# Probing black hole charge with gravitational microlensing of gravitational
waves

Uddeepta Deka<EMAIL_ADDRESS>International Centre for Theoretical
Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
Sumanta Chakraborty<EMAIL_ADDRESS>Indian Association for the Cultivation
of Science, 2A & 2B Raja S C Mullick Road, Kolkata 700032, India  Shasvath J.
Kapadia<EMAIL_ADDRESS>The Inter-University Centre for Astronomy
and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India  International
Centre for Theoretical Sciences, Tata Institute of Fundamental Research,
Bangalore 560089, India  Md Arif Shaikh<EMAIL_ADDRESS>Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bangalore 560089, India  Parameswaran Ajith<EMAIL_ADDRESS>International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bangalore 560089, India

(August 28, 2024)

###### Abstract

Gravitational microlensing of gravitational waves (GWs) opens up the exciting
possibility of studying the spacetime geometry around the lens. In this work,
we investigate the prospects of constraining the ‘charged’ hair of a black
hole (BH) from the observation of a GW signal microlensed by the BH. The
charge can have electromagnetic or modified gravity origin. We compute the
analytic form of the lensing potential with charge and construct the lensed
waveforms for a range of BH mass, charge and impact parameters, assuming non-
spinning BHs. Using an approximate likelihood function, we explore how future
observations of microlensed GWs can constrain the charge of the BH lens. We
find that positive values of the charge parameter (that can be of
electromagnetic or modified gravity origin) can be tightly constrained using
lensed GW signals, while the constraints on negative values of the charge
parameter (modified gravity origin) are modest.

##  I Introduction

The LIGO and Virgo gravitational-wave (GW) detectors have observed $\sim 90$
compact binary coalescence (CBC) events during its first three observing runs
(O1, O2, O3) Abbott  _et al._ (2021a). Most of them are binary black hole
(BBH) mergers. The remaining are mergers of binary neutron stars (BNSs) Abbott
_et al._ (2017a, 2020a) and neutron star-black hole (NSBH) binaries Abbott
_et al._ (2021b).

These detections have enabled some of the most unique and stringent tests of
general relativity (GR) in the strong-field regime Abbott  _et al._ (2021c).
These include a model-agnostic residual test that studies the statistical
properties of the strain data after subtracting out the expected GW signal
buried therein to ascertain if the residual is consistent with noise
Scientific  _et al._ (2016); Abbott  _et al._ (2019a, 2020b); an inspiral-
merger-ringdown consistency test that checks if the GW waveform is consistent
with GR’s prediction of the same by comparing the binary’s intrinsic
parameters inferred from the low- and high-frequency portions of the waveform
Ghosh  _et al._ (2016, 2018); a test that compares the speed of GWs with
respect to the speed of light Abbott  _et al._ (2019b), as well as one that
probes signatures of velocity dispersion as a consequence of a finite graviton
mass Will (1998); and a test of GW polarizations that searches for
polarizations of GWs beyond the two predicted by GR Abbott  _et al._ (2017b,
2021d); Takeda  _et al._ (2021); Wong  _et al._ (2021).

The anticipated observations of gravitationally lensed GWs in future observing
runs 111No confirmed detection of GW lensing have thus far been reported
Hannuksela  _et al._ (2019); Dai  _et al._ (2020); McIsaac  _et al._ (2020);
Abbott  _et al._ (2021e, 2023); Janquart  _et al._ (2023); Goyal  _et al._
(2023a). Some arguments claiming the observation of lensed GWs – based on the
larger BH masses uncovered by GW observations relative to those inferred from
galactic X-ray binaries – have been made Broadhurst  _et al._ (2018, 2022).
promise to enable additional novel tests of GR (see, e.g., Goyal  _et al._
(2021); Fan  _et al._ (2017); Hernandez (2022); Finke  _et al._ (2021); Narola
_et al._ (2023); Chung and Li (2021); Goyal  _et al._ (2023b); Mukherjee  _et
al._ (2020); Collett and Bacon (2017)), while also shedding light on a number
of questions in astrophysics (see, e.g., Basak  _et al._ (2023); Singh  _et
al._ (2023); Magare  _et al._ (2023)) and cosmology (see, e.g., Basak  _et
al._ (2022); Jana  _et al._ (2022); Hannuksela  _et al._ (2020)). As with
electromagnetic (EM) waves, lensing of GWs results when large agglomerations
of matter lead to deviations in the trajectories of these waves Ohanian
(1974); Deguchi and Watson (1986); Wang  _et al._ (1996); Nakamura (1998);
Bartelmann (2010).

Unlike the case involving EM waves, GW lensing is typically studied in two
different regimes. Lensing by galaxies or clusters can be analyzed using the
geometric optics (ray-optics) approximation since the wavelengths of the GWs
(detectable by LIGO-Virgo) are much smaller than the gravitational radii of
such lenses Ng  _et al._ (2018); Dai  _et al._ (2017); Smith  _et al._ (2018).
Strong lensing in this regime results in the production of multiple copies of
the GWs Kormann  _et al._ (1994); Koopmans  _et al._ (2009) separated by time
delays spanning minutes to months Haris  _et al._ (2018); More and More
(2022); Magare  _et al._ (2023). On the other hand, lensing of GWs by massive
compact objects with gravitational radii comparable to the GW wavelength will
incur wave optics effects Takahashi and Nakamura (2003); Jung and Shin (2019);
Diego  _et al._ (2019); Basak  _et al._ (2022); Urrutia and Vaskonen (2022).
The result is the production of a single image with a modulated GW shape
Nakamura (1998). This shape carries with it imprints of the properties of the
lens, and can possibly be used to probe the nature of the compact lens.

Wave-optics effects modulating the GW waveform have been studied extensively
for simple lensing potential models, in particular the one corresponding to
the point mass lens 222Indeed, most searches for microlensing signatures
incurred by wave-optics effects in detected GW events assume a point-mass lens
model Basak  _et al._ (2022); Hannuksela  _et al._ (2019); Abbott  _et al._
(2023). Nakamura (1998); Nakamura and Deguchi (1999); Takahashi and Nakamura
(2003). In this model, the frequency-dependent amplitude modulations of the GW
waveform are exclusively determined by the (redshifted) mass of the lens, and
the impact parameter (i.e, the location of the source in the lens plane)
Takahashi and Nakamura (2003). However, if the compact object has additional
hairs, they are also going to show up in the lensed GW waveform, and can
possibly be detected as and when microlensed GW signals are observed. The
additional hairs could point to the specific nature of the compact object or
point to a deviation from GR.

Besides the mass, the next obvious hair that a static and spherically
symmetric compact object can have is the electric charge $q$, which modifies
the radial and temporal components of the spacetime metric by
$\mathcal{O}(Q/r^{2})$, where $Q\equiv q^{2}$. Indeed, it is widely believed
that astrophysical objects should not have any net electric charge, or, even
if there is some electric charge, it should immediately get shielded Feng  _et
al._ (2023); Pina  _et al._ (2022); Bozzola and Paschalidis (2021) or
neutralised Eardley and Press (1975). Intriguingly, such a ‘charge-like’ hair
can arise from several other contexts as well and can have non-negligible
values. For example, in the context of an extra spatial dimension, where we
live in a four dimensional universe embedded in a five dimensional spacetime,
known as the braneworld scenario, the solution of the effective gravitational
field equations on the four-dimensional brane has exactly the same structure,
but with $Q$ being _negative_ Shiromizu  _et al._ (2000); Dadhich  _et al._
(2000); Harko and Mak (2004); Aliev and Gumrukcuoglu (2005); Maeda and Dadhich
(2007). Similarly, a positive value of $Q$ can arise in scalar coupled Maxwell
theories, described by Horndeski theories Babichev  _et al._ (2015);
Barrientos  _et al._ (2017); Babichev and Charmousis (2014). There have
already been extensive searches for both positive as well as negative values
of the ‘charge’ in various contexts — (a) in the weak field regime, e.g.,
through solar system tests Iorio and Saridakis (2012); Capozziello  _et al._
(2013); Chakraborty and SenGupta (2014); Bhattacharya and Chakraborty (2017);
Mukherjee and Chakraborty (2018), (b) using EM waves from accretion on
supermassive BHs Maselli  _et al._ (2015); Stuchlík and Kotrlová (2009);
Banerjee  _et al._ (2021, 2017), (c) using GWs from BBH and BNS coalescence
Barausse and Yagi (2015); Toshmatov  _et al._ (2016); Andriot and Lucena Gómez
(2017); Chakraborty  _et al._ (2018); Chakravarti  _et al._ (2020); Mishra
_et al._ (2022, 2023a); Gupta  _et al._ (2021); Carullo  _et al._ (2022); Gu
_et al._ (2023), and (d) with strong lensing of EM waves and measurement of BH
shadow Chakraborty and SenGupta (2017); Banerjee  _et al._ (2022); Vagnozzi
_et al._ (2023); Banerjee  _et al._ (2020).

Here we explore the possibility of constraining the charge hair of a BH from
the observation of GWs microlensed by it. We study the lensing-induced
modulations on the GW signal by introducing the charge in the lensing
potential. In particular, we model the spacetime surrounding the lens to be
described by a metric that is analogous to the Reisner-Nordström spacetime,
but with $\pm(Q/r^{2})$ term in the temporal and radial metric elements. If a
lensed GW signal suggest a positive value of $Q$, it is to be interpreted
either in the context of EM-charged BH, or, as a BH in modified theories of
gravity. While, if the lensed GW signal prefers a negative value for $Q$, then
it will provide a hint for the existence of extra dimensions. In what follows
we will evaluate the lensing potential associated with the new metric, and
numerically calculate the corresponding frequency-dependent magnification
function that modulates the GW, for a range of the lens masses, impact
parameters (i.e, the angular location of the source in the image plane), and
the charge $Q$.

To assess our ability to constrain the charge $Q$, we quantify the extent to
which the GW signal lensed by a BH of charge $Q$ deviates from another
waveform lensed by a BH of charge $Q_{\mathrm{true}}$. To that end, we compute
matches, defined as the inner product between two waveforms weighted by the
noise power spectral density (PSD) of the GW detector. We then use these to
construct an approximate likelihood on the lens parameters, which we sample to
assess how well we can recover the true value of the charge. We find that
positive charges can be well constrained, where a lens with a charge of
$Q=0.1$ can be identified as a charged lens at $>90\%$ confidence. On the
other hand, negatively charged lenses are weakly constrainable – only large
$|Q|$’s can be constrained well enough to exclude $Q=0$ at $>90\%$ confidence.

The paper is organized as follows: In Section II, we present the analytical
expression for the lensing potential in the presence of the charge $Q$. Using
the expression for the lensing potential, in Section III we numerically
compute the magnification function, as well as the GW waveform lensed by a
charged BH. In Section IV we explore the possibility of constraining the
charge parameter of a BH from the observation of GWs microlensed by it. We
conclude in Section V. Some of the detailed calculations are presented in
Appendix A.

_Notations and Conventions:_ We will set the fundamental constants $G$ and $c$
to unity and will use the mostly positive signature convention, such that the
Minkowski metric in Cartesian coordinate takes the form
$\eta_{\mu\nu}=\textrm{diag}(-1,+1,+1,+1)$. Greek indices $\mu,\nu,\cdots$
denote four-dimensional spacetime coordinates, while uppercase Roman indices
$A,B,\cdots$ denote five-dimensional spacetime indices.

We will refer to $Q$ as the ‘charge’, or ‘charge hair’ of a BH. Note that,
even a negative electric charge $q$ will correspond to a positive value of
$Q$, since $Q\equiv q^{2}$. Also, we will refer to lensing in the wave optics
regime ($GM_{\mathrm{L}}/c^{2}\sim\lambda_{\mathrm{GW}}$) as microlensing.

##  II Lensing magnification by a charged black hole

In this section, we will determine the magnification function due to the
simplest possible hair of a BH, namely a charge that can either have an EM
origin or, may arise from theories of gravity beyond GR. Of course, any EM
charge is expected to be dissipated away with time, or, would be heavily
shielded by surrounding matter. But it would still be interesting Feng  _et
al._ (2023); Pina  _et al._ (2022); Bozzola and Paschalidis (2021) if any
residual electric charge present in an astrophysical BH, whether it could be
tested using gravitational lensing. More interesting is the case when this
charge arises from an alternative theory of gravity. We will provide some
examples of such modified theories below.

First of all, the presence of an extra spatial dimension can induce such a
charge in the four-dimensional spacetime metric outside of a BH. This is
because the effective gravitational field equations on the four-dimensional
vacuum spacetime (known as the brane) take the following form Shiromizu  _et
al._ (2000); Dadhich  _et al._ (2000)

$\displaystyle G_{\mu\nu}+E_{\mu\nu}=0\leavevmode\nobreak\ .$ (1)

Here $G_{\mu\nu}$ is the four-dimensional Einstein tensor and
$E_{\mu\nu}=W_{ABCD}e^{A}_{\mu}n^{B}e^{C}_{\nu}n^{D}$ is the projected five
dimensional Weyl tensor on the brane, with $e^{A}_{\mu}$ being the projector
and $n_{A}$ being the normal to the brane. Owing to the symmetries of the Weyl
tensor, it follows that $E^{\mu}_{\mu}=0$, which is akin to the traceless
property of the energy-momentum tensor of the EM field. Thus the above field
equation for the metric admits the following static and spherically symmetric
solution, $-g_{tt}=g^{rr}=1-(2M/r)+(Q/r^{2})$, with $Q<0$ and thus differing
from the Reissner-Nordström solution by the sign of the $(1/r^{2})$ term. In
the $f(T)$ gravity as well the same solution for the metric components was
obtained Capozziello  _et al._ (2013), however, with a positive contribution
from the charge term $Q$. The Einstein-Gauss-Bonnet theory in higher dimension
also admits the same solution on the brane with $Q<0$, albeit the origin of
the charge term was from the coupling constant in the Gauss-Bonnet invariant
Maeda and Dadhich (2007). Finally, the same solution also arises in the
context of a sub-class of Horndeski theories of gravity, which involves the
following terms: $\beta G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, as
well as $-(\gamma/2)T^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ in the
gravitational action, besides the Ricci scalar and the Maxwell term. In this
case, the charge $Q$ depends on the ratio $(\gamma/\beta)$ and is a positive
quantity Babichev  _et al._ (2015). Thus, we observe that various cases yield
the same metric, with different origin and sign for the charge term. In
summary, there are several possibilities for a positive charge to appear in
the metric — (a) EM charge, (b) charge arising from the $f(T)$ theory of
gravity, (c) charge depending on the non-trivial coupling between gravity and
electromagnetism with scalar, as in Hornsdeski theories. While the negative
charge predominantly arises in the presence of higher dimensions, either from
the bulk Weyl tensor or, from the Gauss-Bonnet coupling. This makes the
charged metric an ideal ground to probe, since it arises in so many different
contexts, with many varieties and hence it is worthwhile to see the effect of
such a charge on the microlensing of GWs. In particular, when the wavelength
of GWs are comparable to the radius of the event horizon of these charged BHs
wave effects must be taken into account. In what follows, we compute the
effect of this charge term on the amplification of GWs due to microlensing.

Figure 1: The time delay contours (in units of $M_{\rm L}$, where $t_{\rm
d}=0$ at the global minima) as a function of the lens plane coordinates
$(x_{1},x_{2})$. The source position is shown by a star ($y=0.1$). The left,
center, and right panels correspond to the cases with $Q/M_{\rm
L}=-0.5,\,0\text{ and }0.5$, respectively. For each of these cases, the
locations of the images and their types are also indicated.

We will work within the thin lens approximation, i.e., the lens is not
considered a three-dimensional object, but rather a two-dimensional one. This
is because the size of the lens is much smaller compared to the distance of
the lens to the source as well as to the observer. Thus the magnification
function of the GW, as measured by an observer reads Takahashi and Nakamura
(2003),

$\displaystyle\mathcal{F}(f)=\frac{D_{\rm S}\xi_{0}^{2}}{D_{\rm L}D_{\rm
LS}}\frac{f}{i}\int d^{2}\vec{x}\exp\left[2\pi ift_{\rm
d}(\vec{x},\vec{y})\right]\leavevmode\nobreak\ ,$ (2)

where $\vec{x}\equiv\vec{\xi}/\xi_{0}$ is the normalized vector on the lens
plane and $\vec{y}=(D_{\rm L}/\xi_{0}D_{\rm S})\vec{\eta}$ is the normalized
vector on the source plane indicating the location of the source (impact
parameter). Further, $D_{\rm L}$ is the angular diameter distance to the lens,
$D_{\rm S}$ is the angular diameter distance to the source and $D_{\rm LS}$ is
the angular diameter distance between the source and the lens. Further, the
quantity $t_{\rm d}$ is the time taken by the GW to reach the observer from
the source when a lens is present, while $f$ is the frequency of the GW
emitted from the source. Since the source and the observer, in general, are
separated by a cosmological distance, the effect of the cosmological expansion
must be included in the above analysis. This modifies the above magnification
function to,

$\displaystyle\mathcal{F}(f)$ $\displaystyle=\frac{D_{\rm
S}\xi_{0}^{2}\left(1+z_{\rm L}\right)}{D_{\rm L}D_{\rm LS}}\frac{f}{i}\int
d^{2}\vec{x}\exp\left[2\pi ift_{\rm
d}(\vec{x},\vec{y})\right]\leavevmode\nobreak\ ,$ (3)

where, $z_{\rm L}$ is the redshift of the lens. Note that the magnification
function $\mathcal{F}(f)$ is so chosen that $|\mathcal{F}(f)|=1$ in the
absence of the lens. The time taken by the GW to arrive at the observer from
the lens is given by,

$\displaystyle t_{\rm d}(\vec{x},\vec{y})=\frac{D_{\rm
S}\xi_{0}^{2}\left(1+z_{\rm L}\right)}{D_{\rm L}D_{\rm
LS}}\left[\frac{|\vec{x}-\vec{y}|^{2}}{2}-\psi(\vec{x})\right]\leavevmode\nobreak\
.$ (4)

Here $\psi(\vec{x})$ is the two-dimensional deflection potential, depending on
the background geometry of the lens, on which the GW is propagating and it
satisfies the following differential equation,

$\displaystyle\nabla_{x}^{2}\psi=\frac{2\Sigma}{\Sigma_{\rm
cr}}\leavevmode\nobreak\ ,$ (5)

where, $\Sigma$ is the surface energy density of the lens on the lens plane
and $\Sigma_{\rm cr}$ is a critical energy density, to be fixed later. For a
point mass lens, one can simply consider, $\Sigma=M_{\rm
L}\delta^{2}(\vec{\xi})$, i.e., we can take the point mass to be at the origin
of the lens plane. In this case, choosing $\Sigma_{\rm cr}=(M_{\rm
L}/\pi\xi_{0}^{2})$, one obtains, $\psi(\vec{x})=\ln|\vec{x}|$. In the present
context, we are interested in a charged lens, which is either sourced by an EM
field or, arises in gravity theories beyond GR. The effects of these modified
theories of gravity are embedded through additional contributions to the
Einstein’s equations and effectively behave as a ‘charged’ field. In each of
these cases, the energy density arising from the extra contribution is given
by, $\rho_{\rm Q}=(Q/r^{4})$, or, the associated field behaves as
$\mathcal{O}(r^{-2})$ in three space dimensions T.Padmanabhan (2010). Thus on
the lens plane, which is a two-dimensional hypersurface, the falloff of the
field will simply become $\mathcal{O}(r^{-1})$ and hence the energy density
becomes $(Q/|\vec{\xi}|^{2})$ Hendi (2011); Hendi  _et al._ (2020); Tang  _et
al._ (2017); Singha  _et al._ (2022). Thus the total energy density on the
two-dimensional surface becomes,

$\displaystyle\Sigma=M_{\rm
L}\delta^{2}(\vec{\xi})+\frac{Q}{|\vec{\xi}|^{2}}\leavevmode\nobreak\ .$ (6)

Given the above surface energy density, which involves both the contribution
from the point-mass lens and from the charge $Q$, the two-dimensional
gravitational potential can be obtained by solving Eq. (5), and that yields
(see Appendix A),

$\displaystyle\psi(\vec{x})=\ln|\vec{x}|\left(1+\frac{\pi Q}{M_{\rm
L}}\ln|\vec{x}|\right)\leavevmode\nobreak\ .$ (7)

Here, we have chosen the critical surface energy density $\Sigma_{\rm cr}$,
such that in the limit $Q\to 0$, we obtain an identical expression to that of
the point mass lens. This is one of the main results of this paper.

Given this gravitational potential, in the geometric optics regime, the
location of the images on the lens plane can be determined by invoking
Fermat’s principle of gravitational lensing Blandford and Narayan (1986).
Fermat’s principle states that the images are formed at the extrema of the
time delay surface, that is, at points on the lens plane satisfying the
condition:

$\displaystyle\frac{\partial t_{\rm d}(\vec{x},\vec{y})}{\partial\vec{x}}=0.$
(8)

Thus, the image locations can be obtained by solving the following algebraic
equation

$\displaystyle\frac{D_{\rm S}\xi_{0}^{2}}{D_{\rm L}D_{\rm LS}}\left(1+z_{\rm
L}\right)\left[|\vec{x}-\vec{y}|-\frac{1}{|\vec{x}|}\left(1+\frac{2\pi
Q}{M_{\rm
L}}\ln|\vec{x}|\right)\right]\frac{\vec{x}}{|\vec{x}|}=0\leavevmode\nobreak\
.$ (9)

Note that GW microlensing happens in the wave optics regime, where the
geometric approximation is not valid. However, an approximate understanding of
the image locations are useful in interpreting the lensing magnification that
we compute in the wave optics regime.

Figure 2: Number of images shown as a function of the impact parameter
$|\vec{y}|$, and the charge $Q/M_{\rm L}$. There exist two images for all
$Q/M_{\rm L}\leq 0$, irrespective of the impact parameter. However, for
$Q/M_{\rm L}>0$, there exists a region in the parameter space in which four
images are formed. See text for discussion.

It turns out that for $Q\leq 0$, the above algebraic equation has two
solutions for $\vec{x}$, the normalized vector on the lens plane. These will
be the image locations in the geometric optics limit. One of these solutions
corresponds to the saddle point, while the other is the minima, as depicted in
the left and centre panels in Fig. 1. Besides the locations of these two
points, Fig. 1 shows the contours of constant $t_{\rm d}(\vec{x},\vec{y})$ for
a fixed source location $\vec{y}$. Interestingly, for certain choices of
$Q>0$, and the impact parameter $|\vec{y}|$, there can even be four solutions
to Eq. (9) for $\vec{x}$, leading to the formation of four images. This can be
seen in Fig. 2 and the right panel in Fig. 1.

The formation of multiple images can be understood using caustics associated
with the lensing potential. For $Q\leq 0$, there are no caustics (except for
the lens location) and hence there are always two images. For any $Q>0$, a
caustic appears at a finite value of $|\vec{y}|$, which decreases as the
charge increases. Therefore, for $Q\lesssim 0.1$, the source will appear
within the caustic, leading to four images. As the charge increases, the
caustic appears at smaller values of $|\vec{y}|$, and for $Q\sim 0.3$, the
caustic reaches $|\vec{y}|\sim 0$. Thus the source goes out of the caustic,
leading to two images. Subsequent increase of $Q$, results in the caustic
forming at a larger value of $|\vec{y}|$, and hence the source again goes
within the caustic, leading to four images. This is precisely what Fig. 2
demonstrates.

Note that the above situation cannot be compared with the scenario involving
the BH shadow, which involves the lensing of EM waves in the strong gravity
regime. There, the source (emission from the accreting gas) is close to the
lens and the light rays experience the full three-dimensional potential of the
lens. In contrast, typical gravitational lensing (as considered here)
considers the weak field regime. Also, the distance between the source and the
lens is so large that the lens can be approximated by a projected two-
dimensional potential (thin lens approximation). Hence the details of the
images are quantitatively different in the two scenarios.

Another interesting feature of Fig. 1 corresponds to the fact that images and
the source appear along a line in the plane spanned by $\vec{x}$, which arises
from the axially symmetric nature of the lens. This is also reflected in the
fact that the lensing potential $\psi(\vec{x})$ depends on $|\vec{x}|$ alone.
Thus it follows that the solution of Eq. (9) depends on $|\vec{y}|$ alone, and
hence throughout the rest of the analysis we will denote $y\equiv|\vec{y}|$,
which is the only input required for determining the image location and the
associated time delay contours.

##  III Numerical computation of the lensing magnification

Figure 3: The time domain magnification function $\widetilde{\mathcal{F}}(t)$
as a function of time $t$ (in units of $M_{\rm L}$, where $t=0$ at the global
minima). The panels from left to right correspond to cases with $Q/M_{\rm
L}=-0.5,\,0\text{ and }0.5$, respectively. The plots in each of these panels
correspond to impact parameter values $y=0.1,\,0.5\text{ and }1.0$. In the
left and the center panels, there are always two images – a minima image and a
saddle image. The minima image lies at $t=0$ while the saddle image can be
identified by the logarithmically diverging peak. On the right panel, for
$y=0.1$, there are four images – one minima image, two saddle images, and one
maxima image. The maxima image and one of the saddle images have similar time
delays, $t\sim 0.3$. These two images can be identified by the logarithmic
divergence followed by a sharp drop in the zoomed-inset figure on the left.
For $y=0.5\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 1.0$, there is a
minima and a saddle image. However, there are features in addition to the
images as seen in the inset figure on the right.

In this section, we will determine the magnification function due to a charged
lens and will find out how the GW waveform is modulated due to lensing. For an
isolated point mass lens, the lensing magnification function $\mathcal{F}(f)$
can be derived analytically Nakamura and Deguchi (1999). Unfortunately, for
the present case of a charged lens, an analytical expression for the
magnification function cannot be obtained and one must resort to numerical
schemes. Implementing such numerical schemes is also difficult in the present
context owing to the highly oscillatory behaviour of the integral, which
renders the traditional numerical integration methods to be computationally
ineffective. Therefore, we developed and implemented a new numerical scheme to
compute the magnification function, which we outline below. This is an
extension of the methods used in Diego  _et al._ (2019), based on the original
work by Ulmer and Goodman (1995). The new ingredient is to use a histogram for
an efficient computation of the diffraction integral Eq.(13) in time domain.

  1. 1.

Fixing the length scale $\xi_{0}$: The time delay contours, as well as the
lensing potential, depend on the length scale $\xi_{0}$, which we fix to be of
the following form:

$\xi_{0}^{2}=4M_{0}\frac{D_{\rm L}D_{\rm{LS}}}{D_{\rm S}}\leavevmode\nobreak\
,$ (10)

where $M_{0}$ is an arbitrary mass scale. In the present problem, the only
relevant mass scale is set by the lens mass and hence we choose $M_{0}=M_{\rm
L}$. With this choice of the length scale $\xi_{0}$, Eq. (4) becomes,

$\displaystyle t_{\rm d}(\vec{x},\vec{y})$ $\displaystyle=4M_{\rm L}(1+z_{\rm
L})\times\left[\frac{|\vec{x}-\vec{y}|^{2}}{2}-\psi(\vec{x})\right]\leavevmode\nobreak\
.$ (11)

In the subsequent analysis, we will use the above expression for the time
delay.

Figure 4: The frequency domain magnification function $\mathcal{F}(f)$ as a
function of the GW frequency (in Hz) for a lens with mass $M_{\rm
L}=100M_{\odot}$, at redshift $z_{\rm L}=0.5$. The left panel corresponds to
the amplitude of $\mathcal{F}(f)$, while the right panel corresponds to its
phase. The plots corresponds to cases with $Q/M_{\rm L}=-0.5,\,0\text{ and
}0.5$ (lines with different thicknesses), for impact parameter values
$y=0.1,0.5\text{ and }1.0$ (lines with different colors). We restrict the
frequencies to the sensitivity band of the ground-based GW detectors.

  2. 2.

Time domain magnification function: As we have already discussed, determining
the magnification function $\mathcal{F}(f)$ in the frequency domain requires
performing the highly oscillatory integral over the time function $t_{\rm
d}(\vec{x},\vec{y})$, which is numerically challenging. Thus, we will first
compute the magnification function in time domain, and then compute
$\mathcal{F}(f)$ using Fourier transform. For this purpose we rewrite Eq. (3)
in the following form:

$\mathcal{F}(f)=C(f)\int d^{2}\vec{x}\,\exp[2\pi ift_{\rm
d}(\vec{x},\vec{y})]\leavevmode\nobreak\ .$ (12)

We define the time domain magnification function $\widetilde{\mathcal{F}}(t)$
as the inverse Fourier transform of the ratio $\\{\mathcal{F}(f)/C(f)\\}$,
which yields:

$\displaystyle\widetilde{\mathcal{F}}(t)$ $\displaystyle\equiv\int
df\,\frac{\mathcal{F}(f)}{C(f)}\,\exp[-2\pi ift]$ $\displaystyle=\int
d^{2}\vec{x}\,\delta[t_{\rm d}(\vec{x},\vec{y})-t]\leavevmode\nobreak\ .$ (13)

As evident from the above expression, the time-domain magnification function
is related to the area between the time-domain contours in the lens plane. In
particular, $\widetilde{\mathcal{F}}(t)dt$ is the area $dS$ between the curves
$t$ and $t+dt$ of constant time delay, i.e.,

$\widetilde{\mathcal{F}}(t)=\frac{dS}{dt}\leavevmode\nobreak\ .$ (14)

Thus by computing the area between two infinitesimal time domain contours, we
can determine the magnification function $\widetilde{\mathcal{F}}(t)$ in the
time domain.

  3. 3.

Histogram of the time delay map: To find the area between the contours of
constant time delay, we place a uniform grid on the lens plane and determine
the time delay at each of the grid points (see, e.g., Fig. 1). This yields the
number of grid points between time delays $t$ and $t+dt$, which leads to a
histogram for the number of grid points at each of the time delay contours
$t$, with bin size $dt$. This number is a proxy for the area between the time
delays $t$ and $t+dt$. Thereby allowing us to compute the time-domain
magnification function $\widetilde{\mathcal{F}}(t)$.

For the case of a charged lens, the above method of determining the area
between time delay contours leads to the desired time-domain magnification
function (Fig. 3). As evident, the amplification is lowest for $Q<0$ and is
the highest for $Q>0$. Also, the magnification function increases as the
impact parameter $y$ is decreased, since the GW passes closer to the lens and
is most affected. Another intriguing feature, already noted earlier, can also
be seen in Fig. 3, namely, the existence of four images for certain positive
values of $Q$ and for specific impact parameter $y$. For $Q/M_{\rm L}=0.5$ and
$y=0.1$, these four images can be seen explicitly. The one at $t=0$ is present
in all the plots, which have not been shown, then two additional images are
clearly visible, while the fourth one (maxima image) is very close to one of
the saddle images. For larger values of $y$, there are two images with
positive $Q$, but some additional features are present. These can be seen in
the inset figures of the right panel in Fig. 3.

  4. 4.

Frequency domain magnification function: Having computed the time-domain
magnification function $\widetilde{\mathcal{F}}(t)$, it is straightforward to
determine the frequency domain magnification $\mathcal{F}(f)$. For this
purpose, we simply perform a fast Fourier Transform of
$\widetilde{\mathcal{F}}(t)$ and multiply by the overall factor $C(f)$,
yielding:

$\mathcal{F}(f)=C(f)\times\mathrm{FFT}\left[\widetilde{\mathcal{F}}(t)\right]\leavevmode\nobreak\
.$ (15)

Though the time-domain magnification function was real, the frequency-domain
amplification is complex. Following this, we have plotted the magnitude
$|\mathcal{F}(f)|$ and the argument $\textrm{arg}[\mathcal{F}(f)]$ in Fig. 4.
As expected, the frequency-domain magnification also is the largest for $Q>0$,
while is the smallest for $Q<0$, and the magnification function decreases as
the impact parameter increases.

Figure 5: GW signals lensed by charged BHs of mass $M_{\rm L}=100M_{\odot}$,
at redshift $z_{\rm L}=0.5$. The panels from the left to the right correspond
to cases with $Q/M_{\rm L}=-0.5,\,0\text{ and }0.5$ respectively, for various
impact parameters (shown in legends). The top and bottom panels show the
amplitude of lensed waveforms in the frequency and time domains, respectively.
The dashed grey lines indicate the unlensed waveforms.

  5. 5.

Lensed waveform: Finally, the GW strain is going to be amplified by the
frequency-dependent magnification factor. In the frequency domain, the
unlensed GW strain $h^{U}(f|\vec{\theta})$ is going to be modified as:

$h^{L}(f;\vec{\theta},\vec{\lambda})=\mathcal{F}(f;\vec{\lambda})\leavevmode\nobreak\
h^{U}(f;\vec{\theta})\leavevmode\nobreak\ .$ (16)

Here $h^{L}(f;\vec{\theta},\vec{\lambda})$ is the lensed waveform,
$\vec{\theta}$ are the source parameters of the GW signal and
$\vec{\lambda}=(M_{\rm L}^{z},y,Q)$ are the lens parameters, where $M_{\rm
L}^{z}\equiv M_{\rm L}(1+z_{\rm L})$ is the redshifted lens mass.

The GW signal lensed by a charged lens has been shown in both the frequency
and the time domain in Fig. 5, along with the corresponding unlensed waveform.
It is evident (in particular, from the time domain plots) that the lensed
waveform for $Q<0$ has the least departure, while for $Q>0$ has the maximum
departure from the unlensed waveform. For $Q<0$, the lensed and unlensed
waveforms are mostly in phase with the unlensed, while for $Q>0$, these
waveforms are out of phase. As we will see, these properties make constraining
positive $Q$ much easier, than constraining negative values of $Q$.

As a verification of the numerical scheme described above, we have employed
the above method for determining the lensed waveform for a point mass
(uncharged) lens. Since the magnification function for a point mass lens can
be computed analytically Nakamura (1998), we can compare it with our numerical
results. Following this, we have evaluated the mismatch between the lensed
waveforms computed using the analytical form of the magnification function for
isolated point mass lens and lensed waveforms computed using our method with
$Q=0$. We find that the mismatch values are within permissible limits
($\lesssim 10^{-5}$). This validates the numerical method described above and
has been used for determining the magnification function for a charged lens.

##  IV Prospective constraints on the black hole charge from lensing
observations

In this section, we will derive prospective constraints on the BH charge from
future observations of microlensed GW signals. We have already noticed that
positive values of the charge $Q$ significantly modify the GW waveform through
lensing and the effect is present in the amplitude, as well as in the phase of
the GW signal (Fig. 5). Thus, we expect that a BH lens without charge can be
efficiently distinguished from a charged BH lens for $Q>0$, while we expect
such a distinction will be less efficient for $Q<0$.

To get a better understanding of the prospective constraints on BH charge from
future observations of microlensed GWs, we estimate the (approximate) Bayesian
posteriors of $Q$ from simulated observations of microlensed GWs. The GW
signal microlensed by a BH can be parameterised by a set of source parameters
$\vec{\theta}$ and lens parameters $\vec{\lambda}$. Their posterior
distributions can be estimated from the observed data $d$ using the Bayes
theorem:

$p(\vec{\theta},\vec{\lambda}|d)=\frac{p(\vec{\theta},\vec{\lambda})p(d|\vec{\theta},\vec{\lambda},\mathcal{H}_{\mathrm{ML}})}{p(d|\mathcal{H}_{\mathrm{ML}})},$
(17)

where $p(\vec{\theta},\vec{\lambda})$ is the prior distribution of the source
and lens parameters,
$p(d|\vec{\theta},\vec{\lambda},\mathcal{H}_{\mathrm{ML}})$ is the likelihood
of getting data $d$ given the parameters $\vec{\theta},\vec{\lambda}$ and the
hypothesis $\mathcal{H}_{\mathrm{ML}}$ that the data contains a microlensed GW
signal. The denominator, $p(d|\mathcal{H}_{\mathrm{ML}})$, is the likelihood
marginalised over all the parameters (called, the evidence of the hypothesis
$\mathcal{H}_{\mathrm{ML}}$).

Figure 6: The top panel shows the 2D likelihood in the impact parameter $y$
and charge $Q/M_{\rm L}$ computed from a GW signal with $\rm{SNR}=25$
microlensed by a charged BH of mass $M_{\rm L}=500M_{\odot}$. The true values
of the impact parameter $(y=0.5)$ and charge ($Q/M_{\rm L}=-0.5,0,0.5$) are
indicated by the dashed red lines in each panel from left to right. The $95$
percentile levels are indicated by white lines. The bottom panel shows the 2D
likelihood in $M_{\rm L}$ and $Q/M_{\rm L}$ for a microlensed GW signal with
$y=0.5$. It can be seen that $Q/M_{\rm L}$ is largely uncorrelated with other
parameters $M_{\rm L}$ and $y$.

The posterior $p(Q|d)$ of the charge can be computed by marginalising
$p(\vec{\theta},\vec{\lambda}|d)$ over all parameters except $Q$. For
simplicity, we will assume that the lens parameters $\vec{\lambda}$ are
largely uncorrelated with the source parameters $\vec{\theta}$. Hence we need
to compute the likelihood only on $\vec{\lambda}$ for estimating prospective
constraints on $Q$. This is a reasonable assumption although recent work has
identified possible correlations between microlensing modulations and
modulations induced by spin-induced precession Mishra  _et al._ (2023b). An
uncharged point mass lens in the background of a macro lens (e.g., a galaxy)
could also introduce more complex modulations in the GW signal Meena and Bagla
(2020); Diego  _et al._ (2019); Cheung  _et al._ (2021); Mishra  _et al._
(2021), potentially mimicking some of the effect of a charged lens. Also, note
that currently we assume non-spining BH lenses. There could be some
correlations between the charge and the spin of a BH (see, e.g., Carullo  _et
al._ (2022)). For the time being, we ignore these additional complexities.

It turns out that the lens parameters $M_{\mathrm{L}},y$ are also largely
uncorrelated with the charge $Q$ (see, Fig. 6 for an illustration). Thus, to
compute the expected bounds on $Q$, to a good approximation one needs to
compute the likelihood in $Q$ only. We employ the following approximation of
the expectation value of the likelihood

$\mathcal{L}(Q)\equiv\langle
p(d|Q,\mathcal{H}_{\mathrm{ML}})\rangle\simeq\exp{\left[-\rho^{2}\,\mathcal{M}(Q_{\mathrm{tr}},Q)\right]}.$
(18)

Above $\rho$ is the signal-to-noise ratio (SNR) of the signal in the data and
$\mathcal{M}(Q_{\mathrm{tr}},Q)$ is the _mismatch_ between the injected (true)
waveform $h^{L}(f,Q_{\mathrm{tr}})$ and the waveform $h^{L}(f,Q)$ with charge
$Q$.

$\mathcal{M}(Q_{\mathrm{tr}},Q)\equiv
1-4\int_{f_{\mathrm{low}}}^{f_{\mathrm{upp}}}\frac{h^{L}(f,Q_{\mathrm{tr}})\,h^{L*}(f,Q)\,df}{S_{n}(f)}.$
(19)

Here ${f_{\mathrm{low}}}$ and ${f_{\mathrm{upp}}}$ denote the lower and upper
frequency cutoff the detector band and $S_{n}(f)$ the one-sided power spectral
density of the detector noise.

When we assume a flat prior in $Q$, the expectation value of the posterior
distribution $P(Q|d)$ is the same as the likelihood $\mathcal{L}(Q)$. Hence we
present $\mathcal{L}(Q)$ and its 90% credible upper limits in Fig. 7 and Fig.
8. In these figures, we have assumed the PSD of the advanced LIGO (aLIGO)
detectors. Additionally, the unlensed signal is assumed to be due to a non-
spinning equal mass BH binary with component masses $20M_{\odot}$ each. We
compute the likelihood assuming true values of $(Q_{\mathrm{tr}}/M_{\rm
L})=0,-0.1\text{ and }-0.5$ in Fig. 7 and true values of
$(Q_{\mathrm{tr}}/M_{\rm L})=0,0.1\text{ and }0.5$ in Fig. 8.

It is evident from Eq.(18) that the likelihood drops when the mismatch between
the true value $Q_{\mathrm{tr}}$ and the chosen value $Q$ is large (i.e., when
the two waveforms are very different) or when the SNR of the signal is large.
These expected features can be seen in these figures. Negative values of $Q$
are least constrained for smaller values of the impact parameter $y$ and are
slightly better constrained for larger impact parameter $y$ (Fig. 7). Note
that for smaller SNR and for smaller impact parameters, negative values of $Q$
are unconstrained. Increasing the impact parameter, as well as for higher lens
mass $M_{\rm L}$, the constraints on $Q$ significantly improve. Increasing the
SNR further improves the situation.

Figure 7: Likelihood plots with $90\%$ credible bounds on $Q/M_{\rm L}$ using
aLIGO PSD, for various values of impact parameter $y$ (shown in the legend).
Panels from left to right correspond to SNR values of $25,50\text{ and }100$
respectively. In the top, middle and bottom panels, the GW signal is lensed by
a BH of charge $Q_{\mathrm{tr}}/M_{\rm L}=-0.5,-0.1$ and $0$ respectively
(indicated by horizontal dashed black lines). Here, we use the prior $-1\leq
Q/M_{\rm L}\leq 0$, that corresponds to situations such as the braneworld
scenario. Figure 8: Same as Fig. 7, except that the GWs are lensed by BHs of
charge $Q_{\mathrm{tr}}/M_{\rm L}=0.5,0.1$ and $0$ respectively (indicated by
horizontal dashed black lines). Here, we use the prior $0\leq Q/M_{\rm L}\leq
1$, which can correspond to a charge of EM or alternative gravity origin.

On the other hand, and quite contrary to the negative $Q$ case, as Fig. 8
demonstrates, even for SNR of 25, and impact parameter $y\sim 0.1$, positive
values of $Q$ can be measured with precision of $\Delta Q/Q\lesssim 0.1$ for
larger lens masses. The situation improves drastically for higher SNR. For
example, with an SNR of 100, values of $Q\gtrsim 0.02$ can be ruled out with
ninety percent confidence.

Thus using gravitational lensing of GWs, it is possible to rule out positive
values of charge (of EM or modified gravity origin) with a percentage level
accuracy. In contrast, for a negatively charged lens (expected in the presence
of extra dimensions), the accuracy is at least an order weaker.

##  V Conclusion

Upcoming observations are expected to detect gravitationally lensed GWs. One
of the possible lenses is compact objects such as BHs. If the gravitational
radii of these lenses are comparable to the wavelength of GWs, lensing will
produce wave optics effects, producing characteristic deformations in the
observed signals. The exact nature of these deformations will depend on the
precise spacetime geometry (lensing potential) of the lens. Thus, lensing
observations can potentially probe the detailed nature of these lensing
objects.

In this paper, we derived an exact expression for the lensing potential of a
‘charged’ BH lens. The charge $Q$ can have an EM origin, in which case it is a
positive definite quantity ($Q$ is the square of the electric charge $q$), but
can also arise in (at least) four different situations, all of which are
beyond GR. These include — (a) the braneworld scenario, where the presence of
an extra spatial dimension modifies Einstein’s equations, and introduces a
charge term $Q$ that is negative; (b) the Gauss-Bonnet theory in higher
dimensions, which also leads to an effective four-dimensional spacetime with a
negative charge; (c) $f(T)$ theories of gravity, as well as (d) a certain
class of Horndeski theories also brings in a charge term in the spacetime
metric, which is a positive quantity. Thus, negative values of the charge
definitely hint at the presence of extra dimensions, while positive values of
the charge can have an EM origin, or can also arise from modified theories of
gravity. If one can detect the charge hair of a BH spacetime, it is possible
to comment on the fundamental questions, e.g., the existence of extra
dimension, as well as the signature of gravity theories beyond GR.

Using the lensing potential of charged BHs that we derived, we computed the
deformation of the lensed GW signals considering wave optics effects. This was
done using a numerical scheme that we recently developed. This scheme can be
used to compute the frequency-dependent lensing magnification for arbitrary
lensing potentials. We noticed interesting new observables in the case of
charged lenses. Owing to the axial symmetry of the lensing potential, the
images and the source always lie on a line on the lens plane (same as the case
of uncharged lens). For a negatively charged lens there are always two images
(same as the uncharged lens), while for a positively charged lens, there can
be two or four images, depending on the value of the charge and the impact
parameter. This new feature can be understood in terms of the (numerically
computed) structure of the caustics of charged lenses. This introduces rich
and complex effects in the lensed GW signals which are absent in those lensed
by uncharged BHs. These additional features will help us to identify the
presence of positively charged BH lenses, if they exist. On the other hand,
negatively charged lenses produce features very similar to those of uncharged
BHs, making it difficult to distinguish them from uncharged BHs.

We then explored the ability of future lensing observations to constrain the
charge of the lens. We consider lensing observations by a single LIGO detector
with sensitivity described by the PSD of the aLIGO observing run. We showed
that the charge can be constrained using observation of lensed GWs. In
particular, positive values of the charge can be constrained to an accuracy of
$10\%\,(1\%)$ if the lensed GW signal is detected with an SNR $\sim
25\,(100)$. On the other hand, negative values of the charge are constrained
to a much lower precision, $15\%$ at best (with SNR $\sim 100$). For smaller
SNRs, practically all possible negative values of $Q$ are allowed. In short,
the presence of an extra dimension is much harder to detect using lensing of
GWs, while there is excellent chance of detecting or constraining the presence
of EM charge, or, theories beyond GR.

Note that the expected constraints that we present here employ an approximate
likelihood, and neglect the possible correlations between the lensing-induced
modulations in the GW signals and modulations induced by other physical
effects such as orbital eccentricity of the binary source. We also ignored the
possible degeneracy between a GW signal lensed by a charged BH and a GW signal
lensed by an uncharged point mass lens in the presence of a macro lens (e.g.,
a galaxy), which can introduce more complex features. Additionally, we also
neglected the effect of the spin of the BH lens. While we expect these broad
conclusions to hold, the precise forecasts of the prospective constraints need
to be revisited in the future considering these additional complexities.

The numerical scheme that we employ to compute lensing magnification for
arbitrary lensing potentials is too expensive to employ in actual GW parameter
estimation that will require a large number of likelihood evaluations. In
order to employ in GW parameter estimation, we need to develop some
surrogate/semi-analytical models that interpolate the numerically computed
lensing magnifications over the parameter space of interest.

In this paper, we have worked with static and spherically symmetric spacetimes
so far, and hence it would be interesting to generalize the same to rotating
spacetimes, as all astrophysical objects are, in general, rotating. One could
explore the possibility of measuring the spin of the compact object from
lensing observations. One could also explore more general spacetimes: here we
have considered the cases where $-g_{tt}=g^{rr}$; it will be useful to
understand how to derive the lensing potential for spacetimes, with
$-g_{tt}\neq g^{rr}$. Another possibility is probing the astrophysical
environment of BHs using lensing observations. We hope to come back to these
issues in future works.

Note that some of the modifications to GR that induce an effective charge on
BHs could also cause other effects in the generation and propagation of GWs,
which we neglect here. Our proposal should be seen as a way of effectively
checking the consistency of the GW signal that is lensed by a BH in GR (in
this paper the Schwarzschild metric). Any observed inconsistency with the
Schwarzschild lens will need to be investigated further in order to ascertain
the nature of the charge. This is similar in spirit to various other tests of
GR using GW observations. In any case, future observations of lensed GWs are
very likely to offer new ways of probing the nature of compact objects.

###### Acknowledgements.

We are grateful to Otto Hannuksela for the careful review of the manuscript
and useful comments. We thank the members of the astrophysical relativity
group at ICTS for their valuable input. We acknowledge the support of the
Department of Atomic Energy, Government of India, under project no. RTI4001.
M.A.S.’s research was, in addition, supported by the National Research
Foundation of Korea under grant No. NRF-2021R1A2C2012473. S.C. thanks the
Albert Einstein Institute for its warm hospitality, where a part of this work
was performed. The visit to the Albert Einstein Institute is funded by the
Max-Planck Society through its Max-Planck-India mobility grant. Computations
were performed using the Alice computing cluster at the International Centre
for Theoretical Sciences.

##  Appendix A The lensing potential and its properties

In this appendix, we will provide the basic steps in deriving the lensing
potential for a charged lens. For this purpose, we start by evaluating the
two-dimensional Laplacian acting on $(\ln|\vec{x}|)^{2}$. Taking a double
derivative with respect to $x_{1}$ yields,

$\displaystyle\partial_{x_{1}}^{2}(\ln|\vec{x}|)^{2}$
$\displaystyle=\partial_{x_{1}}\left(2\ln|\vec{x}|\frac{x_{1}}{|\vec{x}|^{2}}\right)$
$\displaystyle=2\ln|\vec{x}|\frac{1}{|\vec{x}|^{2}}+2\frac{x_{1}^{2}}{|\vec{x}|^{4}}-4\ln|\vec{x}|\frac{x_{1}^{2}}{|\vec{x}|^{4}}\leavevmode\nobreak\
.$ (20)

Along similar lines the derivative with respect to $x_{2}$ becomes,

$\displaystyle\partial_{x_{2}}^{2}(\ln|\vec{x}|)^{2}=2\ln|\vec{x}|\frac{1}{|\vec{x}|^{2}}+2\frac{x_{2}^{2}}{|\vec{x}|^{4}}-4\ln|\vec{x}|\frac{x_{2}^{2}}{|\vec{x}|^{4}}\leavevmode\nobreak\
.$ (21)

Adding these two double derivatives, the two-dimensional Laplacian acting on
$(\ln|\vec{x}|)^{2}$ gives us,

$\displaystyle\nabla_{x}^{2}(\ln|\vec{x}|)^{2}$
$\displaystyle=\partial_{x_{1}}^{2}(\ln|\vec{x}|)^{2}+\partial_{x_{2}}^{2}(\ln|\vec{x}|)^{2}$
$\displaystyle=4\ln|\vec{x}|\frac{1}{|\vec{x}|^{2}}+2\frac{x_{1}^{2}+x_{2}^{2}}{|\vec{x}|^{4}}-4\ln|\vec{x}|\frac{x_{1}^{2}+x_{2}^{2}}{|\vec{x}|^{4}}$
$\displaystyle=\frac{2}{|\vec{x}|^{2}}\leavevmode\nobreak\ .$ (22)

Thus the Laplacian acting on the $(\ln|\vec{x}|)^{2}$ indeed yields the energy
density arising from the charged nature of the gravitational lens. This shows
the origin of the expression for the lensing potential associated with the
charged lens in the main text.

Let us now describe an interesting property of any axially symmetric lens
system, namely the co-linear nature of the source and the images on the lens
plane, which is also applicable in the present context. From the fact that the
spacetime metric is axially symmetric, it follows that the lensing potential
will also satisfy the same symmetry, i.e., $\psi(\vec{x})=\psi(|\vec{x}|)$.
The associated time delay is then given by,

$\displaystyle t_{\rm d}(\vec{x},\vec{y})$
$\displaystyle=\frac{|\vec{x}-\vec{y}|^{2}}{2}-\psi(|\vec{x}|)$
$\displaystyle=\frac{|\vec{x}|^{2}}{2}-|\vec{x}||\vec{y}|\cos\theta+\frac{|\vec{y}|^{2}}{2}-\psi(|\vec{x}|)\leavevmode\nobreak\
,$ (23)

where $\vec{x}=(x_{1},x_{2})=(|\vec{x}|\cos\theta,|\vec{x}|\sin\theta)$, with
$\theta$ being the angle between the vectors $\vec{x}$ and $\vec{y}$. The
location of the images can be found by solving $\nabla_{x}t_{\rm
d}(\vec{x},\vec{y})=0$ as discussed in section II. This boils down to taking
the derivative of the time delay function with respect to $|\vec{x}|$ and
$\theta$. Extremizing the time delay with respect to the angle $\theta$
yields,

$\frac{\partial t_{\rm
d}}{\partial\theta}=|\vec{x}||\vec{y}|\sin\theta=0\leavevmode\nobreak\ ,$ (24)

which has only two solutions $\theta=0,\pi$. This implies that the images will
always lie along the direction of $\vec{y}$. In other words, for any axially
symmetric lensing potential, the lens, the impact parameter, and the images
are always co-linear on the lens plane. This feature can also be seen in Fig.
1.

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