{1}{2}\frac{h^{\prime}(r)f(r)-h(r)f^{\prime}(r)}{f(r)h(r)}$,
′ representing derivative with respect to r. The final expression for the
Gaussian curvature becomes

$K=-\frac{f(r)\sqrt{gr}}{\sqrt{h(r)}}\frac{\partial}{\partial
r}\left(\sqrt{f(r)g(r)}\frac{\partial}{\partial
r}\sqrt{\frac{h(r)}{f(r)}}\right).$ (34)

The above expression lets us compute the Gaussian curvature for the BHs under
consideration. The Gaussian curvature for a KR BH is
$K=\frac{M(3M-2(\alpha+1)r)}{r^{4}}+\mathcal{O}\left(\frac{\alpha^{2}M}{r^{3}}\right)$
and for a BM BH, it is
$K=\frac{(\beta-1)M(2r-3M)}{r^{4}}+\mathcal{O}\left(\frac{\alpha^{2}M}{r^{3}}\right)$.
Assuming $r_{0}$ being the closest approach to the BH, we rewrite Eq. (30) as
ONO1

$\int\int_{{}_{O}^{\infty}\Box_{S}^{\infty}}KdS=\int_{\phi_{S}}^{\phi_{O}}\int_{\infty}^{r_{0}}K\sqrt{\zeta}drd\phi,$
(35)

We intend to obtain higher-order corrections in the deviation angle. To this
end, we first obtain the deflection angle assuming a straight-line trajectory
where $r=\frac{b}{sin\phi}$ and then using that deflection angle and the
trajectory given in CRISNEJO  as

$u=\frac{1}{r}=\frac{\sin\phi}{b}+\frac{M(1-\cos\phi)^{2}}{b^{2}}-\frac{M^{2}(60\phi\,\cos\phi+3\sin
3\phi-5\sin\phi)}{16b^{3}}+\mathcal{O}\left(\frac{M^{2}\alpha}{b^{5}}\right),,$
(36)

we obtain higher-order corrections. The initial deflection angle for a KR BH
is
$\gamma_{D}^{0}=M^{2}\left(\frac{3\pi}{4b^{2}}-\frac{9\pi\alpha}{8b^{2}}\right)+M\left(\frac{4}{b}-\frac{2\alpha}{b}\right)+\mathcal{O}\left(\frac{M^{3}}{b^{3}},\frac{M^{3}\alpha}{b^{3}}\right)$
and for a BM BH is
$\gamma_{D}^{0}=M^{2}\left(\frac{3\pi}{4b^{2}}-\frac{3\pi\alpha}{8b^{2}}\right)+M\left(\frac{4}{b}-\frac{2\alpha}{b}\right)+\mathcal{O}\left(\frac{M^{3}}{b^{3}},\frac{M^{3}\beta}{b^{3}}\right)$.
Now, integrating $\phi$ from $0$ to $\pi+\gamma_{D}^{0}$ and using the
trajectory [36], we obtain the deflection angle for a KR BH as

$\gamma_{D}=\frac{128M^{3}}{3b^{3}}+\frac{15\pi
M^{2}}{4b^{2}}+\alpha\left(-\frac{104M^{3}}{3b^{3}}-\frac{21\pi
M^{2}}{8b^{2}}-\frac{2M}{b}\right)+\frac{4M}{b}+\mathcal{O}\left(\frac{M^{4}}{b^{4}},\frac{M^{4}\alpha}{b^{4}}\right),$
(37)

and the deflection angle for a BM BH is

$\gamma_{D}=\frac{128M^{3}}{3b^{3}}+\frac{15\pi
M^{2}}{4b^{2}}+\beta\left(-\frac{64M^{3}}{3b^{3}}-\frac{15\pi
M^{2}}{8b^{2}}-\frac{2M}{b}\right)+\frac{4M}{b}+\mathcal{O}\left(\frac{M^{4}}{b^{4}},\frac{M^{4}\beta}{b^{4}}\right).$
(38)

One can observe from analytical expressions in Eqs. [21, 21] that the impact
of LV in KR and BM models on the gravitational lensing is the same, i.e., the
deflection angle decreases with $\alpha$ and $\beta$.

Figure 21: Deflection angle for weak gravitational lensing. The left panel is
for KR BH, the middle one is for BM BH, and the right one illustrates the
comparison.

The qualitative nature of variation of deflection angle with the impact
parameter $b$ for different values of KR parameter is illustrated in Fig.
[21], for BM BH it is shown in Fig. [21], and a comparison between KR and BM
BHs is carried out in Fig. [21]. Although the deflection angle decreases for
both types of BHs, the variation is less for a BM BH. The comparison between
KR and BM BHs reveals that a light ray deviates more from its straight-line
path when passing by a BM BH. However, the deviation for both BHs under
consideration is less than the Schwarzschild BH.

##  VIII Conclusions

Probing LV through its impact on various astrophysical observations has been
the subject of intense research for quite some time now. There are various
models where the LV is incorporated into gravity formulation. The Kalb-Ramond
field model and the Bumblebee model are two such models where the Lorentz
symmetry is spontaneously broken due to a field’s non-zero vacuum expectation
value. Our main focus in this endeavor is to conduct a comparative study of
the two models concerning the impacts of LV on different observables. We
delved into an in-depth study exploring differences in observational
signatures of LV arising in KR and BM models. We initiate our inquisition by
first introducing the two metrics. The uniqueness of the two solutions is
illustrated by their Kretschmann scalars, which are different from the
Kretschmann scalar for the Schwarzschild BH for non-zero KR parameter $\alpha$
and BM parameter $\beta$.
The position of the event horizon for a KR BH is $r_{h}=2M(1-\alpha)$,
indicating a decrease in the horizon radius with $\alpha$. The horizon radius
of a BM BH is the same as that of a Schwarzschild BH, i.e., $r_{h}=2M$. We
obtain differential equations and the effective potential for null geodesics
with the Lagrangian formulation. We get the radius of unstable circular orbits
for the two BHs by imposing appropriate conditions on the effective potential
and its derivatives. Similar to the horizon radius, increasing $\alpha$
decreases the photon radius and the associated impact parameter which are
given by $r_{p}=3M(1-\alpha)$ and $\beta_{p}=3\sqrt{3}(1-\alpha)^{(}3/2)$,
respectively and their values for a BM BH are same as those for a
Schwarzschild BH i.e., $r_{p}=3M$ and $b_{p}=3\sqrt{3}M$. Since the photon
radius and the critical impact parameter depend on the nature of the time and
azimuthal components of the metric. In contrast, the LV comes into the picture
in the radial component of the metric in the BM model, the photon radius and
the associated impact parameter do not depend on $\beta$. Qualitative and
quantitative variations of the photon radius and the critical impact parameter
for a KR BH are shown in Fig. [1] and Table [1]. The importance of the
critical impact parameter lies in the fact that a photon with this impact
parameter will be trapped in an unstable circular orbit around the BH. The
photon on a small perturbation from its circular path may either move towards
the BH and never come back or move towards a distant observer, contributing to
the optical appearance of the BH. Next, we turn our attention to investigating
the shadow of BHs with static and infalling accretions.
It is believed that BHs are surrounded by accretion disks. BHs gain mass by
accreting ambient matter. Accretion disks we considered are of two types:
static and infalling. For both types of accretion, we probed the impact of LV
on the observed intensity and the optical appearance of the BH. For static
accretion, the following are the conclusions we can make from our study:
$(i)$ The observed intensity increases with the LV parameter for both the BHs.
However, the intensity peaks at a lower impact parameter for a higher value of
KR parameter $\alpha$, but the position remains the same for BM BH
irrespective of the value of $\beta$. This can be observed from Figs. [III.1,
III.1]. The size of the bright ring decreases with increasing $\alpha$ but
remains the same for BM BH. This is because the intensity peaks at $b_{p}$
which decreases with $\alpha$ but remains invariant for BM BH.
$(ii)$ The intensity variation with $\alpha$ is significant for KR BH but is
found to be marginal with a variation of $\beta$. The brightness of the
circular ring in Fig. [III.1, III.1] is indicative of this observation.
$(iii)$ The comparison of intensities between KR and BM BHs illustrated in
Fig. [III.1] reveals that a KR BH will be brighter than a BM BH.
The observed intensity in infalling accretion is less, the intensity variation
with the impact parameter is smooth, and the central portion of the shadow
within the bright ring is darker than the static accretion. The impact of LV
parameters $\alpha$ and $\beta$ on the optical appearance with infalling
accretion are found to be as follows: $(i)$ Similar to the static accretion,
the intensity for infalling accretion peaks at lower impact parameters for KR
BH but remains the same for BM BH due to the reason already stated.
$(ii)$ Although the brightness of the photon ring increases with $\alpha$
similar to the static case, the impact of $\beta$ is found to be opposite to
that in the static accretion, i.e., the brightness of the photon ring
decreases with$\beta$. This is because the proper length in BM decreases with
$\beta$, causing the brightness to fall with increasing $\beta$.
The above results indicate that the optical appearance with accretion can be
used to differentiate between KR and BM BHs based on observables. We then move
on to the study of QNMs and ringdown waveforms.
When a BH is perturbed due to a field such as scalar or electromagnetic, it
returns to equilibrium by emitting GWs. QNMs are these modes that are
transient and eventually die down. QNMs are complex-valued functions of model
parameters and, as such, provide excellent avenues to probe the impact of
additional parameters arising in a model. We use the $6$th order Padé averaged
WKB method due to its higher accuracy than other WKB methods konoplya2 . We
first reduce the Klein-Gordon equation for the massless scalar field and the
Maxwell equation for the electromagnetic field to a desired
Schr$\ddot{o}$dinger-like equation that yields the potential required to
obtain QNMs. We then briefly discuss the $6$th order Padé averaged WKB method.
We present graphically the variation of frequency of GWs given by the real
part of QNMs with LV parameters in Fig. [3, 4], and the qualitative variation
of the decay rate with LV parameters are shown in Fig. [5, 6]. They reveal
that the frequency and the decay rate increase with the KR parameter $\alpha$
for both perturbations. The decay rate decreases with BM parameter $\beta$ for
scalar and electromagnetic perturbations. However, an increase(decrease) in
frequency is observed for an increase in $\beta$ for electromagnetic(scalar)
perturbation. A comparison of frequency and decay rate between GWs originating
from KR and BM BHs is made in Figs. [7, 8]. It is observed that a KR BH emits
GWs of higher frequencies, and they decay faster than those originating from a
BM BH. Numerical values given in Tables [2, 3] validate these conclusions. We
also provide ringdown waveforms for KR and BM BHs. They reveal that a KR BH,
once perturbed from its equilibrium, will return to equilibrium quicker than a
Schwarzschild or BM BH by emitting GWs of higher frequency. Comparing a BM BH
with a Schwarzschild BH reveals that a BM BH will take longer to return to
equilibrium.
A BH emits radiation, and a temperature is associated with the BH horizon. The
radiation emitted by a BH at the horizon and the radiation received by a
distant observer differ due to the redshift factor. The greybody distribution
provides the radiation perceived by an asymptotic observer, and GF gives the
probability of Hawking radiation transmission. We intend to observe the
impacts of LV on the Hawking temperature, GF, and power emitted. The Hawking
temperature is found to be decreasing with $\beta$ but increasing with
$\alpha$. As far as GF is concerned, it decreases with $\alpha$ and $\beta$.
However, a comparison of GF between two BHs reveals that the probability of
the Hawking radiation getting detected by an asymptotic observer is higher for
a BM BH. In Figs. [17, 18], we illustrate the variation of the power emitted
in the form of Hawking radiation with frequency $\omega$ for different values
of $\alpha$ and $\beta$. It reveals that the energy emitted by a KR BH is more
than that emitted by a BM or Schwarzschild BH. However, a BM BH emits less
energy than a Schwarzschild BH.
Finally, the gravitational lensing in the weak field limit is considered to
differentiate between KR and BM BHs based on observational signature. We
employed the Gauss-Bonnet theorem to obtain the deflection angle. We obtained
higher order correction terms by first obtaining the deflection angle assuming
a straight-line trajectory where $r=\frac{b}{sin\phi}$ and then by using that
deflection angle and the trajectory given in CRISNEJO . Analytical expressions
of deflection angles for the KR and BM BHs and graphical representations in
Fig. [21] reveal that the deflection angle for KR and BM BHs is smaller than
that for Schwarzschild BH. Fig. [21] shows that a light ray deviates more from
its straight-line path due to a BM BH than a KR BH. With possible future
experimental results on QNMs and deflection angle, we will be able to not only
differentiate between KR and BM BHs but also validate or invalidate any of
them.

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