# Gertsenshtein-Zel′dovich gravitational analog can enable detection of solar
mass black holes in Milky way galaxy

Susmita Jana<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Bombay, Mumbai 400076, India.  Rituparno Goswami
<EMAIL_ADDRESS>Astrophysics Research Centre and School of Mathematics,
Statistics and Computer Science, University of KwaZulu-Natal, Private Bag
X54001, Durban 4000, South Africa.  S. Shankaranarayanan<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India.  Sunil D. Maharaj<EMAIL_ADDRESS>Astrophysics Research Centre and
School of Mathematics, Statistics and Computer Science, University of KwaZulu-
Natal, Private Bag X54001, Durban 4000, South Africa.

###### Abstract

The Milky Way galaxy is estimated to be home to ten million to a billion
stellar-mass black holes (BHs). Accurately determining this number and
distribution of BH masses can provide crucial information about the processes
involved in BH formation, the possibility of the existence of primordial BHs,
and interpreting gravitational wave (GW) signals detected in LIGO-VIRGO-KAGRA.
Sahu et al. recently confirmed one isolated stellar-mass BH in our galaxy
using astrometric microlensing Sahu  _et al._ (2022). This work proposes a
novel method to identify such BHs using the gravitational analog of the
Gertsenshtein-Zel′dovich (GZ) effect. We explicitly demonstrate the generation
of GWs when a kilohertz(kHz) electromagnetic (EM) pulse from a pulsar is
intervened by a spherically symmetric compact object situated between the
pulsar and Earth. Specifically, we show that the curvature of spacetime acts
as the catalyst, akin to the magnetic field in the GZ effect. Using the
covariant semi-tetrad formalism, we quantify the GW generated from the EM
pulse through the _Regge-Wheeler tensor_ and express the amplitude of the
generated GW in terms of the EM energy and flux. We demonstrate how GW
detectors can detect stellar-mass BHs by considering known pulsars within our
galaxy. This approach has a distinct advantage in detecting stellar mass BHs
at larger distances since the GW amplitude falls as $1/r$.

###### pacs:

04.20.Cv , 04.20.Dw

##  I Introduction

General Relativity (GR) has been an experimental triumph for over a century.
Einstein’s key predictions have been verified experimentally with remarkable
accuracy Will (2014a). Besides aiding in comprehending certain cosmological
and astrophysical phenomena, this theory finds utility in satellite laser
ranging, celestial navigation, and very long baseline interferometry (VLBI).
However, several phenomena are yet to be discovered due to non-linear
equations governing GR. Some phenomena, although predicted theoretically, have
not been confirmed. One such phenomenon is the Gertsenshtein-Zel′dovich (GZ)
effect Gertsenshtein (1962); Pustovoit and Gertsenshtein (1962); Zel’dovich
(1974); Zheng  _et al._ (2018); Domcke and Garcia-Cely (2021).

Gertsenshtein observed that electromagnetic (EM) waves and gravitational waves
(GWs) have the same propagation speeds and both are linearly related, and
suggested that wave resonance should be present between them Gertsenshtein
(1962); Pustovoit and Gertsenshtein (1962). Using the linearized Einstein
field equations, he showed that EM waves produce GWs via wave resonance when
they pass through a strong magnetic field. Similarly, GWs passing through a
strong magnetic field produce EM waves Zel’dovich (1974); Kolosnitsyn and
Rudenko (2015). In the quantum picture, the GZ effect can be conceptualized as
the constant magnetic field facilitates the transformation of spin-1 particles
(photons) to spin-2 particles (gravitons) Palessandro and Rothman (2023). To
observe this effect in the laboratory requires a very high magnetic field
Stephenson (2005); Aggarwal  _et al._ (2021). Recently, it has been shown that
GZ effectS explain the origin of Fast Radio Bursts Kushwaha  _et al._ (2023).

It is natural to ask if _gravity can act analogous to the magnetic field_. The
metric and Riemann curvature tensors play roles analogous to those of
potentials and field strengths, respectively, in electromagnetism. It is
important to stress that apart from the formal analogy, GR, even the
linearised theory, and electromagnetism are fundamentally different Will
(2014b). In spite of the difference, we explicitly show that a gravitational
analog to the GZ effect exists for a static, spherically symmetric compact
object. In the era of multimessenger astronomy, efforts are being made to
detect the same object or event with either EM/GWs and particles (neutrinos),
EM and GWs, or all three together Mészáros  _et al._ (2019); Murase and Bartos
(2019). As we show, strong gravity is observable from an entirely different
perspective.

It is known that pulsars emit EM waves across the entire spectrum Ostriker and
Gunn (1969); Lorimer and Kramer (2004); Condon and Ransom (2016). They are
observed in radio, optical, X-ray, and $\gamma$-ray Taylor  _et al._ (1993);
Stappers  _et al._ (2011); Johnston  _et al._ (2021). Pulsars were first
identified at 81.5 MHz, and most initial follow-ups were conducted at low
frequencies (below 200 MHz) Hewish (1970). However, the preponderance of
pulsar observations started in the mid-1970s at 350 MHz due to two factors
Kondratiev and LOFAR Pulsar Working Group (2013); Stovall  _et al._ (2015):
Dispersion and scattering effects in the interstellar medium (ISM). According
to Bates et al., pulsars typically have spectral indices in their flux
densities of approximately $-1.41$ Bates  _et al._ (2013). It is generally
accepted that the radio emission (in a narrow band) is due to coherent
processes, and the high-energy (x-ray and $\gamma$-ray) emission from the
pulsar is due to incoherent processes, like free-free emissions Lorimer
(2008). Despite numerous proposed mechanisms attempting to account for
coherent radio and incoherent high-energy emissions, none have successfully
explained all observed pulsar characteristics Lorimer (2008).

The magnetic dipole model is the simplest pulsar model that accounts for many
of the observed properties of pulsars Goldreich and Julian (1969); Ostriker
and Gunn (1969). According to this model, pulsars generate their EM radiation
through the rotational energy of the neutron star Gold (1968). Consequently,
the predominant EM radiation is attributed to magnetic dipole radiation,
mainly at low frequencies. However, the current generation radio telescopes
faces limitations in measuring radio waves below MHz. Nonetheless, the
extremely low-frequency range can offer crucial insights into pulse profile
evolution Lorimer and Kramer (2004), particularly given its significance at
low frequencies. Although frequencies less than a few kHz cannot propagate
through ISM, the frequency range between $3-6$ kHz falls for GW detection in
GW detectors. The proposed mechanism provides an alternative approach to
comprehending pulsars, offering an indirect means of detecting black holes
(BHs) in our galaxy.

It is estimated that the Milky Way harbours around $10^{8}$ isolated BHs with
an average mass of around $14M_{\odot}$, along with $9.3\times 10^{7}$ BHs
engaged in binary systems, boasting an average mass of $19M_{\odot}$Olejak
_et al._ (2020); Brown and Bethe (1994). The detection of solitary BHs poses
considerable challenges using current astrophysical observational instruments.
Using astrometric microlensing, identifying these BHs has become feasible
Kains  _et al._ (2018); Sahu  _et al._ (2022). Given the prevalence of
isolated stellar-mass BHs in our galaxy, the proposed mechanism presents a
novel approach to detect them via GWs.

Precisely, we want to investigate low-frequency electromagnetic (EM) pulses
emitted by pulsars, which, upon interaction with isolated BHs in our galaxy,
generate GWs of the same frequency. As in astrometric microlensing Sahu  _et
al._ (2022), the BH is between the observer and the source. However, the key
difference is that, unlike microlensing, the BH is active and converts the
incoming EM waves to GWs. To quantify the generated GWs, we focus on a
comoving (fictitious) observer near a static spherically symmetric spacetime
and show that a test EM pulse converts to GWs by interacting with the geometry
of the background spacetime. The setup is similar to Susskind’s thought
experiment for EM memory Susskind (2015).

##  II Gravitational GZ

Consider a timelike (fictitious) observer with a worldline $\gamma$, in a
locally spherically symmetric vacuum around a BH of mass ‘$m$.’ By Birkhoff’s
theorem, the spacetime will be locally Schwarzschild for the observer, and by
aligning the 4-velocity along the direction of the timelike Killing vector in
the Schwarzschild spacetime, the observer can always be made static. As shown
in Fig. 1, astrophysical phenomenon generates non-spherical EM pulse of finite
duration. At a later time, the pulse passes the (fictitious) observer. Since
the EM pulse travels at the speed of light, causality conditions ensure that
the observer does not perceive any effect of the pulse before it reaches the
worldline $\gamma$. Interactions occur during the finite duration of the pulse
passage. However, being a test pulse, the components of the energy-momentum
tensor are much smaller than a covariantly defined scale in the spacetime,
namely the central Schwarzschild mass Goswami and Ellis (2012a).

Figure 1: Gravitational analog of the GZ effect

Hence, the spacetime near the observer remains _almost_ spherically symmetric
and _almost_ vacuum. After the pulse passes, the local spacetime returns to a
vacuum state. Nevertheless, the non-sphericity persists, containing the memory
of the non-spherical test pulse Jana and Shankaranarayanan (2023). This non-
sphericity is subsequently radiated away through GWs, described by the
_traceless Regge-Wheeler tensor_ Clarkson and Barrett (2003a).

To rigorously and transparently analyze the outcome of this thought
experiment, we divide $\gamma$ into three phases — _the pre-pulse phase, the
pulse phase, and the post-pulse phase_. We express the field equations in
terms of covariant and gauge invariant variables for each of these phases
independently, up to linear order perturbations. This approach provides a
comprehensive understanding of how the stress-tensor of the EM test pulse
introduces geometrical variables determining linear-level non-sphericity in an
open set around $\gamma$. Since the geometrical quantities of the background
spherically symmetric spacetime (before the arrival of EM pulse) act as the
catalyst in the generation of the GW, we term this phenomenon the
_gravitational analog to the GZ effect_ Pustovoit and Gertsenshtein (1962);
Zel’dovich (1974); Gertsenshtein (1962).

To quantify the observed effect, we employ the covariant $1+1+2$ semitetrad
formalism van Elst and Ellis (1996); Clarkson and Barrett (2003a); Betschart
and Clarkson (2004); Hansraj  _et al._ (2020). The observable quantities are
evaluated w.r.t a comoving observer characterized by a 4-velocity $u^{a}$
satisfying $u^{a}u_{a}=-1$. Additionally, due to spherical symmetry, the
spacetime features a preferred spacelike direction $n^{a}$ with
$n^{a}n_{a}=1$. Thus, the 4-D spacetime metric can be decomposed as

$g_{ab}=-u_{a}u_{b}+n_{a}n_{b}+N_{ab}\,,$ (1)

where $N_{ab}$ is the projection tensor on the 2-D sheets orthogonal to
$u^{a}$ and $n^{a}$. The directional derivative along $u^{a}$ is denoted by
dot derivative ‘($\;\dot{}\;$)’ and the fully projected directional derivative
along $n^{a}$ is denoted by hat derivative ‘($\;\hat{}\;$)’. The complete
system is described by the geometrical variables of these two congruences
(such as expansion, shear, or vorticity) along with the decomposed variables
of the Weyl tensor and EM energy-momentum tensor. For details, see Refs. van
Elst and Ellis (1996); Clarkson and Barrett (2003a); Betschart and Clarkson
(2004); Hansraj  _et al._ (2020) and Appendix A.

###  II.1 The pre-pulse phase

As discussed above and shown in Fig. 1, the causality conditions ensure that
the pre-pulse phase will be spherically symmetric vacuum around an open set
$\mathcal{S}$ containing $\gamma$, and hence will be locally equivalent to a
part of the maximally extended Schwarzschild solution in $\mathcal{S}$. As the
fictitious observer is outside the BH, we can always assume the existence of a
timelike Killing vector $\xi^{a}$, for which the volume expansion ($\Theta$)
and the shear scalar ($\Sigma$) vanish. By aligning the tangent of $\gamma$ to
this Killing vector $\xi^{a}$ at every point in this pre-pulse phase, we can
make the observer static; That is, the directional derivatives of all
geometrical variables along the observer congruence (dot derivatives) vanish.
The only non-zero geometrical variables in the background spacetime are
Clarkson and Barrett (2003b)

$\mathcal{D}_{0}=\\{\phi,{\mathcal{A}},{\mathcal{E}}\\},$ (2)

where $\phi$ is the spatial expansion of the spacelike congruence $n^{a}$,
${\mathcal{A}}$ is the acceleration scalar for the observer, and
${\mathcal{E}}$ is the Weyl scalar extracted from the electric part of the
Weyl tensor. These satisfy the following propagation (hat derivative along
radial coordinate) equations

$\hat{\phi}=-{\mathcal{E}}-\phi^{2}/2\;,\;\;\hat{\cal A}=-\left({\cal
A}+\phi\right){\cal A}\;,$ (3)

with the constraint ${\mathcal{E}}=-{\cal A}\phi$. As the spacetime is
spherically symmetric, the geometry can be described by foliations of
spherical 2-shells at any given instant. The Gaussian curvature of these
spherical shells is

$K=-{\mathcal{E}}+\phi^{2}/4\;.$ (4)

From the above equations, it is clear that the electric part of the Weyl
scalar is proportional to a $(3/2)th$ power of the Gaussian curvature, and the
proportionality constant (which is the Schwarzschild mass ‘$m$’) produces a
covariant scale in the problem. We can also define the areal radius of the
spherical 2-shells $r$, such that the Gaussian curvature is $1/r^{2}$.
Integrating the propagation equations in terms of this variable, we get
Clarkson and Barrett (2003b)

$\displaystyle
K=\frac{1}{r^{2}}~{},~{}\phi=\frac{2}{r}\sqrt{F(r)}~{},~{}{\mathcal{E}}=-\frac{2m}{r^{3}}~{},~{}{\cal
A}=-\frac{{\mathcal{E}}}{\phi}\,,$ (5)

where, $F(r)\equiv 1-2m/r$. These completely specify the pre-pulse geometry.

###  II.2 The pulse phase

During this phase, the worldine $\gamma$ intersects the null cones of the test
pulse for finite proper time. Due to the non-spherical pulse, the open sets
around $\gamma$ will be non-spherical in the linear-order perturbations,
which, as we show, is entirely dependent on the non-spherical energy-momentum
tensor of the test pulse. In this phase, the spacetime is almost vacuum,
spherical locally and, hence, almost Schwarzschild. As shown in Refs. Ellis
and Goswami (2013); Goswami and Ellis (2011, 2012b), the rigidity of the
vacuum spherically symmetric manifold in GR continues even in the perturbed
scenario, and this almost Schwarzschild manifold will continue over the entire
pulse phase. In the semitetrad formalism, the stress tensor of the test EM
pulse is

$\displaystyle T_{ab}$ $\displaystyle=$ $\displaystyle\mu
u_{a}u_{b}+ph_{ab}+2{\mathcal{Q}}\,e_{(a}u_{b)}+2{\mathcal{Q}}_{(a}u_{b)}$ (6)
$\displaystyle+\Pi(e_{a}e_{b}-N_{ab}/2)+2\Pi_{(a}e_{b)}+\Pi_{ab}\;.$

$T_{ab}$ when expressed as $E_{a},B_{a}$ does not provide much insight. For
completeness, they are provided in Appendix B. In order for the pulse not to
back-react with the metric, the ratio of the magnitude of each component of
the above stress-tensor w.r.t the background Gaussian curvature must be less
than the BH mass ($m$). This has been verified in Refs. Ellis and Goswami
(2013); Goswami and Ellis (2011, 2012b). Due to the pulse, the perturbed field
equations contain both the background quantities and the first-order variables
that determine the non-sphericity of the geometry. The total number of
geometrical and EM pulse variables are given by
$\mathcal{D}\equiv\mathcal{D}_{0}\bigcup\mathcal{D}_{1}^{\rm
geom}\bigcup\mathcal{D}_{1}^{\rm EM}$, where

$\displaystyle\mathcal{D}_{1}^{\rm geom}$ $\displaystyle=$
$\displaystyle\left\\{\Theta,\hskip 0.99594pt\Omega,\hskip
0.99594pt\Sigma,\hskip 0.99594pt\mathcal{H},\hskip 0.99594pt\xi,\hskip
0.99594pt\mathcal{A}_{a},\hskip 0.99594pt\Omega_{a},\hskip
0.99594pt\Sigma_{a},\hskip 0.99594pt\alpha_{a},\hskip 0.99594pta_{a},\hskip
0.99594pt\mathcal{E}_{a},\hskip 0.99594pt\mathcal{H}_{a},\right.$ (7)
$\displaystyle\left.\hskip 0.99594pt\Sigma_{ab},\hskip
0.99594pt\zeta_{ab},\hskip 0.99594pt\mathcal{E}_{ab},\hskip
0.99594pt\mathcal{H}_{ab}\right\\}$ $\displaystyle\mathcal{D}_{1}^{\rm EM}$
$\displaystyle=$ $\displaystyle\left\\{\mu,\hskip 0.99594ptp,\hskip
0.99594pt{\mathcal{Q}},\hskip 0.99594pt\Pi,\hskip
0.99594pt{\mathcal{Q}}_{a},\hskip 0.99594pt\Pi_{a},\hskip
0.99594pt\Pi_{ab}\right\\}.$ (8)

Physical observables in GR must be gauge-invariant van Elst and Ellis (1996).
We need to identify the gauge-invariant variables from the above set to
compare the derived quantities with observations. Pulse introduces variations
in $u^{a}$ and $n^{a}$ along spacelike and timelike directions, rendering the
variables in the set $\mathcal{D}_{0}$ non-gauge-invariant. As per the
Stewart-Walker lemma, the quantities that vanish in the background spacetime
are automatically gauge-invariant Stewart and Walker (1974); Ellis and Bruni
(1989). For complete gauge invariance, we replace these variables with the
following three Betschart and Clarkson (2004); Nzioki  _et al._ (2016)

$\mathcal{D}_{1}^{\rm
GI}=\left\\{W_{a}=\delta_{a}{\mathcal{E}},Y_{a}=\delta_{a}\phi,Z_{a}=\delta_{a}{\cal
A}\right\\}.$ (9)

Hence, the complete set of first-order variables $\mathcal{D}_{1}^{\rm
GI}\bigcup\mathcal{D}_{1}^{\rm geom}\bigcup\mathcal{D}_{1}^{\rm EM}$,
determine the perturbed spacetime in a covariant and gauge-invariant way. For
these variables, linearized evolution and propagation equations can be written
as follows

$\displaystyle\mathcal{L}^{(\rho)}\mathcal{D}^{\rm GI}=\mathcal{D}^{\rm
geom}+f(\mathcal{D}^{\rm EM})\,,$ (10)

where, $\mathcal{L}^{(\rho)}$ represents evolution, propagation, or projected
covariant derivatives on $2-$space. For details, see Appendix C. One key
feature of these equations is that the stress-tensor of the EM field (at the
instant when the pulse reaches the worldline $\gamma$) sources these
linearized equations. Since the $\mathcal{D}^{\rm geom}$ variables vanish
before the pulse arrives, the non-trivial initial data that generated these
gauge-invariant variables in the pulse phase are completely supplied by the EM
pulse.

###  II.3 The post pulse phase

As the pulse leaves the fictitious observer, the open set $\mathcal{S}$
containing the $\gamma$ post-pulse will revert to its vacuum state. However,
the non-sphericity will persist. The initial data that evolves the non-
sphericity variables in the post-pulse phase are those that Cauchy evolved
from the EM pulse initial data of the test pulse in the pulse phase. This
clearly shows how the local inherent Killing symmetry of the spacetime becomes
altered as the test pulse passes the open set $\mathcal{S}$ containing
$\gamma$. Physically, we can view this process as a perturbed black hole
relaxing to a stationary black hole by emitting QNMs Chandrasekhar (1992).

To see transparently how the non-sphericity of the open set is radiated away
via GWs, we construct the following dimensionless, covariant, gauge invariant,
frame invariant, transverse trace-free tensor $M_{\\{ab\\}}$ Betschart and
Clarkson (2004); Nzioki  _et al._ (2016)

$M_{ab}=\phi\,r^{2}\,\zeta_{ab}/2-r^{2}\,{\mathcal{E}}^{-1}\,\delta_{\\{a}W_{b\\}}/3\;.$
(11)

$\zeta_{ab}$ represents distortion of the 2-D sheet and quantifies GW
amplitude in free space. $M_{ab}$ is the Regge-Wheeler tensor and obeys the
following closed wave equation for odd and even parity cases

$\ddot{M}_{\\{ab\\}}-\hat{\hat{M}}_{\\{ab\\}}-{\cal
A}\,{\hat{M}}_{\\{ab\\}}+[\phi^{2}+{\mathcal{E}}]M_{ab}-\delta^{2}M_{ab}=0~{}.$
(12)

Interestingly, the tensor $M_{ab}$ gives a measure of sheet deformation via
the electric part of the Weyl scalar and the deformation tensor related to the
preferred spacelike direction Clarkson and Barrett (2003a); Clarkson  _et al._
(2004). In the pre-pulse phase $M_{ab}=0$, while $M_{ab}\neq 0$ in the post-
pulse phase. Thus, the Regge-Wheeler tensor contains the memory of the EM
pulse Jana and Shankaranarayanan (2023). Thus, the background quantities
$(\phi,{\mathcal{E}})$ act as the catalyst, akin to the magnetic field in the
GZ effect.

To investigate the nature of the GWs and their dependence on the EM pulse
stress tensor, we decompose the geometrical variables as an infinite sum of
components relative to a basis of harmonic functions. This allows us to
replace angular derivatives appearing in the equations with a harmonic
coefficient. Following Ref. Clarkson and Barrett (2003b), we introduce a set
of dimensionless spherical harmonics $Q=Q^{(\ell,m)}$ ($m=-\ell,\cdots,\ell$),
defined in the background, as eigenfunctions of the spherical Laplacian
operator: $\delta^{2}Q=-[\ell(\ell+1)/r^{2}]\,Q$, $\hat{Q}=\dot{Q}=0$.
Expanding the first-order scalar ${\Psi}$ in terms of harmonic functions

${\Psi}=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{m=\ell}{\Psi}_{\mathsf{S}}^{(\ell,m)}Q^{(\ell,m)}={\Psi}_{\mathsf{S}}\,Q,$
(13)

where the sum over $\ell$ and $m$ is implicit in the last equality. The
replacements that must be made for scalars when expanding the equations in
spherical harmonics are

$\\!\\!{\Psi}={\Psi}_{\mathsf{S}}\,Q~{},\delta_{a}{\Psi}=r^{-1}{\Psi}_{\mathsf{S}}\,Q_{a}~{},\varepsilon_{ab}\delta^{b}{\Psi}=r^{-1}{\Psi}_{\mathsf{S}}\,\bar{Q}_{a}\,,$
(14)

where the sums over $\ell$ and $m$ are implicit and $\bar{Q}_{a}$ is the odd
parity vector harmonics. Note that the moment the EM pulse arrives,
$\dot{W}_{a}$ is non-zero (even if other non-spherical $\mathcal{D}^{\rm
geom}$ quantities are zero).

$\dot{W}_{a}=-\frac{1}{2}\phi\left(\delta_{a}{\mathcal{Q}}\right)-\frac{2}{3}\delta_{a}\dot{\mu}$
(15)

Since the background space-time is Schwarzschild, we set
$u^{a}\equiv(\,1/\sqrt{F(r)},\,0,\,0,\,0)$. Integrating the above equation
w.r.t the Schwarzschild coordinate time $t$, we get

$\displaystyle W_{a}=-\frac{F(r)^{\frac{1}{2}}}{2}\phi\left(\delta_{a}\int
dt\,{\mathcal{Q}}\right)-\frac{2}{3}\delta_{a}\mu+C_{1}(\theta,\,\phi).$ (16)

Rewriting in terms of spherical and time harmonics as

$\left(\int
dt\,{{\mathcal{Q}}}\right)=T^{(\omega)}{{\mathcal{Q}}}_{\mathsf{S}}\,Q,\;\;\mu=T^{(\omega)}{\mu}_{\mathsf{S}}\,Q,$
(17)

and decomposing $M_{ab}=T^{(\omega^{{}^{\prime}})}{M}_{\mathsf{T}}\,Q_{ab}$
(where $Q_{ab}=\delta_{\\{a}\delta_{b\\}}\,Qr^{2}$ is the electric parity
tensor spherical harmonics, $\omega$ is the frequency of the incoming EM pulse
and $\omega^{{}^{\prime}}$ represents the frequency of the generated GW), Eq.
(11) becomes

$\displaystyle T^{(\omega^{{}^{\prime}})}{M}_{\mathsf{T}}\,Q_{ab}$
$\displaystyle=$
$\displaystyle(r^{2}/6)F(r)^{\frac{1}{2}}r^{2}\,\phi\,{\mathcal{E}}^{-1}\,\delta_{\\{a}\delta_{b\\}}T^{(\omega)}{{\mathcal{Q}}}_{\mathsf{S}}\,Q$
(18)
$\displaystyle+(2r^{2}/9)\,{\mathcal{E}}^{-1}\,\delta_{\\{a}\delta_{b\\}}T^{(\omega)}{\mu}_{\mathsf{S}}\,Q,$

At resonance, $\omega=\omega^{{}^{\prime}}$, the amplitudes of EM and GWs are
related by

$\displaystyle{M}_{\mathsf{T}}\,=-\frac{r^{2}}{6m}\left(1-\frac{2m}{r}\right){{\mathcal{Q}}}_{\mathsf{S}}-\frac{1}{9}\,\frac{r^{3}}{m}{\mu}_{S},$
(19)

where ${\mathcal{Q}}_{\mathsf{S}}$ is the time integral of the amplitude of
the EM Poynting vector component ${\mathcal{Q}}$ and $\mu_{\mathsf{S}}$ is the
EM energy density and can be related to the Luminosity $({\cal L}_{\rm EM})$
via the relation ${\cal L}_{\rm EM}\,\Delta t$ where $\Delta t$ is the pulse
duration.

This is the key result of this work, regarding which we want to discuss the
following points: Firstly, the above relation is only applicable in the close
vicinity of the fictitious observer $\gamma$, while interacting with the EM
pulse in the pulse phase. At the moment of the EM pulse arrives
$\dot{\zeta}_{ab}=0$, while another first order variable $\Sigma_{ab}$ sources
$\ddot{\zeta}_{ab}$. Hence when calculating the Regge Wheeler tensor we can
safely neglect the $\zeta_{ab}$ term.

Secondly, the outgoing GW amplitude depends on the Poynting vector and energy
density Jana and Shankaranarayanan (2023). In other words, the contribution to
the QNMs arises from the EM memory. As we show, this can be observable in the
next generation GW experiments. Thirdly, like the canonical GZ effect, the
amplitude is maximum at the resonance and is suppressed away from the
resonance. Thus, detecting such a GW signal provides information on the
incoming EM wave. If the incoming EM waves are coherent and monochromatic,
this mechanism can provide key properties of various astrophysical processes
that are beyond the reach of telescopes in the EM band. Lastly,
${M}_{\mathsf{T}}$ derived above is for the static observer w.r.t the BH. To
compare with the observations, we need to transform the above quantity to the
static (detector) frame. Since, ${M}_{\mathsf{T}}$ is gauge and frame
invariant Clarkson and Barrett (2003a), in asymptotically flat spacetime
(${\mathcal{E}}\sim 1/r^{3},\phi\sim 1/r$), we have $M_{ab}\sim r\zeta_{ab}$.
$\zeta_{ab}$ is related to plane GW in Minkowski space-time via the relation:
$\zeta_{ab}\sim(1/2)\partial_{z}h_{ab}^{\rm TT}$ Clarkson  _et al._ (2004).

Figure 2: Schematic figure showing the location of Pulsar, spherically
symmetric compact object, and the earth or space-based detector.

##  III Observational implications and conclusions

We now discuss the implications of our results for stellar mass BH
observations in our galaxy. As mentioned earlier, pulsars emit radiation
across the entire electromagnetic spectrum, and the magnetic dipole model
illustrates the origin of pulsar emission from the rotational kinetic energy
of a neutron star Condon and Ransom (2016). While pulsars are stable to a
precision that rivals atomic clocks, many pulsars show sudden spin-up events,
referred to as glitches Haskell and Melatos (2015); Zhou  _et al._ (2022). The
pulse profile associated with these glitches (in Vega and Crab) suggests an
internal origin. However, no mechanism can explain these glitches Haskell and
Melatos (2015); Zhou  _et al._ (2022).

As shown in Fig. 2, consider a BH intervening pulsar and Earth. Let the
distance between the pulsar and the BH be $d$, and the distance between the BH
and the Earth by $D$. A typical pulsar in our galaxy is at a kpc distance from
the Earth Manchester  _et al._ (2005); Lopez-Coto  _et al._ (2022), hence we
set $d\sim D\sim kpc$. The magnetic dipole model Condon and Ransom (2016)
predicts pulsars with intrinsic luminosity ${\cal L}_{\rm
EM}\sim{\Omega_{0}^{4}}\,10^{38}~{}{\rm erg}/s$ where $\Omega_{0}$ is the
dimensionless frequency scaled w.r.t $30~{}{\rm Hz}$, i. e.,
$\Omega_{0}=\Omega_{e}/30$. [Depending on the pulsar, we have
$10^{3}\leq\Omega_{0}\leq 1$ Manchester  _et al._ (2005); Lorimer and Kramer
(2004).] Since the Poynting vector remains the same at the source and the
detector, the flux emanating from the pulsar will be the same as the flux
received by the fictitious observer near the solar mass BH.

Substituting the above Luminosity, $\Delta t\sim 1\,s$ and $r\sim 10^{5}{\rm
cm}$ (Schwarzschild radius) in Eq. (19), we get

$\displaystyle M_{\rm T}\sim{\Omega_{0}^{4}}\,10^{-2}\,.$ (20)

This is the GW amplitude measured by the fictitious observer near the BH.
Details extended in the Appendix D. As mentioned above, the GW amplitude
measured by the GW detector on Earth is given by $\zeta_{ab}$, hence, $h_{\rm
T}\propto M_{\rm T}/D$ Creighton and Anderson (2011). Setting $D\sim{\rm
kpc}$, we have $h_{\rm T}\sim\Omega_{0}^{4}\,10^{-23}$. This amplitude is well
in line with the expected sensitivity of Einstein Telescope and Cosmic
Explorer in the frequency range $3-6~{}$kHz Hild  _et al._ (2011); Maggiore
_et al._ (2020); Evans  _et al._ (2021); Srivastava  _et al._ (2022).

To conclude, the gravitational analog of the GZ effect enables the detection
of solar mass black holes in the Milky Way galaxy. Though our analysis is for
a Solar-mass BH, the analysis is applicable for any non-rotating compact
object, including some exotic compact objects Cardoso and Pani (2019); Völkel
(2020). This approach has one distinct advantage over the astrometric
microlensing Sahu  _et al._ (2022) — it can probe BHs at larger distance. In
the case of microlensing, the detection is through the measurement of photon
energy, which falls off as $1/D^{2}$. However, our approach uses GW amplitude,
which falls as $1/D$. Hence, it can be a definite probe for detecting stellar
mass BHs at further distances. This approach introduces novel possibilities
for identifying and studying stellar-mass BHs through their interactions with
EM pulses from pulsars.

###### Acknowledgements.

The authors thank I. Chakraborty, S. Malik and S. Mandal for comments on the
earlier draft. SJ thanks IITB-IOE travel fund to U. of KwaZulu-Natal where
this work was initiated. RG thanks the South African National Research
Foundation (NRF) for support. SS thanks SERB-CRG for support.

##  Appendix A Semiterad covariant formalism

In this section, we briefly recapitulate the semi-tetrad formalisms developed
in Ellis and van Elst (1999); Clarkson and Barrett (2003b), which enables us
to study the problem geometrically. The key point of these semi-tetrad
decompositions is that they are local decompositions defined on any open set
$\mathcal{S}$. In the first step of this decomposition, the properties of
spacetime are studied with respect to a real or fictitious observer whose
velocity is along the tangent of a timelike congruence. Thereafter, if the
spacetime has certain symmetries like local rotational symmetry, a preferred
spatial direction exists. The spacetime is then further decomposed using this
preferred spatial congruence. The field equations are then recast in terms of
the geometrical variables related to these congruences and the curvature
tensor of the spacetime (suitably decomposed using the congruences).

###  A.1 Semitetrad 1+3 formalism

$1+3$ covariant formalism is a well-known formalism widely used to study
relativistic cosmology in the frame of different models of general
relativity(GR) and cosmology. The spacetime is locally sliced into the
timelike direction $u^{a}\equiv dx^{a}/d\tau$ and spacelike hypersurface,
which is orthogonal to $u^{a}$. Here, $\tau$ is the affine parameter. In the
1+3 formalism Ellis and van Elst (1999), the timelike unit vector ${u^{a}}$
${\left(u^{a}\hskip 0.99594ptu_{a}=-1\right)}$ is used to split the spacetime
locally in the form ${\mathcal{R}\otimes\mathcal{V}}$, where ${\mathcal{R}}$
is the timeline along ${u^{a}}$ and ${\mathcal{V}}$ is the 3-space
perpendicular to ${u^{a}}$. Thus, the metric becomes

$g_{ab}=-u_{a}\hskip 0.99594ptu_{b}+h_{ab},$ (21)

where ${h_{ab}}$ is the projection tensor used to project any vector or tensor
on 3-space perpendicular to $u^{a}$. $h_{ab}$ becomes metric of the $3-$space
iff there is no twist or vorticity in the $3-$space. The covariant time
derivative along the observers’ worldlines, denoted by ‘${\hskip
0.99594pt\hskip 0.99594pt{}^{\cdot}\hskip 0.99594pt\hskip 0.99594pt}$’, is
defined using the vector ${u^{a}}$, as

$\dot{Z}^{a...b}{}_{c...d}=u^{e}\hskip 0.99594pt\nabla_{e}\hskip
0.99594ptZ^{a...b}{}_{c...d},$ (22)

for any tensor ${Z^{a...b}{}_{c...d}}$. The fully orthogonally projected
covariant spatial derivative, denoted by ‘ ${D}$ ’, is defined using the
spatial projection tensor ${h_{ab}}$, as

$D_{e}\hskip 0.99594ptZ^{a...b}{}_{c...d}=h^{r}{}_{e}\hskip
0.99594pth^{p}{}_{c}\hskip 0.99594pt...\hskip 0.99594pth^{q}{}_{d}\hskip
0.99594pth^{a}{}_{f}\hskip 0.99594pt...\hskip 0.99594pth^{b}{}_{g}\hskip
0.99594pt\nabla_{r}\hskip 0.99594ptZ^{f...g}{}_{p...q},$ (23)

with total projection on all the free indices. The covariant derivative of the
4-velocity vector ${u^{a}}$ is decomposed irreducibly as follows

$\displaystyle\nabla_{a}\hskip 0.99594ptu_{b}$ $\displaystyle=$
$\displaystyle-u_{a}\hskip 0.99594ptA_{b}+\frac{1}{3}h_{ab}\hskip
0.99594pt\Theta+\sigma_{ab}+\epsilon_{abc}\hskip 0.99594pt\omega^{c},$ (24)

where ${A_{b}}$ is the acceleration, ${\Theta}$ is the expansion of ${u_{a}}$,
${\sigma_{ab}}$ is the shear tensor, ${\omega^{a}}$ is the vorticity vector
representing rotation and ${\epsilon_{abc}}$ is the effective volume element
in the rest space of the comoving observer. The vorticity vector $\omega^{q}$
is related to vorticity tensor $\omega^{ab}$ as:
$\omega^{a}\equiv(1/2)\,\epsilon^{abc}\,\omega_{bc}$.

Furthermore, the energy-momentum tensor of matter or fields present in the
spacetime, decomposed relative to ${u^{a}}$, is given by

$\displaystyle T_{ab}$ $\displaystyle=$ $\displaystyle\mu\hskip
0.99594ptu_{a}\hskip 0.99594ptu_{b}+p\hskip 0.99594pth_{ab}+q_{a}\hskip
0.99594ptu_{b}+u_{a}\hskip 0.99594ptq_{b}+\pi_{ab},$ (25)

where ${\mu}$ is the effective energy density, ${p}$ is the isotropic
pressure, ${q_{a}}$ is the 3-vector defining the heat flux and ${\pi_{ab}}$ is
the anisotropic stress. Angle brackets denote orthogonal projections of
vectors onto the three space as well as the projected, symmetric and trace-
free (PSTF) part of tensors.

$\displaystyle v_{<a>}$ $\displaystyle=$ $\displaystyle h^{b}{}_{a}\hskip
0.99594pt\dot{V}_{b},$ (26) $\displaystyle Z_{<ab>}$ $\displaystyle=$
$\displaystyle\left(h^{c}{}_{(a}\hskip
0.99594pth^{d}{}_{b)}-\frac{1}{3}h_{ab}\hskip 0.99594pth^{cd}\right)\hskip
0.99594ptZ_{cd}.$ (27)

The Weyl quantities are decomposed as

$\displaystyle E_{ab}$ $\displaystyle=$ $\displaystyle C_{abcd}\hskip
0.99594ptu^{c}\hskip 0.99594ptu^{d}=E_{<ab>},$ (28) $\displaystyle H_{ab}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\varepsilon_{ade}\hskip
0.99594ptC^{de}{}_{bc}\hskip 0.99594ptu^{c}=H_{<ab>}.$ (29)

###  A.2 Semitetrad 1+1+2 formalism

In the 1+1+2 formalism Clarkson and Barrett (2003b), the 3-space
${\mathcal{V}}$ is now further split by introducing the unit vector ${e^{a}}$
orthogonal to ${u^{a}}$ ${\left(e^{a}\hskip 0.99594pte_{a}=1,u^{a}\hskip
0.99594pte_{a}=0\right)}$. The $3-$space now has two parts—one is the
spacelike direction $e^{a}$, and the second is the $2-$space orthogonal to
$e^{a}$ as well as $u^{a}$, which we refer to as the $2-sheets$. The 1+1+2
covariantly decomposed spacetime is given by

$g_{ab}=-u_{a}\hskip 0.99594ptu_{b}+e_{a}\hskip 0.99594pte_{b}+N_{ab},$ (31)

where ${N_{ab}}$ ${\left(e^{a}\hskip 0.99594ptN_{ab}=0=u^{a}\hskip
0.99594ptN_{ab},N^{a}{}_{a}=2\right)}$ projects vectors onto 2-spaces called
‘2-sheets’ , orthogonal to ${u^{a}}$ and ${e^{a}}$. We introduce two new
derivatives for any tensor ${\phi_{a...b}{}^{c...d}}$:

$\displaystyle\hat{\phi}_{a...b}{}^{c...d}$ $\displaystyle\equiv$
$\displaystyle e^{f}\hskip 0.99594ptD_{f}\hskip
0.99594pt\phi_{a...b}{}^{c...d},$ (32)
$\displaystyle\delta_{f}\phi_{a...b}{}^{c...d}$ $\displaystyle\equiv$
$\displaystyle
N_{f}{}^{j}N_{a}{}^{l}...N_{b}{}^{g}N_{h}{}^{c}...N_{i}{}^{d}D_{j}\phi_{l...g}{}^{h...i}.$
(33)

The 1+3 kinematical and dynamical quantities and anisotropic fluid variables
are split irreducibly as

$\displaystyle A^{a}$ $\displaystyle=$ $\displaystyle\mathcal{A}\hskip
0.99594pte^{a}+\mathcal{A}^{a},$ (34) $\displaystyle\omega^{a}$
$\displaystyle=$ $\displaystyle\Omega\hskip 0.99594pte^{a}+\Omega^{a},$ (35)
$\displaystyle\sigma_{ab}$ $\displaystyle=$
$\displaystyle\Sigma\left(e_{a}\hskip 0.99594pte_{b}-\frac{1}{2}\hskip
0.99594ptN_{ab}\right)+2\hskip 0.99594pt\Sigma_{(a}\hskip
0.99594pte_{b)}+\Sigma_{ab},$ (36) $\displaystyle q_{a}$ $\displaystyle=$
$\displaystyle{\mathcal{Q}}\hskip 0.99594pte_{a}+{\mathcal{Q}}_{a},$ (37)
$\displaystyle\pi_{ab}$ $\displaystyle=$ $\displaystyle\Pi\left(e_{a}\hskip
0.99594pte_{b}-\frac{1}{2}\hskip 0.99594ptN_{ab}\right)+2\hskip
0.99594pt\Pi_{(a}\hskip 0.99594pte_{b)}+\Pi_{ab},$ (38) $\displaystyle E_{ab}$
$\displaystyle=$ $\displaystyle\mathcal{E}\left(e_{a}\hskip
0.99594pte_{b}-\frac{1}{2}\hskip 0.99594ptN_{ab}\right)+2\hskip
0.99594pt\mathcal{E}_{(a}\hskip 0.99594pte_{b)}+\mathcal{E}_{ab},$ (39)
$\displaystyle H_{ab}$ $\displaystyle=$
$\displaystyle\mathcal{H}\left(e_{a}\hskip 0.99594pte_{b}-\frac{1}{2}\hskip
0.99594ptN_{ab}\right)+2\hskip 0.99594pt\mathcal{H}_{(a}\hskip
0.99594pte_{b)}+\mathcal{H}_{ab}\,.$ (40)

The fully projected 3-derivative of ${e^{a}}$ is given by

$\displaystyle D_{a}\hskip 0.99594pte_{b}$ $\displaystyle=$ $\displaystyle
e_{a}\hskip 0.99594pta_{b}+\frac{1}{2}\hskip 0.99594pt\phi\hskip
0.99594ptN_{ab}+\xi\hskip 0.99594pt\varepsilon_{ab}+\zeta_{ab},$ (41)

where traveling along ${e^{a}}$, ${a_{a}}$ is the sheet acceleration, ${\phi}$
is the sheet expansion, ${\xi}$ is the vorticity of ${e^{a}}$ (the twisting of
the sheet) and ${\zeta_{ab}}$ is the shear of ${e^{a}}$ (the distortion of the
sheet).

The 1+1+2 split of the full covariant derivatives of ${u^{a}}$ and ${e^{a}}$
are as follows

$\displaystyle\nabla_{a}\hskip 0.99594ptu_{b}$ $\displaystyle=$
$\displaystyle-u_{a}\left(\mathcal{A}\hskip
0.99594pte_{b}+\mathcal{A}_{b}\right)+e_{a}\hskip
0.99594pte_{b}\left(\frac{1}{3}\hskip
0.99594pt\Theta+\Sigma\right)+e_{a}\left(\Sigma_{b}+\varepsilon_{bc}\hskip
0.99594pt\Omega^{c}\right)+\left(\Sigma_{a}-\varepsilon_{ac}\hskip
0.99594pt\Omega^{c}\right)e_{b}$ (42)
$\displaystyle+N_{ab}\left(\frac{1}{3}\hskip 0.99594pt\Theta-\frac{1}{2}\hskip
0.99594pt\Sigma\right)+\Omega\hskip 0.99594pt\varepsilon_{ab}+\Sigma_{ab},$
$\displaystyle\nabla_{a}\hskip 0.99594pte_{b}$ $\displaystyle=$
$\displaystyle-\mathcal{A}\hskip 0.99594ptu_{a}\hskip
0.99594ptu_{b}-u_{a}\hskip 0.99594pt\alpha_{b}+\left(\frac{1}{3}\hskip
0.99594pt\Theta+\Sigma\right)e_{a}\hskip
0.99594ptu_{b}+\left(\Sigma_{a}-\varepsilon_{ac}\hskip
0.99594pt\Omega^{c}\right)u_{b}+e_{a}\hskip 0.99594pta_{b}$ (43)
$\displaystyle+\frac{1}{2}\hskip 0.99594pt\phi\hskip 0.99594ptN_{ab}+\xi\hskip
0.99594pt\varepsilon_{ab}+\zeta_{ab}.$

We can now immediately see that the Ricci identities and the doubly contracted
Bianchi identities, which specify the evolution of the complete system, can
now be written as the time evolution and spatial propagation and spatial
constraints of an irreducible set of geometrical and electromagnetic (EM)
variables. The irreducible set of geometric variables

$\displaystyle\mathcal{D}_{geom}=\\{\Theta,\hskip 0.99594pt\mathcal{A},\hskip
0.99594pt\Omega,\hskip 0.99594pt\Sigma,\hskip 0.99594pt\mathcal{E},\hskip
0.99594pt\mathcal{H},\hskip 0.99594pt\phi,\hskip 0.99594pt\xi,\hskip
0.99594pt\mathcal{A}_{a},\hskip 0.99594pt\Omega_{a},\hskip
0.99594pt\Sigma_{a},\hskip 0.99594pt\alpha_{a},\hskip 0.99594pta_{a},\hskip
0.99594pt\mathcal{E}_{a},\hskip 0.99594pt\mathcal{H}_{a},\hskip
0.99594pt\Sigma_{ab},\hskip 0.99594pt\zeta_{ab},\hskip
0.99594pt\mathcal{E}_{ab},\hskip 0.99594pt\mathcal{H}_{ab}\\}$ (44)

together with the irreducible set of EM variables

$\displaystyle\mathcal{D}_{\rm EM}$ $\displaystyle=$
$\displaystyle\left\\{\mu,\hskip 0.99594ptp,\hskip
0.99594pt{\mathcal{Q}},\hskip 0.99594pt\Pi,\hskip
0.99594pt{\mathcal{Q}}_{a},\hskip 0.99594pt\Pi_{a},\hskip
0.99594pt\Pi_{ab}\right\\},$ (45)

make up the key variables in the 1+1+2 formalism.

##  Appendix B The energy momentum tensor of the EM test pulse

The test pulse contains the electromagnetic field that perturbs the background
spacetime. By virtue of the semitetrad 1+1+2 splitting, the electric and
magnetic field vectors can be written in the following way:

$\displaystyle E^{a}$ $\displaystyle=$
$\displaystyle\mathscr{E}e^{a}+\mathscr{E}^{a}\;,$ (46) $\displaystyle B^{a}$
$\displaystyle=$ $\displaystyle\mathscr{B}e^{a}+\mathscr{B}^{a}\;,$ (47)

where $\mathscr{E}$ and $\mathscr{B}$ are scalars and $\mathscr{E}^{a}$ and
$\mathscr{B}^{a}$ are projected 2-vectors on the 2-shells that break the
sphericity of these shells. The energy-momentum tensor $T_{ab}$ of the field
can then be split into a scalar, 2-vector, PSTF 2-tensor parts as

$\displaystyle T_{ab}=\mu
u_{a}u_{b}+ph_{ab}+2{\mathcal{Q}}\,e_{(a}u_{b)}+2{\mathcal{Q}}_{(a}u_{b)}+\Pi(e_{a}e_{b}-\frac{1}{2}N_{ab})+2\Pi_{(a}e_{b)}+\Pi_{ab}\;,$
(48)

where,

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathscr{E}^{2}+\mathscr{B}^{2}+\mathscr{E}^{a}\mathscr{E}_{a}+\mathscr{B}^{a}\mathscr{B}_{a})\;$
(49) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{6}(\mathscr{E}^{2}+\mathscr{B}^{2}+\mathscr{E}^{a}\mathscr{E}_{a}+\mathscr{B}^{a}\mathscr{B}_{a})\;$
(50) $\displaystyle{\mathcal{Q}}$ $\displaystyle=$
$\displaystyle\varepsilon_{ab}\,\mathscr{E}^{b}\mathscr{B}^{c}\;$ (51)
$\displaystyle{\mathcal{Q}}_{a}$ $\displaystyle=$
$\displaystyle\varepsilon_{ac}(\mathscr{B}\mathscr{E}^{c}-\mathscr{E}\mathscr{B}^{c})\;$
(52) $\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left[-2(\mathscr{E}^{2}+\mathscr{B}^{2})+(\mathscr{E}^{a}\mathscr{E}_{a}+\mathscr{B}^{a}\mathscr{B}_{a})\right]\;$
(53) $\displaystyle\Pi_{a}$ $\displaystyle=$
$\displaystyle-\left(\mathscr{E}\mathscr{E}_{a}+\mathscr{B}\mathscr{B}_{a}\right)\;$
(54) $\displaystyle\Pi_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathscr{E}^{c}\mathscr{E}_{c}+\mathscr{B}^{c}\mathscr{B}_{c})N_{ab}-(\mathscr{E}_{a}\mathscr{E}_{b}+\mathscr{B}_{a}\mathscr{B}_{b})\;.$
(55)

For the pulse not to back-react on the metric, the ratio of the magnitude of
each component of the above stress-tensor w.r.t the background Gaussian
curvature must be less than the BH mass ($m$). This has been verified in Refs.
Ellis and Goswami (2013); Goswami and Ellis (2011, 2012b). The above energy-
momentum tensor must follow the conservation equations up to the first-order
perturbations on the background Schwarzschild manifold of the pre-pulse phase,
which are given as

$\displaystyle\dot{\mu}+\hat{{\mathcal{Q}}}=$
$\displaystyle-\delta_{a}{\mathcal{Q}}^{a}-(\phi+2\mathcal{A}){\mathcal{Q}}$
(56)
$\displaystyle\dot{{\mathcal{Q}}}+\hat{p}+\hat{\Pi}=\delta_{a}\Pi^{a}-\left(\frac{3}{2}\phi+{\cal
A}\right)\Pi-(\rho+p){\cal A}$ (57)
$\displaystyle\dot{{\mathcal{Q}}}_{\bar{a}}+\hat{\Pi}_{\bar{a}}=$
$\displaystyle-\delta_{a}p+\frac{1}{2}\delta_{a}\Pi-\delta^{b}\Pi_{ab}-\left(\frac{3}{2}\phi+{\cal
A}\right)\Pi_{a}$ (58)

where the bar on the indices denotes the projected part on the perturbed
2-shells.

##  Appendix C Pulse phase equations

In the pulse phase, the first-order evolution equations for the above-defined
gauge invariant first-order variables depicting the non-sphericity of the
manifold can be written as follows. Here, the curly brackets denote the
projected symmetric trace-free part of the tensor on the 2-sheet. For
simplicity for the readers, we have put all the EM contributions within the
square bracket of the RHS of each equation. This is to transparently show
which terms will identically vanish at the next (post-pulse) phase despite the
continuing non-sphericity.

$\displaystyle\dot{W}_{a}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\phi\,{\mathcal{E}}\left(\alpha_{a}+\Sigma_{a}-\varepsilon_{ab}\Omega^{b}\right)+\frac{3}{2}{\mathcal{E}}\left(\delta_{a}\Sigma-\frac{2}{3}\delta_{a}\Theta\right)+\,\varepsilon_{bc}\delta_{a}\delta^{b}{\mathcal{H}}^{c}{}$
(59)
$\displaystyle-\left[\frac{1}{2}\delta_{a}\dot{\Pi}+\frac{1}{2}Y_{a}{\mathcal{Q}}+\frac{1}{2}\phi\left(\delta_{a}{\mathcal{Q}}\right)+\frac{1}{3}\delta_{a}\dot{\mu}+\delta_{a}\delta_{b}{\mathcal{Q}}^{b}\right]~{},$
$\displaystyle\dot{Y}_{a}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\phi^{2}+{\mathcal{E}}\right)\left(\alpha_{a}+\Sigma_{a}-\varepsilon_{ab}\Omega^{b}\right)+\delta_{a}\delta_{c}\alpha^{c}+\,\left(\frac{1}{2}\phi-{\cal
A}\right)\left(\delta_{a}\Sigma-\frac{2}{3}\delta_{a}\Theta\right)+\left[\delta_{a}{\mathcal{Q}}\right]~{}.$
(60)

The shear evolution equations give

$\displaystyle\dot{\Sigma}_{a}-\textstyle\frac{1}{2}\hat{\cal A}_{a}$
$\displaystyle=$ $\displaystyle\textstyle\frac{1}{2}\delta_{a}{\cal
A}+\left({\cal A}-\textstyle\frac{1}{4}\phi\right){\cal
A}_{a}+\textstyle\frac{1}{2}{\cal
A}a_{a}-\textstyle\frac{3}{2}\Sigma\alpha_{a}-{\mathcal{E}}_{a}+\left[\frac{1}{2}\Pi_{a}\right]\;,$
(61) $\displaystyle\dot{\Sigma}_{\\{ab\\}}$ $\displaystyle=$
$\displaystyle\delta_{\\{a}{\cal A}_{b\\}}+{\cal
A}\zeta_{ab}-{\mathcal{E}}_{ab}+\left[\frac{1}{2}\Pi_{ab}\right]\;.$ (62)

Evolution equation for $\hat{e}_{a}$ is:

$\displaystyle\hat{\alpha}_{a}-\dot{a}_{a}$ $\displaystyle=$
$\displaystyle\left(\textstyle\frac{1}{2}\phi-{\cal
A}\right)\left(\Sigma_{a}+\varepsilon_{ab}\Omega^{b}\right)-\left(\textstyle\frac{1}{2}\phi+{\cal
A}\right)\alpha_{a}-\varepsilon_{ab}{\mathcal{H}}^{b}+\left[\frac{1}{2}{\mathcal{Q}}_{a}\right].$
(63)

Electric Weyl evolution gives

$\displaystyle\dot{\mathcal{E}}=\left(\textstyle\frac{3}{2}\Sigma-\Theta\right){\mathcal{E}}+\varepsilon_{ab}\delta^{a}{\mathcal{H}}^{c}+\left[\frac{1}{3}\left(\frac{1}{2}\phi-2{\cal
A}\right){\mathcal{Q}}-\frac{1}{2}\dot{\Pi}-\frac{1}{3}\hat{{\mathcal{Q}}}+\frac{1}{6}\delta_{a}{\mathcal{Q}}^{a}\right]$
(64)
$\displaystyle\dot{\mathcal{E}}_{a}+\textstyle\frac{1}{2}\varepsilon_{ab}\hat{\mathcal{H}}^{b}$
$\displaystyle=$
$\displaystyle\textstyle\frac{3}{4}\varepsilon_{ab}\delta^{b}{\mathcal{H}}+\textstyle\frac{1}{2}\varepsilon_{bc}\delta^{b}{\mathcal{H}}^{c}_{~{}a}-\textstyle\frac{3}{4}{\mathcal{E}}(\Sigma_{a}+2\alpha_{a})+\textstyle\frac{3}{4}{\mathcal{E}}\varepsilon_{ab}\Omega^{b}-\left(\textstyle\frac{1}{4}\phi+{\cal
A}\right)\varepsilon_{ab}{\mathcal{H}}^{b}$ (65)
$\displaystyle-\left[\frac{1}{2}\dot{\Pi}_{a}+\frac{1}{4}\hat{{\mathcal{Q}}}_{a}+\frac{1}{4}\delta_{a}{\mathcal{Q}}-\frac{1}{2}(\frac{1}{4}\phi-{\cal
A}){\mathcal{Q}}_{a}\right]\;,$
$\displaystyle\dot{\mathcal{E}}_{\\{ab\\}}-\varepsilon_{c\\{a}\hat{\mathcal{H}}_{b\\}}^{~{}~{}c}$
$\displaystyle=$
$\displaystyle-\varepsilon_{c\\{a}\delta^{c}{\mathcal{H}}_{b\\}}-\textstyle\frac{3}{2}{\mathcal{E}}\Sigma_{ab}+\left(\textstyle\frac{1}{2}\phi+2{\cal
A}\right)\varepsilon_{c\\{a}{\mathcal{H}}_{b\\}}^{~{}~{}c}-\left[\frac{1}{2}\dot{\Pi}_{\\{ab\\}}+\frac{1}{2}\delta_{\\{a}{\mathcal{Q}}_{b\\}}\right]\;.$
(66)

Magnetic Weyl evolution gives

$\displaystyle\dot{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-\varepsilon_{ab}\delta^{a}{\mathcal{E}}^{b}-3\xi{\mathcal{E}}+\left[\frac{1}{2}\varepsilon_{ab}\delta^{a}\Pi^{b}\right]\;,$
(67)
$\displaystyle\dot{\mathcal{H}}_{a}-\textstyle\frac{1}{2}\varepsilon_{ab}\hat{\mathcal{E}}^{b}$
$\displaystyle=$
$\displaystyle-\textstyle\frac{3}{2}{\mathcal{E}}\varepsilon_{ab}{\cal
A}^{b}+\textstyle\frac{3}{4}{\mathcal{E}}\varepsilon_{ab}a^{b}-\textstyle\frac{1}{2}\varepsilon_{bc}\delta^{b}{\mathcal{E}}^{c}_{~{}a}+\left(\textstyle\frac{1}{4}\phi+{\cal
A}\right)\varepsilon_{ab}{\mathcal{E}}^{b}-\textstyle\frac{3}{4}\varepsilon_{ab}\delta^{b}{\mathcal{E}}$
(68)
$\displaystyle-\left[\frac{1}{4}\varepsilon_{ab}\hat{\Pi}^{b}-\frac{3}{8}\varepsilon_{ab}\delta^{b}\Pi-\frac{1}{4}\varepsilon_{bc}\delta^{b}\Pi^{c}\,_{a}\right]\;,$
$\displaystyle\dot{\mathcal{H}}_{\\{ab\\}}+\varepsilon_{c\\{a}\hat{\mathcal{E}}_{b\\}}^{~{}~{}c}$
$\displaystyle=$
$\displaystyle+\textstyle\frac{3}{2}{\mathcal{E}}\varepsilon_{c\\{a}\zeta_{b\\}}^{~{}~{}c}-\left(\textstyle\frac{1}{2}\phi+2{\cal
A}\right)\varepsilon_{c\\{a}{\mathcal{E}}_{b\\}}^{~{}~{}c}+\varepsilon_{c\\{a}\delta^{c}{\mathcal{E}}_{b\\}}$
(69)
$\displaystyle+\left[\frac{1}{2}\varepsilon_{c\\{a}\hat{\Pi}_{b\\}}\,^{c}-\frac{1}{2}\varepsilon_{c\\{a}\delta^{c}\Pi_{b\\}}\right]\;.$

The time evolution equations for $\xi$ and $\zeta_{\\{ab\\}}$ are

$\dot{\xi}=\left({\cal
A}-\textstyle\frac{1}{2}\phi\right)\Omega+\textstyle\frac{1}{2}\varepsilon_{ab}\delta^{a}\alpha^{b}+\textstyle\frac{1}{2}{\mathcal{H}}\;,$
(70) $\displaystyle\dot{\zeta}_{\\{ab\\}}$ $\displaystyle=$
$\displaystyle\left({\cal
A}-\textstyle\frac{1}{2}\phi\right)\Sigma_{ab}+\delta_{\\{a}\alpha_{b\\}}-\varepsilon_{c\\{a}{\mathcal{H}}_{b\\}}^{~{}~{}c}\;.$
(71)

The vorticity evolution equations are

$\displaystyle\dot{\Omega}$ $\displaystyle=$
$\displaystyle\textstyle\frac{1}{2}\varepsilon_{ab}\delta^{a}{\cal
A}^{b}+{\cal A}\xi\;,$ (72)
$\displaystyle\dot{\Omega}_{a}+\textstyle\frac{1}{2}\varepsilon_{ab}\hat{\cal
A}^{b}$ $\displaystyle=$
$\displaystyle\textstyle\frac{1}{2}\varepsilon_{ab}\left(-{\cal
A}a^{b}+\delta^{b}{\cal A}-\textstyle\frac{1}{2}\phi{\cal A}^{b}\right)\;.$
(73)

Sheet expansion evolution is given by:

$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\left(\textstyle\frac{2}{3}\Theta-\Sigma\right)\left({\cal
A}-\textstyle\frac{1}{2}\phi\right)+\delta_{a}\alpha^{a}\,+\left[{\mathcal{Q}}\right].$
(74)

The Raychaudhuri equation is

$\displaystyle\hat{\cal A}-\dot{\Theta}=-\delta_{a}{\cal A}^{a}-\left({\cal
A}+\phi\right){\cal A}\,+\left[\frac{1}{2}(\mu+3p)\right].$ (75)

The propagation equations of $\xi$ and $\zeta_{\\{ab\\}}$ are:

$\displaystyle\hat{\xi}$ $\displaystyle=$
$\displaystyle-\phi\xi+\textstyle\frac{1}{2}\varepsilon_{ab}\delta^{a}a^{b}\;,$
(76) $\displaystyle\hat{\zeta}_{\\{ab\\}}$ $\displaystyle=$
$\displaystyle-\phi\zeta_{ab}+\delta_{\\{a}a_{b\\}}-{\cal
E}_{ab}-\left[\frac{1}{2}\Pi_{ab}\right]\;.$ (77)

The shear divergence is given by :

$\displaystyle\hat{\Sigma}-\textstyle\frac{2}{3}\hat{\Theta}$ $\displaystyle=$
$\displaystyle-\textstyle\frac{3}{2}\phi\Sigma-\delta_{a}\Sigma^{a}-\left[{\mathcal{Q}}\right]$
(78) $\displaystyle\hat{\Sigma}_{a}-\varepsilon_{ab}\hat{\Omega}^{b}$
$\displaystyle=$
$\displaystyle\textstyle\frac{1}{2}\delta_{a}\Sigma+\textstyle\frac{2}{3}\delta_{a}\theta-\varepsilon_{ab}\delta^{b}\Omega-\textstyle\frac{3}{2}\phi\Sigma_{a}-\textstyle\frac{3}{2}\Sigma
a_{a}$ (79) $\displaystyle+\left(\textstyle\frac{1}{2}\phi+2{\cal
A}\right)\varepsilon_{ab}\Omega^{b}-\delta^{b}\Sigma_{ab}-\left[{\mathcal{Q}}_{a}\right]\;,$
$\displaystyle\hat{\Sigma}_{\\{ab\\}}$ $\displaystyle=$
$\displaystyle\delta_{\\{a}\Sigma_{b\\}}-\varepsilon_{c\\{a}\delta^{c}\Omega_{b\\}}-\textstyle\frac{1}{2}\phi\Sigma_{ab}$
(80) $\displaystyle-\varepsilon_{c\\{a}{\mathcal{H}}_{b\\}}^{~{}~{}c}\;.$

The vorticity divergence equation is:

$\displaystyle\hat{\Omega}$ $\displaystyle=$
$\displaystyle-\delta_{a}\Omega^{a}+\left({\cal A}-\phi\right)\Omega\;.$ (81)

The Electric Weyl divergence is

$\displaystyle\hat{\mathcal{E}}$ $\displaystyle=$
$\displaystyle-\delta_{a}{\mathcal{E}}^{a}-\textstyle\frac{3}{2}\phi{\mathcal{E}}+\left[\frac{1}{3}\hat{\mu}-\frac{1}{2}\hat{\Pi}-\frac{1}{2}\delta_{a}\Pi^{a}-\frac{3}{4}\phi\Pi\right]\;,$
(82) $\displaystyle\hat{\mathcal{E}}_{a}$ $\displaystyle=$
$\displaystyle\textstyle\frac{1}{2}\delta_{a}{\mathcal{E}}-\delta^{b}{\mathcal{E}}_{ab}-\textstyle\frac{3}{2}{\mathcal{E}}a_{a}-\textstyle\frac{3}{2}\phi{\mathcal{E}}_{a}+\left[\frac{1}{3}\delta_{a}\mu+\frac{1}{4}\delta_{a}\Pi-\frac{1}{2}\hat{\Pi}_{a}-\frac{1}{2}\delta^{b}\Pi_{ab}-\frac{3}{4}\phi\Pi_{a}\right]\;.$
(83)

The Magnetic Weyl divergence is:

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=$
$\displaystyle-\delta_{a}{\mathcal{H}}^{a}-\textstyle\frac{3}{2}\phi{\mathcal{H}}-3{\mathcal{E}}\Omega-\left[\frac{1}{2}\varepsilon_{ab}\delta^{a}{\mathcal{Q}}^{b}\right]\;,$
(84) $\displaystyle\hat{\mathcal{H}}_{a}$ $\displaystyle=$
$\displaystyle\textstyle\frac{1}{2}\delta_{a}{\mathcal{H}}-\delta^{b}{\mathcal{H}}_{ab}-\textstyle\frac{3}{2}{\mathcal{E}}\varepsilon_{ab}\Sigma^{b}+\textstyle\frac{3}{2}{\mathcal{E}}\Omega_{a}$
(85)
$\displaystyle+\textstyle\frac{3}{2}\Sigma\varepsilon_{ab}{\mathcal{E}}^{b}-\textstyle\frac{3}{2}\phi{\mathcal{H}}_{a}-\left[\frac{1}{2}\varepsilon_{ab}\delta^{b}{\mathcal{Q}}-\frac{1}{4}\varepsilon_{ab}\phi{\mathcal{Q}}^{b}-\frac{1}{2}\varepsilon_{ab}\hat{{\mathcal{Q}}}^{b}\;\right].$

The sheet expansion propagation is:

$\displaystyle\hat{\phi}$ $\displaystyle=$
$\displaystyle-\textstyle\frac{1}{2}\phi^{2}+\delta_{a}a^{a}-{\mathcal{E}}-\left[\frac{2}{3}\mu+\frac{1}{2}\Pi\right]\;.$
(86)

We also have the following constraints:

$\displaystyle\delta_{a}\Omega^{a}+\varepsilon_{ab}\delta^{a}\Sigma^{b}=\left(2{\cal
A}-\phi\right)\Omega+{\mathcal{H}}~{},$ (87)
$\displaystyle\frac{1}{2}\delta_{a}\phi-\varepsilon_{ab}\delta^{b}\xi-\delta^{b}\zeta_{ab}$
$\displaystyle=$
$\displaystyle-{\mathcal{E}}_{a}-\left[\frac{1}{2}\Pi_{a}\right]~{},$ (88)
$\displaystyle\delta_{a}\Sigma-\frac{2}{3}\delta_{a}\Theta+2\varepsilon_{ab}\delta^{b}\Omega+2\delta^{b}\Sigma_{ab}$
$\displaystyle=$
$\displaystyle-\phi\left(\Sigma_{a}-\varepsilon_{ab}\Omega^{b}\right){}$ (89)
$\displaystyle-2\varepsilon_{ab}{\mathcal{H}}^{b}-\left[{\mathcal{Q}}_{a}\right].$

##  Appendix D Amplitude of gravitational waves on Earth

As shown in Fig. (2), a kilohertz(kHz) electromagnetic (EM) pulse from a
pulsar is intervened by a spherically symmetric compact object situated along
the path between the pulsar and Earth. In this part, we evaluate the GW
amplitude received by the detector on the Earth due to this process.

To go about this, first, using Eq. (19), we obtain the GW amplitude as
detected by the fictitious observer close to the BH. Note that
${\mathcal{Q}}_{\mathsf{S}}$ is the time integral of the amplitude of the
component of the EM Poynting vector (${\mathcal{Q}}$) along the radial
direction. $\mu_{\mathsf{S}}$ is the EM energy density, which is related to
the luminosity $({\cal L}_{\rm EM})$ via the relation ${\cal L}_{\rm
EM}\,\Delta t$ where $\Delta t$ is the pulse duration.

Using the continuity equation of the energy density $\mu$ and radial component
of current ${\mathcal{Q}}$,

$\displaystyle\frac{\partial\mu}{\partial
t}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}{\mathcal{Q}}\right).$
(90)

Integrating the above equation w.r.t time $t$ leads to,

$\displaystyle\mu=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\int
dt\,{\mathcal{Q}}\right).$ (91)

Rewriting $\mu,\,\int dt\,{\mathcal{Q}}$, in terms of time harmonicsClarkson
and Barrett (2003a) — $\mu=Q\,T^{(\omega)}\mu_{\rm S}$, $\int
dt{\mathcal{Q}}=Q\,T^{(\omega)}{\mathcal{Q}}_{\rm S}$, where $Q$,
$T^{(\omega)}$ are spin weighted spherical harmonics and time harmonics,
respectively, — we obtain:

$\displaystyle\mu_{\rm S}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}{\mathcal{Q}}_{\rm S}\right)\,.$ (92)

If the intervening region is diffuse (like in ISM), the Poynting vector
remains the same, and hence, we can set ${\mathcal{Q}}_{\rm S}$ to be
independent of radial distance. This gives the following relation between
$\mu_{\rm S}$ and ${\mathcal{Q}}_{\rm S}$:

$\displaystyle\mu_{\rm S}\sim{\mathcal{Q}}_{\rm S}/r\,.$ (93)

As shown in Fig. (2), we take $d$ as the distance between the pulsar and $D$
as the distance between the BH and the Earth. As discussed in the main text,
the EM pulse close to the BH perturbs the most. Assuming that the EM energy
perturbs the BH at a distance $r_{0}$ from the center of the BH, the amplitude
of the Regge-wheeler tensor

$\displaystyle{M}_{\mathsf{T}}\,=-\frac{r_{0}^{3}}{m}\left[\frac{1}{6}\left(1-\frac{2m}{r_{0}}\right)+\frac{1}{9}\right]\mathcal{L}_{\rm
EM}\Delta t$ (94)

Note that the above relation is only applicable in the close vicinity of the
fictitious observer $\gamma$ while interacting with the EM pulse in the pulse
phase. At the moment of EM pulse arrival $\dot{\zeta}_{ab}=0$, however from
the evolution equation of $\Sigma_{ab}$ (62), we can show
$\ddot{\zeta}_{ab}\neq 0$. Hence, the contribution from $\zeta_{ab}$ into
$M_{ab}$ requires $\dot{\Sigma}_{ab}\neq 0$, which will occur later than the
generation of ${W}_{a}$. The contribution from $\zeta_{ab}$ can be considered
the second order, whereas the contribution from $W_{a}$ is the first order.

For BH mass $m\sim M_{\odot}$ where, $M_{\odot}$ is mass of solar mass BH and
$r_{0}\sim a_{0}r_{\rm s}$, where $r_{\rm s}$ is Schwarzschild radius of the
BH, reinstating the constants ($G,\,c$), and rewriting
$2GM_{\odot}/c^{2}\equiv r_{\rm s}$, we obtain,

$\displaystyle{M}_{\mathsf{T}}\,=-\frac{2a_{0}^{3}r_{\rm
s}^{2}\,G}{c^{4}}\left[\frac{1}{6}\left(1-\frac{1}{a_{0}}\right)+\frac{1}{9}\right]\mathcal{L}_{\rm
EM}\Delta t$ (95)

Choosing $r_{s}\sim 1~{}$km, we obtain,

$\displaystyle|{M}_{\mathsf{T}}|\,\sim
a_{0}^{3}\left[\frac{1}{6}\left(1-\frac{1}{a_{0}}\right)+\frac{1}{9}\right]\mathcal{L}_{\rm
EM}\Delta t\times 10^{-40}$ (96)

As mentioned above, the above relation is only valid close to the compact
object. Hence, setting $a_{0}\sim\mathcal{O}(1)$ in the above expression, we
have:

$\displaystyle|{M}_{\mathsf{T}}|\,\sim\mathcal{L}_{\rm EM}\Delta t\times
10^{-40}$ (97)

The key physical input required now to compute the GW amplitude is the
information about the incoming source. As we have mentioned, we assume that
the low-frequency EM pulse of a Pulsar Condon and Ransom (2016). The magnetic
dipole model is the simplest pulsar model that accounts for many of the
observed properties of the pulsar Ostriker and Gunn (1969); Condon and Ransom
(2016). As the pulsar rotates at a frequency $\Omega$ (or periodicity
$P=1/\Omega$), the rotational kinetic energy is:

$E=\frac{2\pi^{2}I_{\rm MI}}{P^{2}}\,,$

where $I_{\rm MI}$ is moment of inertia of the pulsar. The pulsar EM radiation
extracts this kinetic energy, and the rate of change of $E$ matches with
$P_{\text{rad }}$ Condon and Ransom (2016)

$\displaystyle P_{\text{rad}}\sim-\dot{E}=\frac{4\pi^{2}I_{\rm
MI}\dot{P}}{P^{3}}$ (98)

For magnetic dipole rotating with frequency, the radiated energy is Condon and
Ransom (2016):

$P_{\text{rad
}}=\frac{2}{3}\frac{\left(\Omega^{2}m_{\perp}\right)^{2}}{c^{3}}=\frac{2}{3c^{3}}\left(BR^{3}\sin\alpha\right)^{2}\,\Omega^{4}=\frac{2}{3c^{3}}\left(BR^{3}\sin\alpha\right)^{2}\,\left(\frac{2\pi}{P}\right)^{4}$

The maximum radiated energy will be at the rotation frequency $\leq 1~{}{\rm
kHz}$. However, the low-frequency waves can not propagate through the
interstellar medium. Interestingly, EM waves with frequencies around 2 kHz or
more can propagate through the interstellar medium. However, these can not be
observed with the radio telescopes. Substituting the values of a typical
pulsar in the above expression, the intrinsic luminosity of the pulsar is
Condon and Ransom (2016):

$\mathcal{L}_{\rm EM\sim}\Omega_{0}^{4}\,10^{38}~{}{\rm erg/s}\,,$

where $\Omega_{0}\sim\Omega_{\rm e}/30$. Substituting the above expression in
Eq. (97), the amplitude of the GW generated for the fictitious static observer
near the BH is:

$\displaystyle|{M}_{\mathsf{T}}|\,\sim{h}_{\mathsf{T}}\sim 10^{-2}$ (99)

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