# Gertsenshtein-Zel′dovich effect: A plausible explanation for fast radio
bursts?

Ashu Kushwaha *  |1 , Sunil Malik *  |1,2,3 , S. Shankaranarayanan *  |1
1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India
2Institute fur Physik und Astronomie Universitat Potsdam, Golm Haus 28,
D-14476 Potsdam, Germany
3Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen,
Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>(SM)
(Corresponding Author<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a novel model that may provide an interpretation for a class of
non-repeating FRBs — short ($<1~{}\rm{s}$), bright ($0.1-1000~{}\rm{Jy}$)
bursts of MHz-GHz frequency radio waves. The model has three ingredients —
compact object, a progenitor with effective magnetic field strength around
$10^{10}~{}{\rm Gauss}$, and high frequency (MHz-GHz) gravitational waves
(GWs). At resonance, the energy conversion from GWs to electromagnetic waves
occurs when GWs pass through the magnetosphere of such compact objects due to
the Gertsenshtein-Zel’dovich effect. This conversion produces bursts of
electromagnetic waves in the MHz-GHz range, leading to FRBs. Our model has
three key features: (i) predict peak-flux, (ii) can naturally explain the
pulse width, and (iii) coherent nature of FRB. We thus conclude that the
neutron star/magnetar could be the progenitor of FRBs. Further, our model
offers a novel perspective on the indirection detection of GWs at high-
frequency beyond detection capabilities. Thus, transient events like FRBs are
a rich source for the current era of multi-messenger astronomy.

###### keywords:

Transient phenomena, Fast Radio Bursts, Neutron stars, Pulsars, magnetic
fields, Gravitational Waves

##  1 Introduction

Technological advancement has fuelled research in high-energy astrophysical
phenomena at larger redshift ranges, and we are in a position to address some
unresolved questions starting from pulsar emission mechanism Melrose et al.
(2021) to short bursts such as Gamma-ray bursts (GRBs) Levan et al. (2016),
Fast radio bursts (FRBs) Lorimer (2008); Cordes & Chatterjee (2019); Platts et
al. (2019). To date, more than 600 FRBs have been reported in various
catalogues Petroff & et al (2016); Platts et al. (2019); Pastor-Marazuela et
al. (2021); Rafiei-Ravandi et al. (2021). $99\%$ of these FRBs have the
following three characteristic features: observed peak flux ($S_{\nu}$) varies
in the range $0.1~{}{\rm Jy}<S_{\nu}<700~{}{\rm Jy}$, coherent radiation and
the pulse width is less than a second Rafiei-Ravandi et al. (2021); Petroff &
et al (2016). These observations have posed the following questions: What
causes these extreme high-energy transient radio-bursts from distant galaxies,
lasting only a few milliseconds each Lorimer (2008); Cordes & Chatterjee
(2019); Platts et al. (2019)? Why do some FRBs repeat at unpredictable
intervals, but most do not Platts et al. (2019)? Does strong gravity provide
an active role?

Naturally, many models have been proposed to explain the origin of FRBs. All
these models try to provide a physical mechanism that results in large amount
of coherent radiation in a short time Lorimer (2008); Cordes & Chatterjee
(2019); Petroff & et al (2016); Platts et al. (2019); Pastor-Marazuela et al.
(2021); Rafiei-Ravandi et al. (2021). Since the time scale of these events is
less than a second, and the emission is coherent, the astrophysical processes
that explain these events _cannot_ be thermal Lorimer (2008).

Broadly, these models can be classified into two categories Zhang (2022): FRBs
created by interaction of an object with a pulsar/magnetar and FRBs created
from the magnetar/pulsar itself Lorimer (2008); Cordes & Chatterjee (2019);
Petroff & et al (2016); Platts et al. (2019); Pastor-Marazuela et al. (2021);
Rafiei-Ravandi et al. (2021). The first category can further be classified
into two broad classes. In the first class, the energy powering FRBs comes
from the neutron star magnetosphere/wind themselves, and an orbiting object
converts this energy into radiation. In the second class, the object falls
onto the neutron star, and its gravitational energy partly gets converted to
FRBs. In the second class, many models involving non-thermal processes such as
Synchrotron radiation Lorimer & Kramer (2005), black hole super-radiance
Conlon & Herdeiro (2018), evaporating primordial black hole Rees (1977); Carr
& Kuhnel (2020), spark from cosmic strings Vachaspati (2008), Quark Novae
Shand et al. (2016), synchrotron maser shock model Wu et al. (2020), radiation
from reconnecting current sheets in the far magnetosphere Lyubarsky (2020),
curvature emission from charge bunches Wang et al. (2022) have been proposed.
Several classes of FRB models predict prompt multiwavelength counterparts and
specify the ratio between the energy emitted by the counterpart and by the FRB
Zhang (2017); Metzger et al. (2019).

However, despite the use of exotic new physics, no single model has provided a
universal explanation for the enormous energy released in these events. It is
important to note that all these mechanisms require electromagnetic
interaction to generate FRBs. Due to the nature of electromagnetic
interaction, small-scale emission mechanisms usually predominate over large-
scale coherent electromagnetic processes (like astrophysical masers and pulsar
radio emission). In this work, we provide an alternative framework that
overcomes this and can explain the observed coherence in FRBs.

As shown below, one key missing ingredient is the _dynamics of strong-
gravity_. The Spatio-temporal changes in the strong-gravity regime —
oscillons, phase transitions, plasma instability, primordial black holes,
reheating — generate gravitational waves (GWs) in a broad range of frequencies
($10^{-15}-10^{15}$ Hz) Carr & Hawking (1974); Anantua et al. (2009); Kuroda
et al. (2015); Ejlli et al. (2019); Aggarwal et al. (2021); Chen et al.
(2020); Pustovoit et al. (2021). Like EM waves, GWs are generated by the time-
varying quadrupole moment Sathyaprakash & Schutz (2009); Schutz (2000). Since
all masses have the same gravitational sign and tend to clump together, they
produce large coherent bulk motions that generate _energetic, coherent GWs_
Hendry & Woan (2007). Thus, if a mechanism that converts incoming coherent GWs
to EM waves exists, we can explain the extremely energetic, coherent nature of
FRBs Popov et al. (2018); Lieu et al. (2022). In this work, we construct a
model that uses this feature.

Since FRBs are highly energetic, an attentive reader might wonder do incoming
GWs carry such large energies. GWs carry an enormous amount of energy. For
example, typical GWs from a compact binary collapse with amplitude $h\sim
10^{-22}$ carry the energy of the order of $10^{20}~{}\rm{Jy}$ Sathyaprakash &
Schutz (2009); Schutz (2000). Also, stellar mass binary black hole collision
can lead to peak gravitational-wave luminosity of $10^{56}~{}\rm{ergs/s}$
Schutz (2022); Abbott et al. (2016a). If the GWs indeed carry a lot of energy,
can this energy transform into other observable forms of energy? Currently,
there is evidence of GWs in the frequency range $10^{-9}-10^{4}~{}{\rm Hz}$
from LIGO-VIRGO-KAGRA and PTA observations Abbott et al. (2016b); Agazie et
al. (2023). While most of the current effort has focused on these frequency
ranges, there is a surge in activity for the possibility of detecting GWs in
the MHz-GHz frequency range Aggarwal et al. (2021). New physics beyond the
standard model of particle physics, like an exotic compact object, can produce
observable GW signals in this frequency ranges Aggarwal et al. (2021); Chen et
al. (2020); Pustovoit et al. (2021).

Figure 1: Schematic depiction of GZ effect. The externally generated GWs at
distance $D$ from the NS is converted to EM Waves in the magnetosphere.The
blue region corresponds to NS with radius $r_{*}$ and the yellow region around
NS corresponds to magnetosphere of radius $R_{\rm LC}$. The black curves
correspond to the magnetic field lines.

A physics maxim is that energy can be transformed between different forms.
Although the total energy is conserved, the efficiency of the transformation
depends on the energy scale, background dynamics, and external conditions
(parameters). Energy transformation is one way to probe strong gravity regions
like the early universe, black-holes, and NS. In this work, we propose a novel
approach that uses the energy conversion from incoming, coherent GWs to
electromagnetic (EM) waves that can _explain milli-second bursts, like non-
repeating FRBs_.

GWs get converted to EM waves in the presence of strong transverse magnetic
fields — Gertsenshtein-Zel’dovich (GZ) effect Gertsenshtein (1962); Zel’dovich
(1974); Zheng et al. (2018); Domcke & Garcia-Cely (2021). Gertsenshtein
postulated the existence of wave resonance between EM waves and GWs on the
basis of their identical propagation velocities and the equations describing
them are linear. By employing linearized Einstein’s field equations, he
demonstrated that EM waves are generated through wave resonance when
gravitational waves traverse a strong magnetic field. In the same way, light
passing through a strong magnetic field generates gravitational waves
Zel’dovich (1974); Kolosnitsyn & Rudenko (2015). Appendix A contains the
detailed calculations reproducing the key results.

To understand the GZ effect, consider coherent GWs with a frequency
$\omega_{g}$ passing through a region with a high transverse static magnetic
field ${\bf B}$. The propagation of GWs leads to compression and stretching of
the magnetic field proportional to $h{\bf B}$ ($h$ is the amplitude of GWs),
which acts as a source leading to the generation of EM waves Gertsenshtein
(1962); Zel’dovich (1974). The induced (resultant) EM waves generated will
have maximum amplitude at resonance, i.e., the frequency of EM waves is
identical to $\omega_{g}$. (For details, see appendix A)

In quantum mechanical language, the GZ effect is analogous to the mixing of
neutrino flavors — the external field _catalyzes_ a resonant mixture of photon
and graviton states Palessandro & Rothman (2023). The external magnetic field
provides the extra angular momentum necessary for the spin-1 (photon) field to
mix with the spin-2 (graviton) field. Thus, the GZ mechanism involves the
transfer of energy from the incoming, coherent GWs to emitted, coherent EM
radiation in the presence of the background magnetic field. The maximum
efficiency of this conversion can be achieved if the background field is
strong at the resonance frequency (when the frequency of the emitted EM
radiation is the same as the incoming GWs). Hence, the background magnetic
field acts as a catalyst in this mechanism.

We show that the GZ effect may provide an interpretation for a class of _non-
repeating FRBs_. To explain the energy bursts in FRBs, we propose a model with
the following two realistic assumptions: (i) the astrophysical object is
compact and has a strong gravity environment, and (ii) the object possesses a
small time-dependent magnetic field on top of the large, effective static,
transverse magnetic field. These two assumptions principally lead us to
stellar remnants, such as NS and magnetars with magnetic field strength
ranging from $10^{8}-10^{15}~{}\rm{G}$ Andersen et al. (2020); Belvedere et
al. (2015); Cordes & Chatterjee (2019); Lorimer (2008). The small time-
dependent magnetic fields arise due to the rotation of the NS about its axis
with frequency $\omega_{B}$ Belvedere et al. (2015); Melikidze & Gil (2008);
Platts et al. (2019); Pons & Viganò (2019); Pons, J. A. et al. (2012). We
consider $\omega_{B}$ in the range $[1,10^{3}]~{}{\rm Hz}$ Lorimer & Kramer
(2005). As a result, the effective magnetic field at a given point in the NS
magnetosphere is ${\bf B}(t)={\bf B}^{(0)}+\delta{\bf B}\sin(\omega_{B}t)$. It
has been noted that $|\delta{\bf B}/{\bf B}^{(0)}|$ can be as large as $0.1$
Pons, J. A. et al. (2012). Here, we take $|\delta{\bf B}/{\bf B}^{(0)}|\sim
10^{-2}$.

##  2 Model

1 gives the schematic depiction of the physical model to explain the energy
burst in FRBs. Consider GWs generated due to exotic compact objects (such as
Boson stars, Oscillons, gravastars) Aggarwal et al. (2021); Chen et al.
(2020); Pustovoit et al. (2021) passing through the magnetosphere of NS at a
distance $D$. In the figure, the magnetosphere is depicted as a cylinder. GZ
effect converts GWs to EM waves as they pass through the magnetosphere
Gertsenshtein (1962); Zel’dovich (1974). This conversion occurs at all points
in the magnetosphere. Therefore, a faraway observer will see the integrated
effect happening in the entire magnetosphere in this short duration. For
example, the light cylinder radius ($R_{\rm LC}$) for a typical NS is $\sim
10^{7}-10^{9}~{}{\rm cm}$, implying that the GWs take less than one second to
cover the entire magnetosphere. This is one of the primary ingredients
supporting our analysis for the FRB observations.

To compute the GZ effect at a point in the magnetosphere, we consider source-
free Maxwell’s equations on the background space-time with GW fluctuations
(see details in Appendix (A)). The linearized Einstein’s equations, up to
first order in the space-time perturbations are highly accurate. In this
approximation, the effects of GWs on the stress-tensor (Riemann tensor) are
negligible. Hence, we consider background space-time to be Minkowski (in
cartesian coordinates) Misner et al. (1973)). The two polarizations of GW
(with frequency $\omega_{g}$ and wave-vector $k_{g}$) propagating along the
z-direction are:

$\displaystyle
h_{+}=A_{+}\,e^{i\left(k_{g}z-\omega_{g}t\right)},h_{\times}=iA_{\times}\,e^{i\left(k_{g}z-\omega_{g}t\right)},$
(1)

where $A_{+}$ and $A_{\times}$ are the constant amplitudes of the GWs. We
assume that both the modes of GWs are generated with an equal amount of
energy, i.e., $|A_{+}|=|A_{\times}|$ — the isospectrality condition in general
relativity Chandrasekhar (1998). Taking the distance between coherent GW
source, such as an exotic compact object, and NS to be $D=1\,\rm{kpc}$, gives
$h\simeq A_{+}=10^{-20}$ at $1\,\rm{GHz}$ (see, Aggarwal et al. (2021) and
Appendix E). In this work, we have assumed $h$ to be three orders smaller
($\sim 10^{-23}$) near the NS.

As mentioned above, the key requirement of the GZ-effect is the presence of
the transverse magnetic field to the direction of propagation of coherent GWs.
Besides, the time taken by the GWs to pass through the entire magnetosphere is
much smaller than the rotation period of the millisecond pulsar/magnetar
($\omega_{B}^{-1}$). Given the direction of propagation of GWs along the
z-axis, the effective time-dependent transverse magnetic field is taken to be
$\textbf{B}(t)=\left(0,B^{(0)}_{y}+\delta B_{y}\sin(\omega_{B}t),0\right)$
Pons & Viganò (2019); Pons, J. A. et al. (2012).

Although the effective magnetic field depends on the distance from the surface
of the object Lorimer (2008); Belvedere et al. (2015); Cordes & Chatterjee
(2019); Andersen et al. (2020), we assume that ${\bf B}(t)$ is independent of
the distance from the surface up to $R_{\rm LC}$ Lorimer & Kramer (2005). In
appendix (B), we explicitly show that the above assumption that the background
magnetic field can be treated as a constant in the entire magnetosphere gives
_identical results_ to that of the background field decreasing radially, i.
e.,

$\displaystyle\left(B_{r},B_{\theta},B_{\phi}\right)=B_{*}\,\left(\frac{r_{*}}{r}\right)^{3}\,\left(2\cos\theta,\sin\theta,0\right)$
(2)

where $B_{*}$ is the magnetic field on the NS surface. More specifically,
assuming that the NS magnetic field is dipolar, we show in appendix (B) that
we can approximate the average magnetic field at any point in the
magnetosphere to be constant. In other words, the total conversion factor we
obtain using the above assumption mimics the realistic NS regions.

Given the above setup, we now evaluate the GZ-effect in the magnetosphere of
the NS and compare it with observational quantities in two steps:

  1. 1.

First step involves evaluating the conversion from coherent GWs to EM waves at
a typical point inside the magnetosphere. This is referred to as _conversion
factor_ ($\alpha$). We then obtain the total conversion factor ($\alpha_{\rm
tot}$) inside the entire magnetosphere _at resonance_ (the frequency of EM
waves is identical to $\omega_{g}$). This conversion factor includes only
those contributions that are along the direction of line-of-sight and
coinciding with the incoming GWs. Also, only the transverse magnetic field
component to the direction of propagation at each point of the magnetosphere
will contribute to the emitted EM waves. This can potentially explain the
coherent nature of FRBs Katz (2014); Kumar et al. (2017).

  2. 2.

For a given conversion factor, we obtain the Poynting vector of the resultant
EM waves along the direction of propagation ($S_{z}$). Then, we compare the
theoretically derived Poynting vector with the observation of peak flux with
the reported FRBs.

The Poynting vector is a well-defined quantity for photons that travel from
the source to the observer without any hindrance Carroll & Ostlie (2017);
Condon & Ransom (2016); Zhang (2018). More specifically, assuming there is no
absorption/emission of the photons during the entire journey, the Poynting
vector (of the EM waves) $S_{z}$ is conserved (independent of the distance
between the magnetosphere and observer) and is valid for emissions from
compact sources. Hence, the Poynting vector estimated at a small angle (along
the direction of the incoming gravitational waves) remains the same at the
source and the detector. Furthermore, as shown in 1, the incoming, coherent GW
is along the $z-$axis in the entire magnetosphere. Therefore, the cumulative
effect of the emitted EM waves is in the same direction.

To evaluate the conversion factor, solving the linearized covariant Maxwell’s
equations leads to the following electric and magnetic fields induced due to
GWs, i.e., $\tilde{E}_{x}$ and $\tilde{B}_{y}$ as:

$\displaystyle\tilde{E}_{x}$
$\displaystyle\simeq-\frac{A_{+}}{4}\,B^{(0)}_{y}\left(1-\xi\,\omega_{B}t\,\,\right)\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}$
(3) $\displaystyle\tilde{B}_{y}$
$\displaystyle\simeq-\frac{A_{+}}{4}\,B^{(0)}_{y}\left(1+2\xi\,\omega_{g}t\,\,\right)e^{i\left(k_{g}z-\omega_{g}t\right)}\,,$
(4)

where $\xi\equiv\delta B_{y}/B_{y}^{(0)}$. Note that the amplitude of
$\tilde{B}_{y}$ has a dependence on $\omega_{g}$, while the amplitude of
$\tilde{E}_{x}$ has $\omega_{B}$ dependence on $\omega_{g}$. This is because
the induced electric field arises due to the time-varying magnetic field (see
appendix A).

The conversion factor ($\alpha$) — ratio of the energy density of EM wave and
GWs — gives the efficiency of the process at resonance. $\alpha$ for this
process is

$\displaystyle\\!\\!\\!\alpha\equiv\frac{\rho_{\rm EM}}{\rho_{\rm
GW}}\simeq\frac{G|B^{(0)}_{y}|^{2}}{4c^{2}}\left[2\left(\frac{\xi
z}{c}\right)^{2}+2\frac{\xi}{\omega_{g}}\frac{z}{c}+\frac{1}{\omega_{g}^{2}}\right]$
(5)

where $z$ refers to the radial distance in the magnetosphere. For details, see
appendix C.

The above expression is the conversion factor at a single point on the
magnetosphere. Assuming that there is no cross-correlation of GZ-effect at two
distinct points in the magnetosphere, we obtain the total conversion by
integrating over the entire magnetosphere (from the surface of the compact
object to the light cylinder $R_{\rm LC}$). (The cross-correlation corresponds
to the induced EM waves at two distinct points affecting each other. This
physically corresponds to higher-order effects
($A_{+}\partial_{z}\tilde{B}_{y}$) which are neglected.). Thus, the total
conversion factor is

$\displaystyle\alpha_{\rm{tot}}\simeq\frac{\pi
G|B^{(0)}_{y}|^{2}}{c^{2}}\left[\frac{2}{3}\left[\frac{\xi R_{\rm
LC}}{c}\right]^{2}\\!\\!+\frac{\xi R_{\rm
LC}}{\omega_{g}c}+\frac{1}{\omega_{g}^{2}}\right].$ (6)

See appendix C for details. It is important to note that $\alpha_{\rm tot}$ is
_independent of the amplitude_ of GWs. To understand the variation of
$\alpha_{\rm tot}$ with $\omega_{g}$, in 2, we have plotted the conversion
factor for magnetar and NS/milli-second pulsar.

Figure 2: Log-Log plot of $\alpha_{\rm tot}$ versus $\omega_{g}$ for typical
magnetar (left plot) and typical NS (right plot). For magnetar, we set
$B^{(0)}_{y}=10^{15}\,{\rm G},R_{\rm LC}=10^{9}~{}\,{\rm
cm},\omega_{B}=1\,{\rm Hz}$. For NS, we set $B^{(0)}_{y}=10^{10}{\rm G},R_{\rm
LC}=10^{7}{\rm cm},\omega_{B}=1{\rm kHz}$.

From the plots, we infer that the total conversion factor is almost
insensitive at higher frequencies ($>1~{}{\rm MHz}$) for both types of compact
objects. This is because the second and third terms in RHS of Eq. (6) are
inversely proportional to $\omega_{g}$ and, hence, the contribution to
$\alpha_{\rm tot}$ is only from the first term. To elaborate, 3 plots each of
these terms in Eq. (6) which shows a clear distinction between milli-second
Pulsar and magnetar. In both cases, the cross-over occurs below $1~{}{\rm
MHz}$. Since we are interested in radio frequency in the MHz-GHz range, the
total conversion factor (6) is independent of the incoming, coherent GW
frequency.

This leads to the important question: What is the efficiency of the GZ-effect
near magnetar and NS? Table (1) lists the total conversion factor (4th column)
for magnetar and NS for different frequencies. We want to emphasize the
following points: First, as mentioned earlier, we have assumed the amplitude
of the GWs, i.e., $A_{+}=A_{\times}=1.4\times 10^{-23}$ Kuroda et al. (2015);
Aggarwal et al. (2021). However, as we show below, even with this conservative
value, the model explains the peak flux of FRBs. Second, $\alpha_{\rm tot}$ is
very high near the magnetar for low-frequency GWs. Specifically, the
conversion is $0.06\%$ and $0.25\%$ at 1000 Hz and 500 Hz, respectively (for
the magnetar $B_{y}^{(0)}=10^{15}\,{\rm G}$). For high-frequency GWs, as we
show below, even a total conversion factor $(\alpha_{\rm tot})$ of $10^{-19}$
can lead to appreciable energy in the radio frequency (MHz-GHz range).

Figure 3: Log-Log plot of the three terms in the RHS of Eq. (6) versus
$\omega_{g}$ for typical magnetar (top plot) and typical NS (bottom plot). For
magnetar, we have set $B^{(0)}_{y}=10^{15}{\rm G},R_{\rm LC}=10^{9}~{}{\rm
cm},\omega_{B}=1{\rm Hz}$. For NS/milli-second pulsar, we have set
$B^{(0)}_{y}=10^{10}{\rm G},R_{\rm LC}=10^{7}{\rm cm},\omega_{B}=1{\rm kHz}$.

Further, to compare the generated EM energy density in the entire
magnetosphere with the observations, we compute the flux of the induced EM
waves (3, 4) by calculating the Poynting vector Rybicki & Lightman (1979).
Rewriting Eq. (6) as a quadratic equation in $\xi R_{\rm LC}/c$ provides the
functional dependence in-terms of $\alpha_{\rm tot}$. This leads to:

$\displaystyle
S_{z}\simeq\frac{A_{+}^{2}|B_{y}^{(0)}|^{2}c}{256\pi}\left[\sqrt{\frac{24c^{2}\omega_{g}^{2}\alpha_{\rm
tot}}{\pi G|B^{(0)}_{y}|^{2}}-15}-\frac{12c^{2}\omega_{g}\omega_{B}\alpha_{\rm
tot}}{\pi G|B^{(0)}_{y}|^{2}}-1\right].$ (7)

Note that $\alpha_{\rm{tot}}$ in RHS of the above expression is a function of
$\omega_{g}$ and the parameters of magnetar/NS. See appendix D for details.
Thus, the Poynting vector of the induced EM waves in the vicinity of
magnetar/NS can be obtained by substituting the the parameters of magnetar/NS,
$\omega_{g}$ and $A_{+}$. To compare with the peak flux of FRBs, 4 contains
the plot of the Poynting vector per unit frequency $(S_{z}/\omega_{g})$ as a
function of $\omega_{g}$ for magnetar and NS.

Figure 4: Log-Log plot of $S_{z}/\omega_{g}$ versus $\omega_{g}$ for typical
magnetar (top plot) and typical NS (bottom plot). For magnetar, we have set
$B^{(0)}_{y}=10^{15}{\rm G},R_{\rm{LC}}=10^{9}~{}\,{\rm
cm},\omega_{B}=1\rm{Hz}$. For NS/milli-second pulsar, we have set
$B^{(0)}_{y}=10^{10}\,{\rm G},R_{\rm{LC}}=10^{7}{\rm
cm},\omega_{B}=1\,\rm{kHz}$. For both plots we have set $A_{+}=1.4\times
10^{-23}$ corresponding to a typical GW source Aggarwal et al. (2021).

The last column in Table (1) contains the spectral flux density (Poynting
vector per unit frequency) for generic parameter ranges of $R_{\rm LC}$ and
$B_{y}^{(0)}$. Given the parameters listed in the table (1), our model
predicts a range of spectral flux density that can be as small as $0.1~{}{\rm
Jy}$ (milli-second Pulsar) or $10^{-24}{\rm erg~{}s^{-1}cm^{-2}Hz^{-1}}$ and
can be as large as $10^{11}~{}{\rm Jy}$ (Magnetar) or $10^{-12}{\rm
erg~{}s^{-1}cm^{-2}Hz^{-1}}$. [ 1 Jy in CGS units is
$1\rm{Jy}=10^{-23}erg~{}s^{-1}cm^{-2}Hz^{-1}$.]. We are now in a position to
compare the results of the model with radio observations in MHz-GHz frequency.

##  3 A plausible explanation for FRBs

As mentioned earlier, more than 600 (non-repeating) FRBs are reported in
various catalogues Petroff & et al (2016); Platts et al. (2019); Pastor-
Marazuela et al. (2021); Rafiei-Ravandi et al. (2021) $99\%$ of these FRBs
were found to have the following three characteristic features: peak flux
($S_{\nu}$) varies in the range $0.1~{}{\rm Jy}<S_{\nu}<700~{}{\rm Jy}$, the
pulse width is less than one second and coherent radiation Rafiei-Ravandi et
al. (2021); Petroff & et al (2016). In Ref. Petroff et al. (2016), the authors
classified the quantities associated with FRBs into two types — observed and
derived. Peak flux and width are observed quantities, while Luminosity and
Luminosity distance are derived quantities. The uncertainty in the Dispersion
Measure versus redshift relation can have a cascading effect on the derived
quantities of the FRBs, leading to uncertainty in the origin of these events
Pol et al. (2019). Hence, to reduce the systematic bias, we focus on observed
Peak flux and estimate the same using the Poynting vector of the emitted EM
waves ($S_{z}/\omega_{g}$).

Table 1: The table contains numerical values of the total conversion factor ($\alpha_{\rm tot}$), energy density of EM waves ($\rho_{\rm EM}$), and spectral flux density (Poynting vector per unit frequency). The first two rows are for a typical Magnetar and the last three rows are for a typical NS. We have set $A_{+}=10^{-23}$ corresponding to a typical GW source Aggarwal et al. (2021). Note that if we use $\xi\equiv\delta B_{y}/B_{y}^{(0)}\sim 0.1$, $\alpha_{\rm tot},\rho_{\rm EM}$, and the value of Poynting vector will increase by two-orders of magnitude. $R_{\rm LC}$ | $B_{y}^{(0)}$ | $\omega_{g}$ | $\alpha_{\rm tot}$ | $\rho_{\rm EM}\,$ | $\frac{S_{z}}{\omega_{g}}$
---|---|---|---|---|---
(cm) | (Gauss) | (MHz) |  | (${\rm{erg\,cm^{-3}}}$) | (Jy)
$10^{9}$ | $10^{15}$ | $1$ | $1.74\times 10^{-5}$ | ${9.3\times 10^{-13}}$ | $9.95\times 10^{11}$
$10^{9}$ | $10^{12}$ | $500$ | $1.72\times 10^{-11}$ | ${2.3\times 10^{-13}}$ | $9.94\times 10^{5}$
$10^{8}$ | $10^{11}$ | $1400$ | $1.72\times 10^{-15}$ | ${1.8\times 10^{-16}}$ | $961.57$
$10^{7}$ | $10^{10}$ | $1400$ | $1.72\times 10^{-19}$ | ${1.8\times 10^{-20}}$ | $0.99$
$10^{8}$ | $10^{9}$ | $1400$ | $1.72\times 10^{-19}$ | ${1.8\times 10^{-20}}$ | $0.09$

As mentioned earlier, $R_{\rm LC}$ of a typical NS is $\sim
10^{7}-10^{9}~{}\rm{cm}$ and it takes less than one second for the GWs to pass
through the entire magnetosphere. This directly implies that the induced EM
waves due to GZ-effect will appear as a burst lasting for less than one
second. Thus, our model provides a natural explanation for FRBs lasting less
than a second. Further, we see from the last column of Table (1) that our
model predicts the burst of EM wave with the flux $<1000~{}\rm{Jy}$. Thus, our
model naturally explains the observed peak flux and pulse width of $99\%$ of
the reported FRBs.

Having estimated the peak flux and pulse width for our model, we can now
calculate other quantities associated with FRBs. In particular, we calculate
two such quantities — Fluence (${\cal F})$ Keane & Petroff (2015); Macquart &
Ekers (2018) and Isotropic Equivalent Luminosity (IEL) Lu & Kumar (2019). The
Fluence is defined as the product of the burst width $\Delta t=(2R_{LC})/c$
and the peak flux ($S_{z}/\omega_{g}$) Keane & Petroff (2015); Macquart &
Ekers (2018):

${\cal F}=(S_{z}/\omega_{g})\times\Delta t.$

Using $S_{z}$ from the table 1, we see that our model predicts the Fluence in
the range $1<{\cal F}({\rm Jy~{}ms})<10^{7}$. We now rewrite the energy
density $(\rho_{\rm EM})$ of emitted EM waves in terms of the isotropic
equivalent luminosity ($L_{\rm IEL}$). Assuming that the progenitor is at a
distance $d$ from the observer, $L_{\rm IEL}$ is given by Lu & Kumar (2019):

$\displaystyle\rho_{\rm EM}=\frac{L_{\rm IEL}}{4\pi d^{2}c}.$ (8)

The above expression assumes that $d\gg$ size of the progenitor is justified
for the reported FRBs. Due to the lack of confirmed detection in other
wavelengths, the distance of the FRBs is not well constrained Zhang (2022).
The dispersion measure of the FRB gives one indirect estimation Thornton et
al. (2013a); Ravi et al. (2016). It is expected that the distance of the FRBs
is in the range $[10~{}{\rm kpc}-1~{}{\rm Gpc}]$.

From the FRB catalogs Petroff & et al (2016); Platts et al. (2019); Pastor-
Marazuela et al. (2021); Rafiei-Ravandi et al. (2021), we see that FRBs have a
peak flux of around $100~{}{\rm Jy}$. For our model, this corresponds to the
energy density of the emitted EM waves to be $\sim 9.2\times 10^{-18}~{}{\rm
erg\,cm}^{-3}$. Taking the above distance range ($[10~{}{\rm kpc}-1~{}{\rm
Gpc}]$), the above energy density translates to $L_{\rm IEL}$ in the range
$[3.1\times 10^{39}-10^{49}]~{}{\rm erg/s}$. Thus, our model can explain the
inferred luminosity of FRBs Law et al. (2017).

Given this, we can now identify the possible progenitors of FRBs. To identify
this, we consider two FRBs — FRB120127 Thornton et al. (2013b) and FRB011025
Burke-Spolaor & Bannister (2014). These two sources represent a typical FRB in
the catalogs. The observations of these two FRBs at
$1.5~{}\rm{and}~{}1.3~{}\rm{GHz}$, show a typical peak flux of
$0.62^{+0.35}_{-0.10}~{}\rm{Jy}$ and $0.54^{+0.11}_{-0.07}~{}\rm{Jy}$,
respectively. From the last row of the Table (1), we see that our model
predicts the progenitor should be a millisecond pulsar with an effective
magnetic field strength of $10^{10}~{}{\rm G}$ and $R_{\rm{LC}}\sim
10^{7}~{}\rm{cm}$. Fluence for these two FRBs is around $0.66~{}\rm{Jy\,ms}$
and isotropic equivalent luminosity $(L_{\rm IEL})$ at a distance
${d=10~{}{\rm Mpc}}$ is $6\times 10^{42}\rm{erg\,s^{-1}}$.

While many FRBs are considered to be extragalactic, recently, there have been
confirmed galactic FRBs Bochenek et al. (2020); CHIME/FRB Collaboration et al.
(2020), Lu & Kumar (2019). Specifically, FRB 200428 is confirmed to be a
galactic FRB Bochenek et al. (2020). Recent observations indicate that the
magnetar SGR 1935+2154 residing in the Milky Way is associated with the FRB
200428 with a fluence of $>1.5\times 10^{6}~{}{\rm Jy~{}ms}$ in the
$1.28$–$1.4$ GHz band detected by the STARE2 radio array Bochenek et al.
(2020); CHIME/FRB Collaboration et al. (2020). It is reported that this
magnetar has a surface dipole magnetic field of $\sim 2.2\times 10^{14}$ G,
which can be deduced from the period slow-down rate of $3.24~{}{\rm s}$. Given
these observational quantities, we can confirm whether our model can explain
the origin of this galactic FRB. To do this, we substitute the constant
magnetic field approximation $B_{y}^{(0)}\sim 2.2\times 10^{14}~{}{\rm G}$
with $R_{\rm LC}\sim 10^{9}~{}{\rm cm}$ and $A_{+}\sim 10^{-26}$ in Eq.(7),
our model predicts peak flux $\sim 4.8\times 10^{4}~{}{\rm Jy}$ and Fluence to
be $3.2\times 10^{6}{\rm Jy~{}ms}$. Thus our model predicts isotropic
equivalent luminosity at $~{}10\ {\rm kpc}$ to be $L_{\rm IEL}\sim 2.8\times
10^{43}\rm{erg\,s^{-1}}$. From these, it appears that our model confirms that
the FRB 200428 is a galactic burst and originated from the GZ mechanism.
Consequently, our model has the potential to explain these observations and
play a crucial role in any future FRB and progenitor association (if any).

We can do a similar analysis for all the FRBs in the catalog with a pulse
width of less than one second Rafiei-Ravandi et al. (2021); Petroff & et al
(2016). Our model predicts that the progenitor can be a NS with an effective
magnetic field strength in the range $10^{9}-10^{11}~{}{\rm G}$ and rotation
frequency $1<\omega_{B}<1000$. Our model can provide an explanation for the
observed peak flux for a class of _non-repeating_ FRBs and predicts FRBs with
extremely low and high peak flux. Note that our model is not sensitive to the
galactic environment. Such future detections will further strengthen the
model.

As mentioned earlier, the model uses the GZ mechanism to explain the origin of
FRBs. GZ mechanism involves the transfer of energy from the incoming GWs to
emitted EM radiation in the presence of the background magnetic field.
Although our model falls in the first category where FRBs are generated from
the magnetar/NS, electromagnetic radiation is generated when GWs pass through
the magnetosphere. In other words, the background magnetic field acts as a
catalyst in this mechanism. We can understand the energy budget of this
process by looking at it in two different ways:

  1. 1.

We evaluate the ratio of the energy density of the outgoing EM waves
($\rho_{\rm EM}$) with the energy density of the background magnetic field
($\rho_{B}$). For a magnetic field strength of $10^{10}{\rm G}$, we get

$\frac{\rho_{\rm EM}}{\rho_{B}}\sim 2.2\times 10^{-39}.$

  2. 2.

Second, the ratio of the incoming GWs to the emitted EM waves. Rewriting the
energy density of the incoming GWs in terms of the Luminosity and energy
density of the emitted EM waves in terms of the Luminosity, we have:

$L_{\rm EM}\sim\alpha_{\rm tot}L_{\rm GW}\left(\frac{R_{LC}}{D}\right)^{2},$
(9)

where $L_{\rm GW}$ is given by Schutz (2022); Sathyaprakash & Schutz (2009);
Abbott et al. (2017):

$L_{\rm GW}=L_{0}\left(\frac{\omega_{g}Dh}{c}\right)^{2}$

where $L_{0}\sim 10^{59}~{}{\rm erg/s}$ is referred to as fundamental
luminosity Sathyaprakash & Schutz (2009), $\omega_{g}$ is the frequency of the
GWs, and $h$ is the GW amplitude. The detected BH mergers in the LIGO-VIRGO
band are of the order $\sim 10^{56}$ erg/s Abbott et al. (2016a). Therefore it
is certainly possible to have such large luminosity GW sources. In fact, for
highly-relativistic systems, $L_{\rm GW}\sim L_{0}$Sathyaprakash & Schutz
(2009).
From the above two expressions and taking the values we have used in the
earlier section, we see that our model predicts $L_{\rm EM}\propto
10^{36}~{}{\rm erg/s}$. However, if we use $\xi\equiv\delta
B_{y}/B_{y}^{(0)}\sim 0.1$, our model predicts $\alpha_{\rm tot}\sim 10^{-3}$
and can lead to larger value of $L_{\rm EM}$.
Note that $L_{\rm EM}$ derived is theoretical Luminosity based on the
Luminosity of the incoming GW and is not equivalent to $L_{\rm IEL}$ defined
in Eq. 8. From the above discussions, it s clear that our mechanism is
energetically expensive and may not be a frequent phenomenon.

##  4 Discussions

Our mechanism requires that the emitted GWs pass through the NS, resulting in
_non-repeating_ FRBs along the line of sight of the observer. In other words,
we have assumed that the EM emission due to the GZ-mechanism in the pulsar
magnetosphere is directional dependent. The intervening medium does not impact
the radiation from this mechanism. More importantly, the mechanism requires
that the emitted GWs pass through the NS, resulting in FRBs along the line of
sight of the observer. The probability of this event is a product of the
probability that the GW passes through the NS and the probability that the
emitted EM is along the line of sight of the observer. Hence, the probability
of seeing such an event is not very high. More importantly, any existing
NS/Magnetar in any galaxy can produce FRB. Interestingly, recently GZ effect
was employed by Kalita & Weltman (2023) to compare the expected output of the
GZ signal with two known FRBs and their potential detection in GW detectors.
They demonstrated that any continuous GW signal detected by the suggested GW
detectors from the FRB location would instantly suggest that the merger-like
theories are unable to account for all FRBs, hence offering strong evidence in
favor of the GZ hypothesis.

Further, it is estimated that around $10^{8}-10^{9}$ NSs are present in the
Milky Way galaxy, roughly $1\%$ of the total number of stars in the galaxy
Diehl et al. (2006); Sartore, N. et al. (2010). Also, it is estimated that the
magnetar formation rate is approximately $1-10$ percent of all pulsars Gullón
et al. (2015); Beniamini et al. (2019). One of the basic assumption of our
model is that, given GW signal in MHz-GHz frequency, all NSs can act as source
of FRBs at all times. This assumption translates to the fact that the maximum
FRB events per day can be $10^{8}-10^{9}$. However, the observed FRB rate is
$10^{3}$ for the entire sky per day. This can be attributed to the fact that
the probability of this event is a product of the probability that the GW
passes through the NS times the probability that the emitted EM is along the
line of sight of the observer. Consequently, our model predicts the observed
FRB event rate and the coherent nature of FRBs Katz (2014). Since, GWs can be
generated up to 14 GHz Aggarwal et al. (2022); Ito et al. (2020), our model
naturally does not have high-energy counterpart.

As mentioned in the Introduction, the co-planar property of the EM wave
emitted due to the GZ mechanism is due to the coherent nature of the incoming
GWs Hendry & Woan (2007). Due to the co-planar property, it can maintain the
flux Lieu et al. (2022). Hence, the three key features of non-repeating FRBs
are naturally explained in our model. Extending the calculation for repeating
will be discussed in future work.

A variety of processes generate GWs in a broad range of frequencies Carr &
Hawking (1974); Anantua et al. (2009); Kuroda et al. (2015); Ejlli et al.
(2019); Aggarwal et al. (2021); Pustovoit et al. (2021). However, it is
possible to detect these waves only in a limited range. The proposed model
provides an indirect mechanism to detect GWs at high frequencies.
Interestingly, our model also provides a way to test modified theories of
gravity. In this work, we have focused on GWs in general relativity. Certain
modified theories like Chern-Simons gravity lead to birefringence Alexander et
al. (2006) which can explain the polarized nature of FRBs. However, exact
rotating solutions in these theories are unknown and require sophisticated
numerics. Thus, the high-frequency GWs will provide a unique view of the
Universe — it’s birth and evolution.

## Acknowledgments

We thank the referee for the valuable comments that significantly improved the
manuscript. The authors thank Sushan Konar and Surjeet Rajendran for
discussions. The authors thank Koustav Chandra, S. M. Chandran, Archana Pai,
and T. R. Seshadri for their comments on the earlier draft. AK thanks L. F.
Wei for clarifications of reference. AK is supported by the MHRD fellowship at
IIT Bombay. This work is supported by ISRO Respond grant.

## Data Availability

We have not utilized any separate dataset for our calculation. The data used
for the plots are from the equations given in the paper.

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##  Appendix A Gertsenshtein-Zel′dovich effect and induced electromagnetic
waves

Consider a background space-time with GWs Misner et al. (1973):

$\displaystyle g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}\,,\qquad
g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$ (10)

where $h_{\alpha\beta}\ll 1$ is the GW fluctuation, and
$\eta_{\alpha\beta}=\rm{diag}(1,-1,-1,-1)$ is Minkowski space-time. The
background space-time is generally curved; however, the results derived for
Minkowski space-time carry through for the conversion factor computations. For
the generic curved background, the Riemann corrections contribution is tiny
Misner et al. (1973). Hence, we only report the results for the Minkowski
space-time.

For the GWs propagating in the $z-$axis, we have $h_{xx}=-h_{yy}=h_{+}$,
$h_{xy}=h_{\times}$. Eq. (1) corresponds to a monochromatic circularly
polarized GW Misner et al. (1973); Zheng et al. (2018). As mentioned earlier,
the presence of magnetic field transverse to the direction of propagation of
GWs acts as a catalyst for the conversion process Zel’dovich (1974). As
discussed, we consider the total magnetic field in the magnetosphere to be

$\displaystyle\textbf{B}(t)=\left(0,B^{(0)}_{y}+\delta
B_{y}\sin(\omega_{B}t),0\right)$ (11)

where $B^{(0)}_{y}$ is static magnetic field and $B_{y}\sin(\omega_{B}t)$ is
the alternating (time-varying) magnetic field with frequency $\omega_{B}$ and
$|\delta B_{y}|<|B^{(0)}_{y}|$. Also, we assume that the amplitude of the
alternating magnetic field is two orders lower than the static magnetic field,
i.e., $|{B_{y}}/{B^{(0)}_{y}}|\approx 10^{-2}$, so that time-varying magnetic
field has significant effects on the conversion (Pons, J. A. et al. (2012);
Pons & Viganò (2019)). The induced electric field due to the time-varying
magnetic field is [Jackson (1998)]

$\displaystyle\textbf{E}(z,t)=\left(-\frac{z\,\delta\omega_{B}B_{y}}{c}\cos(\omega_{B}t),0,0\right)\,.$
(12)

In the presence of GWs, the EM field tensor is:

$\displaystyle
F_{\alpha\beta}=F^{(0)}_{\alpha\beta}+F^{(1)}_{\alpha\beta}=\begin{pmatrix}0&\mathcal{E}_{x}&\tilde{E}_{y}&\tilde{E}_{z}\\\
-\mathcal{E}_{x}&0&-\tilde{B}_{z}&\mathcal{B}_{y}\\\
-\tilde{E}_{y}&\tilde{B}_{z}&0&-\tilde{B}_{x}\\\
-\tilde{E}_{z}&-\mathcal{B}_{y}&\tilde{B}_{x}&0\end{pmatrix},$ (13)

where $F^{(1)}_{\alpha\beta}$ is the field tensor induced due to the GWs that
needs to be determined. Similarly, the induced electric
$[\tilde{\textbf{E}}=\left(\tilde{E}_{x},\tilde{E}_{y},\tilde{E}_{z}\right)]$
and magnetic field vectors
$[\tilde{\textbf{B}}=\left(\tilde{B}_{x},\tilde{B}_{y},\tilde{B}_{z}\right)$]
due to GWs are to be determined. Note that $\mathcal{B}_{y}=B^{(0)}_{y}+\delta
B_{y}\sin(\omega_{B}t)+\tilde{B}_{y}$ and $\mathcal{E}_{x}=-(z\,\delta
B_{y}\omega_{B}/{c})\cos(\omega_{B}t)+\tilde{E}_{x}$. The covariant Maxwell’s
equations (in the source-free region) are:

$\displaystyle\partial_{\mu}\left(\sqrt{-g}F^{\mu\nu}\right)=0;~{}~{}\partial_{\mu}\left(\sqrt{-g}\tilde{F}^{\mu\nu}\right)=0$
(14)

where $\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$ is
the dual of EM field tensor,
$\epsilon^{\mu\nu\alpha\beta}=\eta^{\mu\nu\alpha\beta}/{\sqrt{-g}}$ and
$\eta^{\mu\nu\alpha\beta}$ is defined as $\eta^{0123}=1=-\eta_{0123}$ is an
antisymmetric tensor. Substituting Eq. (13) in Eq. (14), and treating
$F^{(1)}_{\alpha\beta}$ and $h_{\alpha\beta}$ as first-order perturbations, we
have

$\displaystyle{}\frac{1}{c}\partial_{t}\tilde{E}_{x}-\partial_{y}\tilde{B}_{z}+\partial_{z}\tilde{B}_{y}+\left(B^{(0)}+\delta
B_{y}\sin(\omega_{B}t)\right)\,\partial_{z}h_{+}$
$\displaystyle{}-\frac{z\,\delta
B_{y}\omega_{B}}{c^{2}}\frac{\partial}{\partial
t}\left(h_{+}\,\cos(\omega_{B}t)\right)=0$ (15a)
$\displaystyle{}\frac{1}{c}\partial_{t}\tilde{B}_{y}-\partial_{x}\tilde{E}_{z}+\partial_{z}\tilde{E}_{x}=0\,,$
(15b)

where we have expressed in terms of electric and magnetic fields. Since GWs
(and EM waves) propagate along the z-direction, we have
$\tilde{E}_{z}=\tilde{B}_{z}=0$. Substituting Eq. (1) in the above wave
equations lead to the following wave equations for
$\tilde{E}_{x},\tilde{B}_{y}$:

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\tilde{E}_{x}}{\partial
t^{2}}-\partial_{z}^{2}\tilde{E}_{x}$
$\displaystyle=-f_{E}(z^{\prime},t^{\prime})$ (16a)
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\tilde{B}_{y}}{\partial
t^{2}}-\partial_{z}^{2}\tilde{B}_{y}$
$\displaystyle=-f_{B}(z^{\prime},t^{\prime})$ (16b)

where $f_{E/B}(z^{\prime},t^{\prime})$ are the forcing functions and are given
by:

$\displaystyle f_{E}(z^{\prime},t^{\prime})=\frac{iA_{+}\delta
B_{y}k_{g}}{2c}\left[\omega_{+}e^{i(k_{g}z^{\prime}-\omega_{+}t^{\prime})}-\omega_{-}e^{i(k_{g}z^{\prime}-\omega_{-}t^{\prime})}\right]$
$\displaystyle+\frac{z^{\prime}A_{+}\delta
B_{y}\omega_{B}}{2c^{3}}\left[\omega_{+}^{2}e^{i(k_{g}z^{\prime}-\omega_{+}t^{\prime})}+\omega_{-}^{2}e^{i(k_{g}z^{\prime}-\omega_{-}t^{\prime})}\right]$
$\displaystyle+\frac{A_{+}B^{(0)}k_{g}\omega_{g}}{c}e^{i(k_{g}z^{\prime}-\omega_{g}t^{\prime})}$
(17a) $\displaystyle f_{B}(z^{\prime},t^{\prime})=\frac{iA_{+}\delta
B_{y}k_{g}^{2}}{2}\left[e^{i\left(k_{g}z^{\prime}-\omega_{+}t^{\prime}\right)}-e^{i\left(k_{g}z^{\prime}-\omega_{-}t^{\prime}\right)}\right]$
$\displaystyle+\frac{iA_{+}\delta
B_{y}\omega_{B}}{2c^{2}}\left[\omega_{+}e^{i\left(k_{g}z^{\prime}-\omega_{+}t^{\prime}\right)}-\omega_{-}e^{i\left(k_{g}z^{\prime}-\omega_{-}t^{\prime}\right)}\right]$
$\displaystyle-\frac{z^{\prime}A_{+}\delta
B_{y}\omega_{B}k_{g}}{2c^{2}}\left[\omega_{+}e^{i(k_{g}z^{\prime}-\omega_{+}t^{\prime})}-\omega_{-}e^{i(k_{g}z^{\prime}-\omega_{-}t^{\prime})}\right]$
$\displaystyle+A_{+}B^{(0)}k_{g}^{2}e^{i\left(k_{g}z^{\prime}-\omega_{g}t^{\prime}\right)}$
(17b)

The solutions to the wave equations (16a) and (16b) are given by:

$\displaystyle F_{E/B}(z,t)=\int\int
dz^{\prime}dt^{\prime}G(t,t^{\prime};z,z^{\prime})f_{E/B}(t^{\prime},z^{\prime})$
(18)

where $F_{E/B}(z,t)$ denotes the corresponding solution of the forcing
function $f_{E/B}(z^{\prime},t^{\prime})$ and $G(t,t^{\prime};z,z^{\prime})$
is the retarded Green’s function corresponding to the wave equation and is
given by:

$\displaystyle
G(t,t^{\prime};z,z^{\prime})=\frac{c}{2}\Theta\left(\,\,c(t-t^{\prime})-|z-z^{\prime}|\,\,\right)$
(19)

where $\Theta-$function is non-zero only for $t\geq
t^{\prime}+\frac{|z-z^{\prime}|}{c}$. Substituting the forcing functions for
the electric (17) and the magnetic (17) fields, in the integral equation (18),
leads to:

$\displaystyle\tilde{E}_{x}$
$\displaystyle\simeq-\frac{B^{(0)}_{y}A_{+}}{4}\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}$
(20) $\displaystyle+\frac{\delta
B_{y}A_{+}\omega_{B}t}{2}\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}+\frac{\delta
B_{y}A_{+}}{4i}\,\left(\frac{\omega_{B}}{\omega_{g}}\right)\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}$
$\displaystyle\tilde{B}_{y}$
$\displaystyle\simeq-\frac{B^{(0)}_{y}A_{+}}{4}\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}$
(21) $\displaystyle-\frac{\delta
B_{y}A_{+}\omega_{g}t}{2}\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}+\frac{\delta
B_{y}A_{+}}{4i}\,\left(\frac{\omega_{B}}{\omega_{g}}\right)^{2}\,\,e^{i\left(k_{g}z-\omega_{g}t\right)}\,\,.$

We want to mention the following points regarding the above expressions:
First, in obtaining the above expressions, we have assumed that
$\omega_{B}\ll{\omega_{g}}$. This is valid in our case because the conversion
from GWs to EM waves occurs in the MHz-GHz frequency range. This approximation
leads to $\omega^{n}_{\pm}\simeq\omega^{n}_{g}\left(1\pm
n\frac{\omega_{B}}{\omega_{g}}\right)$ and allows us to approximate
$\omega_{\pm}\approx\omega_{g}$ in the exponentials. Note that we have also
ignored the terms with $e^{i\omega_{g}t}$ since they will lead to wave
propagating along negative z-direction i.e.,
$e^{i\left(k_{g}z+\omega_{g}t\right)}$, and hence are not relevant to our
analysis. Second, the last term in both expressions is tiny and can be
neglected leading to Eqs. (3, 4). Lastly, GZ effect is a pure gravitational
effect (due to incoming GWs) and the generation of EM waves does not require
any source term (plasma or charged particles). Hence, if the incoming GWs are
coherent, the emitted EM waves are coherent at resonance.

##  Appendix B Evaluating the radius of the magnetosphere

We show that the assumption that the background magnetic field can be treated
as a constant in the entire magnetosphere gives identical results to that of
the background field decreasing radially.

In evaluating $\alpha_{\rm tot},$ we have assumed that the background magnetic
field is a constant until light cylinder radius $R_{\rm LC}$. In this section,
we show that this assumption is physically consistent.

To do that, we consider the magnetic field of the pulsar magnetosphere in
vacuum to be dipolar. We evaluate the average dipolar magnetic field in the
magnetosphere. In spherical polar coordinates, the dipolar magnetic field is
given by Cerutti & Beloborodov (2017); Pétri (2016):

$\displaystyle\left(B_{r},B_{\theta},B_{\phi}\right)=B_{*}\,\left(\frac{r_{*}}{r}\right)^{3}\,\left(2\cos\theta,\sin\theta,0\right)$
(22)

where $B_{*}$ is the magnetic field on the NS surface.

We now compute the average magnetic field on the volume between the radius of
the NS ($r_{*}$) and $R_{\rm{LC}}$:

$\\!\\!\bar{\textbf{B}}(r,\theta,\phi)=\frac{1}{V}\int_{r_{*}}^{R_{\rm{LC}}}\\!\\!\\!\\!\\!\\!dr\int_{0}^{\pi}\\!\\!\\!\\!d\theta\int_{0}^{2\pi}\\!\\!\\!\\!d\phi\,r^{2}\sin\theta\,\textbf{B}(r,\theta,\phi)$
(23)

where $V$ is the volume enclosed by both the surfaces:

$\displaystyle V=\frac{4\pi}{3}\left(R_{\rm{LC}}^{3}-r_{*}^{3}\right).$ (24)

From Eqs. (23, 24), we define the integral $\mathcal{I}$ as:

$\displaystyle\\!\\!\\!\\!\mathcal{I}\equiv
V\,\bar{\textbf{B}}(r,\theta,\phi)=\int_{r_{*}}^{R_{\rm{LC}}}\\!\\!\\!\\!dr\int_{0}^{\pi}\\!\\!\\!\\!d\theta\int_{0}^{2\pi}\\!\\!\\!\\!d\phi\,r^{2}\sin\theta\textbf{B}(r,\theta,\phi).$
(25)

Note that the volume average of the radial component of the magnetic field
vanishes. The non-zero contribution comes from the angular variation of the
magnetic field, i.e.,

$\displaystyle\bar{\textbf{B}}_{\theta}$ $\displaystyle=\frac{3\pi
B_{*}r_{*}^{3}}{4\left(R_{\rm{LC}}^{3}-r_{*}^{3}\right)}\ln\left|\frac{R_{\rm{LC}}}{r_{*}}\right|.$
(26)

From Eqs. (25, 26), we obtain

$\displaystyle\mathcal{I}$
$\displaystyle=\pi^{2}B_{*}r_{*}^{3}\ln\left|\frac{R_{\rm{LC}}}{r_{*}}\right|$
(27)

We now compare this with the assumption that the magnetic field is
approximately constant until $R_{\rm LC}$. To do this, we substitute the
average magnetic field obtained in Eq. (26) inside the integral in Eq.(25).
Doing the radial integral between $r_{*}$ to $\mathcal{R}$ leads to:

$\displaystyle\mathcal{I}$
$\displaystyle=\pi^{2}B_{*}r_{*}^{3}\frac{(\mathcal{R}^{3}-r_{*}^{3})}{\left(R_{\rm{LC}}^{3}-r_{*}^{3}\right)}\,\ln\left|\frac{R_{\rm{LC}}}{r_{*}}\right|.$
(28)

Setting $\mathcal{R}=R_{\rm{LC}}$ in the above expression, we see that the two
expressions (27, 28) are approximately the same. Thus, even if there is a
radial and angular variation in the magnetic field in the magnetosphere, we
can approximate the background field to be approximately constant in the
region $r_{*}\leq r\leq R_{\rm{LC}}$. Thus, our expression for the total
conversion factor mimics the realistic NS regions.

##  Appendix C Conversion factor from entire magnetosphere

In this appendix, we estimate the total conversion factor from the entire
magnetosphere of the compact object. The energy density carried by these
induced EM waves is Jackson (1998)

$\displaystyle\rho_{\rm{EM}}\simeq\frac{|A_{+}|^{2}|B^{(0)}_{y}|^{2}}{64\pi}\left[1+2\,\xi^{2}\omega_{g}^{2}\,t^{2}+2\,\xi\omega_{g}\,t\,\,\right]\,.$
(29)

where $\xi=\delta B_{y}/B_{y}^{(0)}$. As mentioned earlier, the conversion is
maximum when EM waves are approximately the same as the incoming waves. Having
obtained the energy carried by the induced EM waves, we need to obtain what
fraction of wave energy is converted to EM waves Zel’dovich (1974)? To do
this, we calculate the energy density carried by the GWs Misner et al. (1973),
i. e.:

$\displaystyle\rho_{\rm{GW}}=\frac{c^{2}\omega_{g}^{2}}{32\pi
G}\left(|A_{+}|^{2}+|A_{\times}|^{2}\right)=\frac{c^{2}\omega_{g}^{2}}{16\pi
G}\,\,|A_{+}|^{2}\,,$ (30)

wherein second equality, we have used the fact that both the modes of GWs are
generated with an equal amount of energy, i.e., $|A_{+}|=|A_{\times}|$ also
referred to as isospectrality relation Chandrasekhar (1998). From Eqs. (29,
30), we obtain Eq. (5).

To obtain the conversion factor in the entire magnetosphere, we define the
following dimensionless parameter:

$\displaystyle X=\frac{r}{R_{\rm{LC}}}$ (31)

where $r$ is the distance from the surface of neutron star to a point in the
magnetosphere, i.e., $r_{*}\lesssim\,r\lesssim\,R_{\rm{LC}}$, and
$r_{*}(=10^{6}\rm{cm})$ is the radius of the Neutron star (NS). In the case of
Magnetar $R_{\rm LC}=10^{9}~{}{\rm cm}$ and for NS $R_{\rm LC}=10^{7}~{}{\rm
cm}$, hence, $X_{*}=\frac{r_{*}}{R_{\rm{LC}}}<1$. In other words, the range of
$X$ is $0.001\lesssim\,X\lesssim\,1$.

Since we are interested in the EM waves reaching the observer, we are
interested in evaluating the conversion factor along the observer’s line of
sight. Thus, we have

$\displaystyle z=r=XR_{\rm{LC}}.$ (32)

where $\theta=0$ corresponds to the direction along the line-of-sight of the
observer. Thus, the total conversion factor is given by the integral

$\displaystyle\alpha_{\rm{tot}}=\Omega\int_{X_{*}}^{1}\alpha\,\,dX,$ (33)

where $\Omega$ is the total solid angle which is an overall constant factor
because Eq.(33) is independent of the angular coordinates.

##  Appendix D Poynting vector

In astrophysical observations of compact objects, a quantity of interest is
the energy flux density which is the Poynting vector. This section evaluates
the Poynting vector for magnetar and neutron star in SI unit and Jansky Hz, a
widely used unit for spectral flux density in radio observations.

The Poynting vector of the induced electric field (3) and induced magnetic
field (4) is Jackson (1998):

$\displaystyle S_{z}$
$\displaystyle=\frac{c}{8\pi}\tilde{E}_{x}\times\tilde{B}^{*}_{y}$ (34)
$\displaystyle\simeq\frac{A_{+}^{2}\,|B_{y}^{(0)}|^{2}\,c}{128\pi}\left[1+2\omega_{g}\xi\frac{R_{\rm{LC}}}{c}-2\omega_{g}\omega_{B}\xi^{2}\left(\frac{R_{\rm{LC}}}{c}\right)^{2}\right]$

where $\tilde{B}^{*}_{y}$ is the complex conjugate of the induced magnetic
field $\tilde{B}_{y}$. As mentioned above, the frequency of the alternating
magnetic field is $1\rm{Hz}$ for the magnetar and $10^{3}\,\rm{Hz}$ for the
millisecond pulsar. Here again, we have assumed that
$\omega_{B}\ll\omega_{g}$. Rewriting the above Poynting vector in terms of
$\alpha_{\rm{tot}}$, we get Eq. (7).

##  Appendix E Current status of high-frequency GWs

Primordial black-holes, Exotic compact objects and early Universe can generate
high-frequency GW (HFGW) in MHz to GHz Akutsu et al. (2008); Nishizawa et al.
(2008); Chou et al. (2016, 2017); Ito et al. (2020); Domènech (2021);
Goryachev et al. (2021); Aggarwal et al. (2022); Domcke et al. (2022). These
can generate GWs up to 14 GHz Ito et al. (2020). Over the last decade, many
HFGW detectors are proposed, and some of them are operational. For instance,
the Japanese 100 MHz detector with a $0.75~{}{\rm m}$ armlength interferometer
has been operational for a decade Akutsu et al. (2008); Nishizawa et al.
(2008), Holometer detector has put some limit on GWs at MHz Chou et al. (2016,
2017). A GHz GW detector is also proposed Ito et al. (2020). These detectors
are ideally suited for searching for physics beyond the standard model (SM),
like primordial black-holes, exotic compact objects and the early Universe.
For instance, exotic compact objects with characteristic strain $h$ Aggarwal
et al. (2021):

$h\lesssim 10^{-19}C^{5/2}\left(\frac{\rm MHz}{f}\right)\left(\frac{\rm
Mpc}{D}\right)$

lead to the following coherent GWs with amplitudes:

$h_{\rm 1.4GHz,10kpc}\lesssim 10^{-21}\,,h_{\rm 1.4GHz,1Mpc}\lesssim 10^{-23}\
.$

After 153 days of operation, the Bulk Acoustic GW detector experiment recently
reported two MHz events Goryachev et al. (2021). According to Goryachev et al.
Goryachev et al. (2021), the data corresponding to two MHz events fits best a
single energy depositing event. The authors mention that the strongest
observed signal results in a required characteristic strain amplitude of
$h_{c}\approx 2.5\times 10^{-16}$, corresponding to a PBH merger of
$m_{\mathrm{PBH}}<4\times 10^{-4}M_{\odot}$ (which gives a maximum frequency
at inspiral of $5.5\ \mathrm{MHz}$), at a distance of $D\approx
0.01\mathrm{pc}$. It is important to note that these are not conclusive
enough. However, these detections also imply that these events are not rare.

As mentioned above, many HFGW detectors are proposed, and better estimates
will be available as more and more detectors will be operational in the coming
decades. This can provide better estimate of these events in the next decade.