# Gravitational Faraday rotation of gravitational waves by a Kerr black hole

Zhao Li<EMAIL_ADDRESS>Department of Astronomy, University of
Science and Technology of China, Hefei, Anhui 230026, China,
and School of Astronomy and Space Science, University of Science and
Technology of China, Hefei 230026, China  Jin Qiao  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing, 210023, P.R. China,
and School of Astronomy and Space Science, University of Science and
Technology of China, Hefei 230026, China  Wen Zhao<EMAIL_ADDRESS>Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China,
and School of Astronomy and Space Science, University of Science and
Technology of China, Hefei 230026, China  Xinzhong Er  South-Western Institute
for Astronomy Research, Yunnan University, Kunming, Yunnan, 650000, China

###### Abstract

Gravitational Faraday Rotation (GFR) is a frame-dragging effect induced by
rotating massive objects, which is one of the important, yet studied
characteristics of lensed gravitational waves (GWs). In this work, we
calculate the GFR angle $\chi_{g}$ of GWs in the weak deflection limit,
assuming it is lensed by a Kerr black hole (BH). We find that the GFR effect
changes the initial polarization state of the lensed GW. Compared with the
Einstein deflection angle, the dominant term of the rotation angle $\chi_{g}$
is a second-order correction to the polarization angle, which depends on the
light-of-sight component of BH angular momentum. Such a rotation is tiny and
degenerates with the initial polarization angle. In some critical cases, the
GFR angle is close to the detection capability of the third-generation GW
detector network, although the degeneracy has to be broken.

##  I Introduction

According to Einstein’s General Relativity (GR), a massive rotating object
changes the spacetime structure and drags the inertial frame around it
Chandrasekhar (1983), which causes various detectable effects on test
particles Weinberg (1972); Misner  _et al._ (1971); Poisson and Will (2014).
Detecting these effects provides an excellent opportunity to study the
fundamental properties and the internal structure of gravitational sources
Poisson and Will (2014). For instance, Refs. Thirring (1918); Lense and
Thirring (1918) described an additional precession of a gyroscope moving in a
weak and slowly-rotating spacetime, precisely measured by Gravity Probe B
experiment Everitt  _et al._ (2011). The measured frame-dragging drift rate is
$-37.2\pm 7.2\ {\rm marcsec/yr}$, which is consistent with the prediction of
GR.

Gravitational Faraday rotation (GFR) Dehnen (1973); Piran and Safier (1985);
Wang (1991) is a crucial dragging effect, a general-relativistic effect
analogous to the Faraday rotation Rybicki and Lightman (2008). The
polarization plane of light rotates around the propagation direction when it
propagates through the magnetic field. The rotation angle is proportional to
the magnetic field along the line of sight (L.O.S.). In GR, the gravitational
field generated by spinning massive objects is analogous to a magnetic field,
interacting with electromagnetic waves (EMWs) and gravitational waves (GWs).
The GFR effect of EMWs, induced by a Kerr black hole (BH), has been studied in
the geometrical optical approximation Connors and Stark (1977); Connors  _et
al._ (1980); Chandrasekhar (1983); Ishihara  _et al._ (1988); Nouri-Zonoz
(1999); Asada  _et al._ (2003); Sereno (2004, 2005); Brodutch  _et al._
(2011); Chen  _et al._ (2015); Deriglazov (2021); Chakraborty (2022).

Like EMWs, the GWs are deflected by the gravitational field of massive
objects, e.g., a galaxy or a BH Lawrence (1971); Bray (1986); Schneider  _et
al._ (1993); Sereno and De Luca (2006); Bozza (2010); Nambu (2013); Hou  _et
al._ (2019). With the discovery of GW events generated by the mergers of
compact binaries Abbott  _et al._ (2016, 2017, 2020, 2019, 2021a), the
gravitationally-lensing effects of GW signals attract more and more attention
in both observation Hannuksela  _et al._ (2019); Abbott  _et al._ (2021b);
Piórkowska  _et al._ (2013); Li  _et al._ (2018) and theory Bray (1986);
Baraldo  _et al._ (1999); Takahashi and Nakamura (2003); Hou  _et al._ (2019).
In addition to the lensing effects such as multiple images, magnification,
phase shift, interference, and diffraction Nakamura (1998); Dai  _et al._
(2018); Meena and Bagla (2019); Mishra  _et al._ (2021); Cheung  _et al._
(2021); Nakamura and Deguchi (1999); Takahashi and Nakamura (2003); Guo and Lu
(2020); Baraldo  _et al._ (1999); Shan  _et al._ (2022), the polarization
plane of GW can be dragged by the angular momentum of the lens, i.e., the GFR
of GWs Wang (1991); Fan  _et al._ (2017); Liao  _et al._ (2017); Hou  _et al._
(2019). The lensing effects of GWs have been explicitly studied in the
geometric optics or diffraction theory framework Born and Wolf (1959), except
for the GFR effect. In this work, we will present the explicit result of the
GFR angle of GWs induced by a Kerr BH by solving the geodesic equations of
massless particles Ishihara  _et al._ (1988); Chandrasekhar (1983); Bray
(1986); Sereno and De Luca (2006). We focus on the GFR effect in the weak
deflection limit (WDL) of strong gravitational lensing. The geometric optical
approximation is valid since the wavelength of the GW is assumed to be not
comparable with the gravitational radius of the lens in our studies.

The organization of this article is as follows. Section II is a brief
introduction to the Walker-Penrose theorem for Kerr spacetime. Then, we
describe a suitable definition of the GW metric in Section III and present our
derivation of the GFR angle of GW in Section IV. Finally, Section V shows the
evaluation result of the GFR angle of GW. We introduce the observational
effect of GFR in gravitationally-lensed GW signals in Section VI. Section VII
concludes and discusses the main results of this paper. In this article, we
will use units $c=G=1$, where $c$ is the speed of light in the vacuum, and $G$
is the gravitational constant.

##  II Walker-Penrose Theorem

A Kerr BH is fully described by its mass $M$ and angular momentum $a$. In
Boyer-Lindquist coordinates $(t,r,\theta,\varphi)$ Boyer and Lindquist (1967),
the metric of Kerr spacetime $g_{\mu\nu}$ Kerr (1963) is

$ds^{2}=-dt^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}+(r^{2}+a^{2})\sin^{2}\theta
d\varphi^{2}+\frac{2Mr}{\rho^{2}}\left(a\sin^{2}\theta d\varphi-
dt\right)^{2},$ (1)

where $\Delta=r^{2}-2Mr+a^{2}$ and $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$. Let
$x^{\mu}=x^{\mu}(\tau)$ be an affine-parameterized null geodesic and $k^{\mu}$
be the tangent vector of $x^{\mu}(\tau)$, $k^{\mu}\equiv\partial
x^{\mu}/\partial\tau$, where $\tau$ is proper time. $k^{\mu}$ is light-like
and parallel transported along the geodesic, i.e.
$|\bm{k}|^{2}=g_{\mu\nu}k^{\mu}k^{\nu}=0$ and
$k^{\alpha}\nabla_{\alpha}k_{\mu}=0$. The vector $f^{\mu}$ is assumed to be
orthogonal to $k^{\mu}$, and parallel transports along the null geodesic,
i.e., $\bm{k}\cdot\bm{f}=g_{\mu\nu}k^{\mu}f^{\nu}=0$ and
$k^{\alpha}\nabla_{\alpha}f_{\mu}=0$. Since the Kerr geometry is of Petrov
type D Chandrasekhar (1983), one can define Walker-Penrose (WP) conserved
quantity,

$K_{\rm WP}=(A-iB)(r-ia\cos\theta)=K_{1}+iK_{2},$ (2)

WP conserved quantity (2) satisfies the WP theorem Walker and Penrose (1970),

$k^{\mu}\nabla_{\mu}K_{\rm WP}=0,$ (3)

where
$A\equiv(\bm{k}\cdot\bm{l})(\bm{f}\cdot\bm{n})-(\bm{k}\cdot\bm{n})(\bm{f}\cdot\bm{l})$
and
$iB\equiv(\bm{k}\cdot\bm{m})(\bm{f}\cdot\bar{\bm{m}})-(\bm{k}\cdot\bar{\bm{m}})(\bm{f}\cdot\bm{m})$
Chandrasekhar (1983). $\\{\bm{l},\bm{n},\bm{m},\bar{\bm{m}}\\}$ is the Newman-
Penrose tetrad of Kerr BHs Kinnersley (1969). $K_{1}$ and $K_{2}$ are the real
and imaginary parts of $K_{\rm WP}$, respectively. WP theorem has been used to
predict the GFR of EMWs by treating $f^{\mu}$ as the EMW polarization vector
Ishihara  _et al._ (1988); Brodutch  _et al._ (2011); Nouri-Zonoz (1999). One
can always set $f^{t}=0$ without loss of generality. The radiation gauge
causes $\bm{f}$ to always be in the polarization plane. WP theorem constrains
the evolution of polarization vector along a null geodesic. Thus, the
polarization vector $\bm{f}$ rotates on the polarization plane during
propagation.

Far from the BH, the WP conserved quantity (2) is

$\left\\{\begin{aligned} K_{1}&=r\left[-\beta
f^{\theta}-(\gamma\sin\theta)f^{\varphi}\right]\frac{k^{r}}{|k^{r}|}\\\
K_{2}&=r\left[+\gamma
f^{\theta}-(\beta\sin\theta)f^{\varphi}\right]\end{aligned}\right.,$ (4)

where
$\beta\equiv\sqrt{\eta+a^{2}\cos^{2}\theta-\xi^{2}\cot^{2}\theta}{k^{\theta}}/{|k^{\theta}|}$
and $\gamma\equiv\xi\csc\theta-a\sin\theta$. In (4), $\eta$ is the
$z$-component of angular momentum, and $\xi$ is the Carter constant of a
massless particle Carter (1968). Since $K_{1}$ and $K_{2}$ are finite,
$f^{\theta}$ and $f^{\varphi}$ approach $0$ when $r$ is large, and $f^{r}$
also vanishes due to the gauge condition $\bm{k}\cdot\bm{f}=0$. Therefore, it
is convenient to introduce a unit transverse vector $\bm{\mathcal{E}}$ with
components,

$\mathcal{E}^{r}\equiv f^{r},\quad\mathcal{E}^{\theta}\equiv-
rf^{\theta},\quad\mathcal{E}^{\varphi}\equiv-r\sin\theta f^{\varphi}.$ (5)

$\mathcal{E}^{r}$ vanishes for large $r$. Thus, $\bm{\mathcal{E}}$ is a vector
tangent to a large sphere, with two orthogonal components
$\mathcal{E}^{\theta}$ and $\mathcal{E}^{\varphi}$, and conserved norm
$[(\mathcal{E}^{\theta})^{2}+(\mathcal{E}^{\varphi})^{2}]^{1/2}=(K_{1}^{2}+K_{2}^{2})^{1/2}$.
This equality means that the energy carried by EMWs is conserved during
propagation in geometrical optics approximation. Therefore, the asymptotic
form of WP conserved quantity is expressed in $\mathcal{E}^{\theta}$ and
$\mathcal{E}^{\varphi}$,

$\left\\{\begin{aligned}
\frac{k^{r}}{|k^{r}|}K_{1}&=+\beta\mathcal{E}^{\theta}+\gamma\mathcal{E}^{\varphi}\\\
K_{2}&=-\gamma\mathcal{E}^{\theta}+\beta\mathcal{E}^{\varphi}\end{aligned}\right..$
(6)

The WP theorem requires that $K_{1}$ and $K_{2}$ have the same values along
the graviton geodesic. The conservation of $K_{\rm WP}$ provides an exact
translation relation of $\bm{\mathcal{E}}$ between the source and detector
positions. This transformation is given by Ref. Ishihara  _et al._ (1988) and
the following Eq.(10) in Section IV.

##  III Definition of GW Metric

The WP theorem constrains the evolution of the unit transverse vector
$\bm{\mathcal{E}}$ and can also constrain the evolution of GWs. In this
section, the GW metric tensor $h^{ij}$ is defined to prepare to derive the GFR
angle of gravitationally-lensed GWs by a Kerr BH in the next section.

Firstly, the metric tensor $F^{\mu\nu}$ is defined based on vector $\bm{f}$ as
follows,

$F^{\mu\nu}\equiv f^{\mu}f^{\nu},\ (|\bm{f}|\ll 1).$ (7)

Obviously, $F^{\mu\nu}$ is symmetric, orthogonal to the wave vector $\bm{k}$,
and parallel transported along null geodesics. Thus, we conclude that the
definition (7) satisfies the requirements of a metric tensor for the GW.
Secondly, we introduce the traceless condition using transverse-traceless (TT)
projector Maggiore (2008)
$\Lambda_{ij,kl}(\hat{\mathbf{k}})=P_{ik}P_{jl}-(1/2)P_{ij}P_{kl}$ with
$P_{ij}(\hat{\mathbf{k}})=\eta_{ij}-\hat{k}_{i}\hat{k}_{j}$, where
$\eta_{ij}=\text{diag}(1,r^{2},r^{2}\sin^{2}\theta)$ is the spatial part of a
flat background metric at an infinite distance,
$\hat{\mathbf{k}}\equiv\mathbf{k}/|\mathbf{k}|$, and $\mathbf{k}$ is the
spatial part of the wave vector $\bm{k}$. Considering a Schwarzschild lens,
the particle orbit will be in the BH’s equatorial plane, and so will the
background source and the detector. This plane is also called the source-lens-
detector plane. This conclusion is generally invalid for a Kerr lens. However,
under the WDL approximation, the trajectories of gravitons deviate
sufficiently small from the source-lens-detector plane, which is not generally
the equatorial plane of Kerr BH Bray (1986); Ishihara  _et al._ (1988);
Brodutch  _et al._ (2011). Therefore, the source-lens-detector plane can
safely regarded as the orbital plane, presented in Fig. 1. The source and
detector positions are described as coordinates
$(r_{s},\theta_{s},\varphi_{s})$ and $(r_{d},\theta_{d},\varphi_{d})$, which
are sufficiently distant from the BH. Therefore, these two wave vectors are
approximated as $\hat{\mathbf{k}}_{s}=(-1,0,0)$ and
$\hat{\mathbf{k}}_{d}=(1,0,0)$ at these two positions. The subscripts $s$ and
$d$ represent the corresponding quantities evaluated at the source and
detector positions, respectively. TT projector gives an equivalent metric
tensor of (7), $H^{ij}$, in TT gauge,

$H^{ij}=\Lambda_{ij,kl}(\hat{\mathbf{k}}_{s,d})F^{kl}=\left(\begin{array}[]{cc}\frac{1}{2}\left(F^{\theta\theta}-F^{\varphi\varphi}\sin^{2}\theta\right)&F^{\theta\varphi}\\\
F^{\theta\varphi}&\frac{1}{2}\left(F^{\varphi\varphi}-F^{\theta\theta}\csc^{2}\theta\right)\\\
\end{array}\right),$ (8)

which is tangent to an infinitely large sphere. The $r$-component is removed
for simplification. One can find that tensor $H^{ij}$ (8) only has two
independent components, $H^{\theta\theta}$ and $H^{\theta\varphi}$. As a
result, the tensor $H^{ij}$ describes a beam of GW propagating along with
vector $\hat{\mathbf{k}}_{s}$ or $\hat{\mathbf{k}}_{d}$ at infinity.
Furthermore, after transforming $H^{ij}$ into a Cartesian system, the
transformed tensor $\tilde{H}^{ij}$ (where $i,j=x,y,z$, $x\equiv
r\sin\theta\cos\varphi,y\equiv r\sin\theta\sin\varphi,z\equiv r\cos\theta$)
describes GWs propagating along with the direction
$(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ in TT gauge.

In (5), the normalized vector $\bm{\mathcal{E}}$ is introduced. A similar
operation is applied to the metric tensors, which is convenient for
constructing basis vectors and projecting metric tensors. Generally, a vector
$\bm{f}$ can be written as
$\bm{f}=f^{i}\bm{e}_{i}=\mathcal{E}^{i}\hat{\bm{e}}_{i}$, where
$\bm{e}_{i}=\bm{e}_{r},\ \bm{e}_{\theta},\ \bm{e}_{\varphi}$ are basis vectors
and $\hat{\bm{e}}_{r}\equiv\bm{e}_{r},\
\hat{\bm{e}}_{\theta}\equiv-(1/r)\bm{e}_{\theta},\
\hat{\bm{e}}_{\varphi}\equiv-(1/r\sin\theta)\bm{e}_{\varphi}$ are unit basis
vectors in the spherical coordinate system. Similarly, a tensor $\bm{H}$ can
generally be written as
$\bm{H}=H^{ij}\bm{e}_{i}\bm{e}_{j}=h^{ij}\hat{\bm{e}}_{i}\hat{\bm{e}}_{j}$.
Combining the above discussion, we define a new metric tensor with component,
$h^{\theta\theta}\equiv r^{2}H^{\theta\theta}$, $h^{\theta\varphi}\equiv
r^{2}\sin\theta H^{\theta\varphi}$, and $h^{\varphi\varphi}\equiv
r^{2}\sin^{2}\theta H^{\varphi\varphi}$, which is expressed as

$h^{ij}=\left(\begin{array}[]{cc}h^{\theta\theta}&h^{\theta\varphi}\\\
h^{\theta\varphi}&h^{\varphi\varphi}\end{array}\right)=\left(\begin{array}[]{cc}\frac{1}{2}\left[(\mathcal{E}^{\theta})^{2}-(\mathcal{E}^{\varphi})^{2}\right]&\mathcal{E}^{\theta}\mathcal{E}^{\varphi}\\\
\mathcal{E}^{\theta}\mathcal{E}^{\varphi}&-\frac{1}{2}\left[(\mathcal{E}^{\theta})^{2}-(\mathcal{E}^{\varphi})^{2}\right]\end{array}\right).$
(9)

Note that, the metric $h^{ij}$ (9) is an equivalent expression of $F^{ij}$ (7)
in TT gauge and is expressed in unit basis vectors.

In conclusion, the definition of the GW metric comes from the direct product
of two transverse vectors and satisfies all the physical properties of GW. TT
gauge is then applied to remove the residual degree of freedom. The Cartesian-
coordinate form of metric tensor is given in (9).

Figure 1: Demonstration of GFR of lensed GW system. The gravitons emitted from
a source at $(r_{s},\theta_{s},\varphi_{s})$ pass near a Kerr BH with angular
momentum $\bm{a}$, and then arrive at a detector at
$(r_{d},\theta_{d},\varphi_{d})$.

##  IV Gravitational Faraday Rotation

In this section, we derive the GFR angle of the lensed GWs and reproduce the
previous results of the EMW case in Ref. Ishihara  _et al._ (1988) for
comparison. To find the transformation matrix for polarization
$(h^{\theta\theta},h^{\theta\varphi})$ between the source and the detector
positions, we firstly calculate the transformation matrix of the unit
transverse vector $(\mathcal{E}^{\theta},\mathcal{E}^{\varphi})$. The WP
theorem conserves $K_{1}$ and $K_{2}$ at the source and detector positions,
calculated from (6). The conservation gives
$\beta_{s}\mathcal{E}_{s}^{\theta}+\gamma_{s}\mathcal{E}_{s}^{\varphi}=-(\beta_{d}\mathcal{E}_{d}^{\theta}+\gamma_{d}\mathcal{E}_{d}^{\varphi})$
and
$-\gamma_{s}\mathcal{E}_{s}^{\theta}+\beta_{s}\mathcal{E}_{s}^{\varphi}=-\gamma_{d}\mathcal{E}_{d}^{\theta}+\beta_{d}\mathcal{E}_{d}^{\varphi}$.
The electric vectors $\bm{\mathcal{E}}$ at the source and detector positions
are related through a transformation matrix Ishihara  _et al._ (1988)

$\left(\begin{array}[]{c}\mathcal{E}^{\theta}_{d}\\\
\mathcal{E}^{\varphi}_{d}\end{array}\right)=\bm{\mathcal{T}}\left(\begin{array}[]{c}\mathcal{E}^{\theta}_{s}\\\
\mathcal{E}^{\varphi}_{s}\end{array}\right),$ (10)

which can be written explicitly as

$\bm{\mathcal{T}}=-\frac{1}{\beta_{d}^{2}+\gamma_{d}^{2}}\left(\begin{array}[]{cc}\beta_{d}\beta_{s}-\gamma_{d}\gamma_{s}&\beta_{d}\gamma_{s}+\gamma_{d}\beta_{s}\\\
\beta_{d}\gamma_{s}+\gamma_{d}\beta_{s}&\gamma_{d}\gamma_{s}-\beta_{d}\beta_{s}\end{array}\right).$
(11)

Furthermore, the transformation matrix $\bm{\mathcal{T}}$ can be simply
rewritten as

$\bm{\mathcal{T}}=\frac{1}{\sqrt{1+x^{2}}}\left(\begin{array}[]{cc}1&-x\\\
-x&-1\end{array}\right).$ (12)

through a new symbol $x$, defined as

$x\equiv\frac{\gamma_{s}\beta_{d}+\beta_{s}\gamma_{d}}{\gamma_{s}\gamma_{d}-\beta_{s}\beta_{d}}.$
(13)

The quantity $x$ (13) depends on the motion constants and angular coordinates
of the source. In this new form (12), the equality
$\beta^{2}+\gamma^{2}=\eta+(\xi-a)^{2}$ has been considered. The matrix
$\bm{\mathcal{T}}$ describes the full change in the transverse vector. Only
two independent components remain when applying radiation gauge for transverse
vector $\bm{\mathcal{E}}$. The change between the final state
$(\mathcal{E}_{d}^{\theta},\mathcal{E}_{d}^{\varphi})$ and the initial state
$(\mathcal{E}_{s}^{\theta},\mathcal{E}_{s}^{\varphi})$ represents a rotation
on the polarization plane, determined by WP theorem.

###  IV.1 GW case

Above transformation (10) and the definition of $h^{ij}$ (9) give the
transformation of the independent component of GW metric between the source
and detector positions,

$\left(\begin{array}[]{c}h^{\theta\theta}_{d}\\\
h^{\theta\varphi}_{d}\end{array}\right)=\bm{\mathcal{S}}\left(\begin{array}[]{c}h^{\theta\theta}_{s}\\\
h^{\theta\varphi}_{s}\end{array}\right),$ (14)

with transformation matrix

$\bm{\mathcal{S}}=\frac{1}{1+x^{2}}\left(\begin{array}[]{cc}1-x^{2}&-2x\\\
-2x&x^{2}-1\end{array}\right).$ (15)

The matrix $\bm{\mathcal{S}}$ describes the total change in the $\theta\theta$
and $\theta\varphi$ components of the metric tensor.

To present the measurement of the GW polarization, we construct a set of basis
vectors as a reference frame, $\hat{\mathbf{u}}$ and $\hat{\mathbf{v}}$, and
project metric tensor $h^{ij}$ into them. The vectors $\hat{\mathbf{u}}$ and
$\hat{\mathbf{v}}$ need to be orthogonal to $\hat{\mathbf{k}}$ or,
equivalently, on the polarization plane. Without loss of generality,
$\hat{\mathbf{v}}$ is assumed to be orthogonal to the orbital plane and
$\hat{\mathbf{u}}=\hat{\mathbf{v}}\times\hat{\mathbf{k}}$ (Fig. 1). As
mentioned in section III, the wavevectors $\hat{\mathbf{k}}_{s}$ and
$\hat{\mathbf{k}}_{d}$ are on the orbital plane. The normal vector of the
orbital plane, $\mathbf{v}$, is then determined approximately by
$\mathbf{v}=\hat{\mathbf{k}}_{d}\times\hat{\mathbf{k}}_{s}$ and the basis
vector is $\hat{\mathbf{v}}=(v^{r},rv^{\theta},r\sin\theta v^{\varphi})$.
Another basis vector $\hat{\mathbf{u}}$ is given by
$\hat{\mathbf{u}}_{s}=(0,-\hat{v}_{s}^{\varphi},\hat{v}_{s}^{\theta})$ at the
source position, and
$\hat{\mathbf{u}}_{d}=(0,\hat{v}_{d}^{\varphi},-\hat{v}_{d}^{\theta})$ at the
detector position. For simplification, we define
$n_{s}\equiv\hat{v}_{s}^{\varphi}/\hat{v}_{s}^{\theta}$,
$n_{d}\equiv\hat{v}_{d}^{\varphi}/\hat{v}_{d}^{\theta}$ and write them as
following form,

$n_{s}=\sin\theta_{s}\frac{+\cot\theta_{d}-\cot\theta_{s}\cos(\varphi_{d}-\varphi_{s})}{\sin(\varphi_{d}-\varphi_{s})}\quad\text{and}\quad
n_{d}=\sin\theta_{d}\frac{-\cot\theta_{s}+\cot\theta_{d}\cos(\varphi_{d}-\varphi_{s})}{\sin(\varphi_{d}-\varphi_{s})},$
(16)

to represent $\hat{\mathbf{v}}_{s}$ and $\hat{\mathbf{v}}_{d}$, equivalently.
The polarization tensors are then constructed as
$e_{+}^{ij}\equiv\hat{u}^{i}\hat{u}^{j}-\hat{v}^{i}\hat{v}^{j}$ and
$e_{\times}^{ij}\equiv\hat{u}^{i}\hat{v}^{j}+\hat{v}^{i}\hat{u}^{j}$ Maggiore
(2008), which are expressed explicitly as

$\displaystyle e_{+}^{ij}$
$\displaystyle=\left(\begin{array}[]{cc}(\hat{u}^{\theta})^{2}-(\hat{v}^{\theta})^{2}&\hat{u}^{\theta}\hat{u}^{\varphi}-\hat{v}^{\theta}\hat{v}^{\varphi}\\\
\hat{u}^{\theta}\hat{u}^{\varphi}-\hat{v}^{\theta}\hat{v}^{\varphi}&(\hat{u}^{\varphi})^{2}-(\hat{v}^{\varphi})^{2}\\\
\end{array}\right),\quad\text{and}\quad e_{\times}^{ij}$
$\displaystyle=\left(\begin{array}[]{cc}2\hat{u}^{\theta}\hat{v}^{\theta}&\hat{u}^{\theta}\hat{v}^{\varphi}+\hat{v}^{\theta}\hat{u}^{\varphi}\\\
\hat{u}^{\theta}\hat{v}^{\varphi}+\hat{v}^{\theta}\hat{u}^{\varphi}&2\hat{u}^{\varphi}\hat{v}^{\varphi}\end{array}\right),$
(17)

respectively. The polarization tensors are normalized and orthogonal to each
other, $e_{P}^{ij}e_{P^{\prime}}^{ij}=2\delta_{PP^{\prime}}$, where subscripts
${}_{P,P^{\prime}}=+,\times$. The metric $h^{ij}$ is decomposed into plus and
cross modes, $h^{ij}=h^{+}e_{+}^{ij}+h^{\times}e_{\times}^{ij}$. The
polarization modes are $h^{+}=(1/2)h^{ij}e_{ij}^{+}$ and
$h^{\times}=(1/2)h^{ij}e_{ij}^{\times}$, respectively. Rearranging it as a
matrix form, we obtain

$\left(\begin{array}[]{c}h^{+}\\\
h^{\times}\end{array}\right)=\bm{\mathcal{K}}\left(\begin{array}[]{c}h^{\theta\theta}\\\
h^{\theta\varphi}\end{array}\right)$ (18)

with projection matrix

$\bm{\mathcal{K}}=\left(\begin{array}[]{cc}(\hat{u}^{\theta})^{2}-(\hat{v}^{\theta})^{2}&\hat{u}^{\theta}\hat{u}^{\varphi}-\hat{v}^{\theta}\hat{v}^{\varphi}\\\
\hat{u}^{\theta}\hat{v}^{\theta}-\hat{u}^{\varphi}\hat{v}^{\varphi}&\hat{u}^{\varphi}\hat{v}^{\theta}+\hat{u}^{\theta}\hat{v}^{\varphi}\end{array}\right).$
(19)

The metric tensor $h^{ij}$ projects onto two basis tensors through the
$\bm{\mathcal{K}}$ matrix, and two polarization modes $h^{+}$ and $h^{\times}$
of GWs are obtained.

Combining transformation (14) and projection (18), one can obtain the rotation
relation of the polarization modes from the source to the detector,

$\begin{aligned} \left(\begin{array}[]{c}h^{+}_{d}\\\
h^{\times}_{d}\end{array}\right)=\bm{\mathcal{K}}_{d}\left(\begin{array}[]{c}h^{\theta\theta}_{d}\\\
h^{\theta\varphi}_{d}\end{array}\right)=\bm{\mathcal{K}}_{d}\bm{\mathcal{S}}\left(\begin{array}[]{c}h^{\theta\theta}_{s}\\\
h^{\theta\varphi}_{s}\end{array}\right)=\bm{\mathcal{K}}_{d}\bm{\mathcal{S}}\bm{\mathcal{K}}_{s}^{-1}\left(\begin{array}[]{c}h^{+}_{s}\\\
h^{\times}_{s}\end{array}\right)\end{aligned}.$ (20)

The rotation of a polarization plane (denoted by an angle $\chi_{g}$)
transforms the polarization modes of GWs by $h^{+}\rightarrow h^{+}\cos
2\chi_{g}-h^{\times}\sin 2\chi_{g}$ and $h^{\times}\rightarrow h^{+}\sin
2\chi_{g}+h^{\times}\cos 2\chi_{g}$ Maggiore (2008), where the factor $2$
comes from the spin-$2$ symmetry of the graviton. Thus, the production of
three matrices in (20) reduces to a rotation matrix,

$\bm{\mathcal{R}}\equiv\bm{\mathcal{K}}_{d}\bm{\mathcal{S}}\bm{\mathcal{K}}_{s}^{-1}=\left(\begin{array}[]{cc}\cos
2\chi_{g}&-\sin 2\chi_{g}\\\ \sin 2\chi_{g}&\cos 2\chi_{g}\end{array}\right),$
(21)

and the GFR angle $\chi_{g}$ of GWs is given by

$\chi_{g}=\frac{1}{1+x^{2}}\left(\frac{X-x}{1+X^{2}}-\frac{X}{1+X^{2}}x^{2}+Yx\right),$
(22)

where

$X\equiv-\frac{n_{s}+n_{d}}{1-n_{s}n_{d}}\quad{\rm and}\quad
Y\equiv\frac{(n_{s}+n_{d})^{2}}{(1+n_{s}^{2})(1+n_{d}^{2})}.$ (23)

The quantities $X$ and $Y$ depend on the choice of basis vectors
$\hat{\mathbf{u}}$ and $\hat{\mathbf{v}}$. The matrix $\bm{\mathcal{R}}$ is
real orthogonal, ensuring the equality
$(h_{s}^{+})^{2}+(h_{s}^{\times})^{2}=(h_{d}^{+})^{2}+(h_{d}^{\times})^{2}$,
i.e. the energy carried by GW is conserved during propagation.

We should mention that the WP theorem (3) is independent of the reference
frame, governing the GFR effect of light and GW rays. Therefore, the
polarization angle $\psi$ is also frame-dependent, describing the polarization
states of photons and gravitons. Thus, defining the two standard polarization
directions is necessary to investigate the initial and final polarization
states. Therefore, there are frame-dependent terms ($n_{s}$ and $n_{d}$) in
the final result (22). But we only focus on the GFR angle, the difference
between the initial and final polarization state, which is frame-independent.

In the WDL assumption, $\hat{\mathbf{k}}_{s}$ and $\hat{\mathbf{k}}_{d}$ are
approximately coplanar. For convenience, we follow the method proposed by Ref.
Ishihara  _et al._ (1988), requiring the $\hat{\mathbf{v}}$-axis to be
perpendicular to the orbital plane, or equivalently, the
$\hat{\mathbf{u}}$-axis to be in the orbital plane, as shown in Fig. 1. The
GFR effect disappears for a nonspinning lens. One will find that the angle
between the polarization direction and the $\hat{\mathbf{u}}$-axis is constant
during propagation. Thus, the initial and final polarization angles are equal,
and the GFR angle is zero. For the Kerr lens, the GFR angle is obtained in the
reference frame shown in Fig. 1. In another frame (for example, one requires a
nonzero angle between the $\hat{\mathbf{u}}$-axis and orbital plane), the
measured polarization angle is different. However, we can verify that the GFR
angle remains unchanged.

In order to prove the frame covariance of GFR angle, we rotate the
$(\hat{\mathbf{u}}-\hat{\mathbf{v}})$ plane around $\hat{\mathbf{k}}_{s}$ and
$\hat{\mathbf{k}}_{d}$ by an angle $\phi$. At this time, the symbols $n_{s}$
and $n_{d}$ will be transformed as follows,

$\displaystyle n_{s}\rightarrow n_{s}(\phi)=$
$\displaystyle\frac{\cos\phi[\cos\theta_{d}\sin\theta_{s}-\sin\theta_{d}\cos\theta_{s}\cos(\varphi_{d}-\varphi_{s})]+\sin\phi\sin\theta_{d}\sin(\varphi_{d}-\varphi_{s})}{-\sin\phi[\sin\theta_{s}\cos\theta_{d}-\sin\theta_{d}\cos\theta_{s}\cos(\varphi_{d}-\varphi_{s})]+\cos\phi\sin\theta_{d}\sin(\varphi_{d}-\varphi_{s})},$
(24) $\displaystyle n_{d}\rightarrow n_{d}(\phi)=$
$\displaystyle\frac{\cos\phi[\sin\theta_{s}\cos\theta_{s}-\cos\theta_{d}\sin\theta_{s}\cos(\varphi_{d}-\varphi_{s})]+\sin\phi\sin\theta_{s}\sin(\varphi_{d}-\varphi_{s})}{+\sin\phi[\sin\theta_{s}\cos\theta_{s}-\sin\theta_{s}\cos\theta_{s}\cos(\varphi_{d}-\varphi_{s})]-\cos\phi\sin\theta_{s}\sin(\varphi_{d}-\varphi_{s})}.$

According to (23), under the rotation transformation, $X$ and $Y$ will become

$X\rightarrow
X(\phi)=-\frac{(\cos\theta_{d}-\cos\theta_{s})\sin(\varphi_{d}-\varphi_{s})}{\cos\varphi_{d}\cos\varphi_{s}(\cos\theta_{d}\cos\theta_{s}-1)+\sin\varphi_{d}\sin\varphi_{s}(\cos\theta_{d}\cos\theta_{s}-1)+\sin\theta_{d}\sin\theta_{s}},$
(25)

and

$Y\rightarrow
Y(\phi)=\left[\frac{(\cos\theta_{d}-\cos\theta_{s})\sin(\varphi_{d}-\varphi_{s})}{\sin\theta_{d}\sin\theta_{s}\sin\varphi_{d}\sin\varphi_{s}+\sin\theta_{d}\sin\theta_{s}\cos\varphi_{d}\cos\varphi_{s}+\cos\theta_{d}\cos\theta_{s}-1}\right]^{2}.$
(26)

Eqs.(25) and (26) are independent of the rotation angle $\phi$, and are
exactly the same as those given in (23), $X(\phi)=X$ and $Y(\phi)=Y$.
Therefore, $X$ and $Y$ are frame-invariant. Furthermore, the GFR angle is also
frame-invariant.

###  IV.2 EMW case

In this subsection, we present the main results of the GFR of EMWs for
comparison. The unit transverse vector $\bm{\mathcal{E}}$ (5) is seen as an
electric vector of EMWs. The transformation of $\bm{\mathcal{E}}$ between
source and detector positions is given by (10). $\bm{\mathcal{E}}$ is
projected onto frame $(\hat{\mathbf{u}},\hat{\mathbf{v}})$ as shown in Fig. 1
and is decomposed as
$\bm{\mathcal{E}}=\mathcal{E}^{x}\hat{\mathbf{u}}+\mathcal{E}^{y}\hat{\mathbf{v}}$.
These two polarization modes are given by

$\displaystyle\left(\begin{array}[]{c}\mathcal{E}^{x}\\\
\mathcal{E}^{y}\end{array}\right)=\bm{\mathcal{N}}\left(\begin{array}[]{c}\mathcal{E}^{\theta}\\\
\mathcal{E}^{\varphi}\end{array}\right),\quad\bm{\mathcal{N}}\equiv\left(\begin{array}[]{cc}\hat{u}^{\theta}&\hat{u}^{\varphi}\\\
\hat{v}^{\theta}&\hat{v}^{\varphi}\end{array}\right),$ (27)

where the $r$-component has been ignored. Similar to GWs, the polarization
vector of EMWs has only two independent components
$(\mathcal{E}^{\theta},\mathcal{E}^{\varphi})$. The matrix $\bm{\mathcal{N}}$
projects $(\mathcal{E}^{\theta},\mathcal{E}^{\varphi})$ onto two basis
vectors, and gives two polarization modes $(\mathcal{E}^{x},\mathcal{E}^{y})$
of EMWs. Combining (27) and (10), the transformation between initial and final
polarization modes is obtained

$\displaystyle\left(\begin{array}[]{c}\mathcal{E}^{x}_{d}\\\
\mathcal{E}^{y}_{d}\end{array}\right)=\bm{\mathcal{N}}_{d}\left(\begin{array}[]{c}\mathcal{E}^{\theta}_{d}\\\
\mathcal{E}^{\varphi}_{d}\end{array}\right)=\bm{\mathcal{N}}_{d}\bm{\mathcal{T}}\left(\begin{array}[]{c}\mathcal{E}^{\theta}_{s}\\\
\mathcal{E}^{\varphi}_{s}\end{array}\right)=\bm{\mathcal{N}}_{d}\bm{\mathcal{T}}\bm{\mathcal{N}}_{s}^{-1}\left(\begin{array}[]{c}\mathcal{E}^{x}_{s}\\\
\mathcal{E}^{y}_{s}\end{array}\right).$ (28)

The production of these three matrices reduces to a rotation matrix,

$\bm{\mathcal{N}}_{d}\bm{\mathcal{T}}\bm{\mathcal{N}}_{s}^{-1}\equiv\left(\begin{array}[]{cc}\cos\chi_{e}&-\sin\chi_{e}\\\
\sin\chi_{e}&\cos\chi_{e}\end{array}\right).$ (29)

This rotation matrix (29) relates the initial and final state of the lensed
EMW. The change between the initial and final states is a rotation of the
polarization vector in the $(\hat{\mathbf{u}}-\hat{\mathbf{v}})$ plane, that
is, the GFR effect. In the rotation matrix (29), $\chi_{e}$ represents the GFR
angle of the lensed EMWs. Compared with the GW case, the rotation angle of EMW
is two times smaller than that of GWs since the photons are spin-1 particles.
Eq.(29) gives the GFR angle $\chi_{e}$,

$\chi_{e}=\frac{1}{\sqrt{1+x^{2}}}\frac{X-x}{\sqrt{1+X^{2}}}.$ (30)

This result is different from the GFR angle of GWs (22). Interestingly, the
GFR angle of the lensed EMWs and GWs have the same results under the WDL
approximation.

##  V Results

This section aims to determine the value of $x$, $X$, and $Y$ and then
predicts the GFR angle of GWs induced by a Kerr BH. The source, the lens BH,
and the detector are well aligned in a strong lensing scenario. We define the
angular shifts of the gravitons as

$\Delta\theta\equiv\theta_{d}+\theta_{s}-\pi\quad\text{and}\quad\Delta\varphi\equiv\varphi_{d}-\varphi_{s}-\pi.$
(31)

These two quantities can be fully determined by the two motion constants,
$\xi$ and $\eta$, and the final angular position of graviton relative to Kerr
BH, $\theta_{d}$. In this study, as the impact parameter of graviton to the
lens BH, $r_{m}$, is much larger than the gravitational radius of the BH, i.e.
$\tilde{M}\equiv M/r_{m}^{(0)},\tilde{a}\equiv a/r_{m}^{(0)}\sim\epsilon\ll
1$, where $r_{m}^{(0)}=\sqrt{\eta+\xi^{2}}$ is the zero-order approximation of
$r_{m}$. Thus, several quantities are small: $\Delta\theta,\Delta\varphi$,
$r_{m}^{(0)}/r_{s}$, and $r_{m}^{(0)}/r_{d}$, which allows us to integrate
geodesic equations in the WDL. The magnitudes of $r_{m}^{(0)}/r_{s}$ and
$r_{m}^{(0)}/r_{d}$ are denoted by $\nu$, which is different from $\epsilon$.
Without loss of generality, we assume the two have a similar order of
magnitude, $\nu\sim\epsilon$.

We consider a static detector and a static GW source in the asymptotically
flat region of a Kerr BH. The geometric relation between source and detector
positions, $(r_{s},\theta_{s},\varphi_{s})$ and
$(r_{d},\theta_{d},\varphi_{d})$, is derived from integrals

$\int\frac{dr}{\pm\sqrt{R(r)}}=\int\frac{d\theta}{\pm\sqrt{\Theta(\theta)}}$
(32)

and

$\varphi_{d}-\varphi_{s}=\int\frac{2Mar-a^{2}\xi}{\pm\Delta\sqrt{R(r)}}dr+\int\frac{\xi\csc^{2}\theta}{\pm\sqrt{\Theta(\theta)}}d\theta,$
(33)

where the radial function
$R(r)=r^{4}+(a^{2}-\xi^{2}-\eta)r^{2}+2Mr[\eta+(\xi-a)^{2}]-a^{2}\eta$, and
the angular function
$\Theta(\theta)=\eta+(a^{2}-\xi^{2}\csc^{2}\theta)\cos^{2}\theta$. The
integral path is $r_{s}\rightarrow r_{m}\rightarrow r_{d}$ and
$\theta_{s}\rightarrow\theta_{m}\rightarrow\theta_{d}$, with $r_{m}$ being the
largest zero point of function $R(r)$, pericenter radius of gravitons paths.
And $\theta_{m}=\theta_{\min/\max}$ is a root of $\Theta(\theta)=0$, which
corresponds to the maximum or the minimum of the $\theta$ coordinate on the
trajectory. Up to the order of $\epsilon^{3}$, the approximated solutions to
$r_{m}$ and $\theta_{m}$ are

$r_{m}\approx
r^{(0)}_{m}\left(1-\tilde{M}-\frac{3}{2}\tilde{M}^{2}+2\tilde{M}\tilde{a}\tilde{\xi}-\frac{1}{2}\tilde{a}^{2}\tilde{\xi}^{2}-4\tilde{M}^{3}+6\tilde{M}^{2}\tilde{a}\tilde{\xi}-2\tilde{M}\tilde{a}^{2}\tilde{\xi}^{2}\right)$
(34)

and

$\cos\theta_{m}\approx\pm\tilde{\eta}^{1/2}\left(1+\frac{1}{2}\tilde{a}^{2}\tilde{\xi}^{2}\right),$
(35)

where the dimensionless motion constants $\tilde{\xi}$ and $\tilde{\eta}$ are
$\tilde{\xi}\equiv\xi/r_{m}^{(0)}$ and
$\tilde{\eta}\equiv\eta/[r_{m}^{(0)}]^{2}$.

Following the method provided by Bray (1986), we perform the $r$-integral and
$\theta$-integral (32) up to $\epsilon^{3}$. The results are

$\displaystyle\int_{r_{s}}^{r_{d}}\frac{|dr|}{\sqrt{R(r)}}$
$\displaystyle\approx\frac{1}{r_{m}^{(0)}}\left(1+\frac{1}{2}\tilde{a}^{2}\right)\left[\pi\left(1-\frac{3}{4}\tilde{a}^{2}\tilde{\eta}+\frac{15}{4}\tilde{M}^{2}-15\tilde{M}^{2}\tilde{a}\tilde{\xi}\right)\right.$
(36)
$\displaystyle\left.+4\tilde{M}-8\tilde{M}\tilde{a}\tilde{\xi}+\frac{128}{3}\tilde{M}^{3}+10\tilde{M}\tilde{a}^{2}-16\tilde{M}\tilde{a}^{2}\tilde{\eta}\right]-\frac{1}{r_{m}^{(0)}}\left(\mathcal{W}+\mathcal{Q}\right),$

and

$\int_{\theta_{s}}^{\theta_{d}}\frac{|d\theta|}{\sqrt{\Theta(\theta)}}\approx\frac{1}{r_{m}^{(0)}}\left(1-\frac{3}{4}\tilde{a}^{2}\tilde{\eta}+\frac{1}{2}\tilde{a}^{2}\right)\times\left\\{\pi\mp\arctan\left[\left(1+\frac{1}{4}\tilde{a}^{2}\tilde{\eta}\right)\cot\sigma_{s}\right]\mp\arctan\left[\left(1+\frac{1}{4}\tilde{a}^{2}\tilde{\eta}\right)\cot\sigma_{d}\right]\right\\},$
(37)

respectively. In (37), the rescaled angular coordinate is
$\cos\sigma\equiv\cos\theta/|\cos\theta_{m}|$. The upper sign in (37)
corresponds to the ray that passes through $\theta_{\min}$, and the lower sign
corresponds to $\theta_{\max}$. The new symbols $\mathcal{W}$ and
$\mathcal{Q}$ in (36) are defined as

$\mathcal{W}=\frac{r_{m}^{(0)}}{r_{s}}+\frac{r_{m}^{(0)}}{r_{d}},\quad\text{and}\quad\mathcal{Q}=\frac{1}{6}\left[\left(\frac{r_{m}^{(0)}}{r_{s}}\right)^{3}+\left(\frac{r_{m}^{(0)}}{r_{d}}\right)^{3}\right].$
(38)

The last term in (36) is absent in Ref. Ishihara  _et al._ (1988) since they
have assumed that the source and detector are far from the lens object,
equivalently, $\epsilon\gg\nu$.

Inserting the $r$-integral (36) and $\theta$-integral (36) into (32), the
relationship between $\theta_{d}$ and $\theta_{s}$ is obtaind,

$\cos\theta_{s}\approx-\cos\delta\cos\theta_{d}\mp\sin\delta\left\\{\mu+\tilde{a}^{2}\left[\frac{1}{2\mu}\tilde{\eta}\tilde{\xi}^{2}+\frac{\mu}{4}\left(\tilde{\eta}-2\mu^{2}\right)\right]\right\\},$
(39)

where $\mu$ is defined as $\mu=\sqrt{\tilde{\eta}-\cos\theta_{d}}$ and
$\delta$ is

$\displaystyle\delta=4\tilde{M}+\frac{15\pi}{4}\tilde{M}^{2}-8\tilde{M}\tilde{a}\tilde{\xi}+\frac{128}{3}\tilde{M}^{3}-15\pi\tilde{M}^{2}\tilde{a}\tilde{\xi}-13\tilde{M}\tilde{a}^{2}\tilde{\eta}+10\tilde{M}\tilde{a}^{2}-\left(1-\frac{1}{2}\tilde{a}^{2}+\frac{3}{4}\tilde{a}^{2}\tilde{\eta}\right)\mathcal{W}-\mathcal{Q}.$
(40)

Except for the last two terms, this definition is completely consistent with
that provided by Bray (1986); Ishihara  _et al._ (1988). From (39) and (33),
the angular shifts of the gravitons (31) now become

$\displaystyle\Delta\theta\approx$
$\displaystyle\pm\mu\csc\theta_{d}\delta-\frac{1}{2}\tilde{\xi}^{2}\cot\theta_{d}\csc^{2}\theta_{d}\delta^{2}$
(41)
$\displaystyle\qquad\pm\frac{1}{4\mu}\csc\theta_{d}\left(2\tilde{\eta}\tilde{\xi}^{2}+\tilde{\eta}\mu^{2}-2\mu^{4}\right)\tilde{a}^{2}\delta\pm\frac{\mu}{6}\delta^{3}\tilde{\xi}^{2}\csc^{5}\theta_{d}\left(2\mu^{2}+2\tilde{\xi}^{2}-3\right),$

and

$\displaystyle\Delta\varphi$
$\displaystyle=4\tilde{M}\tilde{a}-8\tilde{M}\tilde{a}^{2}\tilde{\xi}+5\pi\tilde{M}^{2}\tilde{a}+\delta\tilde{\xi}\csc^{2}\theta_{d}\pm\delta^{2}\mu\tilde{\xi}\cot\theta_{d}\csc^{3}\theta_{d}+\frac{1}{2}\tilde{a}^{2}\tilde{\xi}\delta$
(42)
$\displaystyle\qquad+\frac{1}{4}\tilde{a}^{2}\delta\Bigg{[}\left(-3\tilde{\eta}^{3}+4\tilde{\eta}^{2}\mu^{2}+6\tilde{\eta}^{2}+\tilde{\eta}\mu^{4}-6\tilde{\eta}\mu^{2}-3\tilde{\eta}-2\mu^{6}+2\mu^{2}\right)\tilde{\xi}\csc^{6}\theta_{d}$
$\displaystyle\qquad\qquad+\frac{1}{2}\left(-\tilde{\eta}^{2}+\tilde{\eta}\mu^{2}+\tilde{\eta}-2\mu^{2}\right)\tilde{\xi}\csc^{4}\theta_{d}\Bigg{]}+\frac{1}{3}\delta^{3}\left(\tilde{\eta}^{2}+\tilde{\eta}\mu^{2}-\tilde{\eta}-2\mu^{4}+2\mu^{2}\right)\tilde{\xi}\csc^{6}\theta_{d}.$

Approximately, we have $\Delta\theta$, $\Delta\varphi\sim\epsilon$ from (41)
and (42) Ishihara  _et al._ (1988); Bray (1986). So far, we have completed the
calculation of the null geodesics equations of gravitons. The angular shifts
are determined by the dimensionless motion constants and detector position.

The quantities $x$ (13), $X$, and $Y$ (23) are expressed in angular positions
of the source and detector, $(\theta_{s},\varphi_{s})$ and
$(\theta_{d},\varphi_{d})$, where $\varphi_{d}$ can be set as $0$ since Kerr
BH has an axisymmetric spacetime. The quantities $x$, $X$, and $Y$ can then be
expressed as functions of $\theta_{d}$ and angular shift, $\Delta\theta$ and
$\Delta\varphi$. We use the angular shifts of gravitons, $\theta_{d}$,
$\Delta\theta$ (41), and $\Delta\varphi$ (42) for $x$ (13), $X$, and $Y$ (23).
Up to the order of $\epsilon^{3}$, they are given by

$\displaystyle x$
$\displaystyle\approx\pm\Bigg{[}\frac{\tilde{\xi}}{\mu}\cot\theta_{d}\Delta\theta+\frac{1}{\mu}\cos\theta_{d}\sin\theta_{d}\tilde{a}\Delta\theta-\frac{1}{2}\tilde{\xi}\left(\frac{\tilde{\xi}^{4}}{\mu^{3}}-\frac{\tilde{\xi}^{2}}{\mu^{3}}+\frac{2\tilde{\xi}^{2}}{\mu}+\mu-\frac{2}{\mu}\right)\csc^{2}\theta_{d}\Delta\theta^{2}$
(43)
$\displaystyle\qquad+\frac{1}{2}\left(\frac{\tilde{\xi}^{3}}{\mu^{3}}-\frac{\tilde{\xi}}{\mu^{3}}+\frac{\tilde{\xi}}{\mu}\right)\sin\theta_{d}\cos\theta_{d}\tilde{a}^{2}\Delta\theta-\frac{1}{2}\left(\frac{\tilde{\xi}^{4}}{\mu^{3}}-\frac{\tilde{\xi}^{2}}{\mu^{3}}-\mu\right)\tilde{a}\Delta\theta^{2}$
$\displaystyle\qquad\qquad-\left(\frac{\tilde{\xi}^{5}}{2\mu^{5}}-\frac{\tilde{\xi}^{3}}{2\mu^{5}}+\frac{\tilde{\xi}^{3}}{\mu^{3}}-\frac{\tilde{\xi}}{\mu^{3}}+\frac{\tilde{\xi}}{6\mu}\right)\cot\theta_{d}\Delta\theta^{3}\Bigg{]},$
$X\approx\cos\theta_{d}\Delta\varphi+\frac{1}{2}\sin\theta_{d}\Delta\theta\Delta\varphi\\\
+\frac{1}{12}\left(1+3\cos^{2}\theta_{d}\right)\cos\theta_{d}\Delta\varphi^{3},$
(44)

and

$Y\approx\cos^{2}\theta_{d}\Delta\varphi^{2}+\sin\theta_{d}\cos\theta_{d}\Delta\theta\Delta\varphi^{2}.$
(45)

In $x$’s expression (43), the $\pm$ symbol comes from the definition of
$\beta$. Gravitons pass the maximum or minimum position of $\theta$,
corresponding to the upper or lower sign.

In (22) and (30), the GFR angle of lensed GWs and EMWs are expressed as
functions of symbols $x$, $X$, and $Y$. Inserting (43), (44), and (45) into
(22), up to $\epsilon^{3}$, the GFR angle is finally obtained,

$\chi_{g}=\tilde{a}\mathcal{W}\cos\theta_{d}+\frac{5}{4}\pi\tilde{M}^{2}\tilde{a}\cos\theta_{d}\pm\mu\left(2\tilde{M}\tilde{a}\mathcal{W}-\frac{1}{2}\tilde{a}\mathcal{W}^{2}\right).$
(46)

The first term is of order $\epsilon^{2}$, while the last three are of order
$\epsilon^{3}$. For $\nu\gtrsim\epsilon$, $\tilde{a}\mathcal{W}\cos\theta_{d}$
is the leading term of GFR angle. Since the dragging effect of Kerr BH causes
GFR, the rotation angle is proportional to the angular momentum of the BH,
where $\cos\theta_{d}$ accounts for the projection along the L.O.S. When
$\cos\theta_{d}<0$ ($>0$), both the BH and the polarization plane rotate
clockwise (anticlockwise). In the leading and subleading order of
approximations, we find the same GFR angle of EMWs. For the $\nu\ll\epsilon$
case, the first third-order term,
$(5/4)\pi\tilde{M}^{2}\tilde{a}\cos\theta_{d}$, will be the leading term,
which has been given in Ref. Ishihara  _et al._ (1988). Additionally, our GFR
angle (46) is frequency-independent because of geometric optical
approximation.

The magnitude of the GFR angle can be estimated from (46). We assume that the
mass of a supermassive BH is $M\sim 10^{9}M_{\odot}\sim 10{\rm AU}$, angular
momentum along L.O.S. is $a\cos\theta_{d}\sim 0.9M$ and the BH-source distance
is $r_{s}\sim 10^{3}{\rm AU}$. Let the pericenter of the graviton path be as
close as possible to the event horizon without exceeding the weak deflection
limit, e.g., $\sim 10^{2}{\rm AU}$. In this case, the condition
$\nu\sim\epsilon$ is satisfied, and the GFR angle can reach $0.5\ {\rm deg}$,
which is close to the detection sensitivity of the third-generation GW
detector network to polarization angle $\psi$.

Additionally, substituting (43) and (44) into (30), we derive the GFR angle of
EMWs after a similar calculation. The result of EMW’s GFR angle is same as the
GW case, i.e., $\chi_{e}=\chi_{g}$, which is the consequence of geometrical
optics approximation. Ref. Ishihara  _et al._ (1988) has studied such effect,
but they only obtained the first third-order term of our result (46). Because
they assume that , the parameter $\nu$ is much smaller than $\epsilon$, this
assumption is not necessary for the real situation. By setting
$\nu\ll\epsilon$ (equivalently, $\mathcal{W}\ll 1$), we reproduce the same
result in Ref. Ishihara  _et al._ (1988).

##  VI Imprints in lensing GW waveform

There are some GFR imprints in gravitationally-lensed GW signals. In the
geometric optical approximation, the lensing GW signal is calculated through

$\tilde{h}^{+,\times}_{\rm len}(f)=F(f)\ \tilde{h}^{+,\times}_{\rm unlen}(f),$
(47)

where $\tilde{h}$ is the frequency-domain waveform, and $f$ is the frequency
of GW without cosmological redshift. The lensed and unlensed waveform is
denoted by subscripts ‘len’ and ‘unlen’. The complex transmission factor
$F(f)$ is a useful tool to describe the wave effects, amplitude enhancement,
and phase shift of gravitationally-lensed GWs. The Kirchhoff integral is
derived from the Huygens principle, and depends on the time delay of all
possible paths of photons or gravitons. The transmission factor is calculated
from Kirchhoff integral Schneider  _et al._ (1993); Guo and Lu (2020); Dai
_et al._ (2018); Takahashi and Nakamura (2003),

$F(f)\equiv\frac{f(1+z_{L})}{i}\frac{d_{L}d_{S}}{d_{LS}}\theta_{E}^{2}\int
d^{2}\bm{\xi}e^{i2\pi f(1+z_{L})\tau(\bm{\xi},\bm{\eta})},$ (48)

where $\bm{\xi}$ is the 2-dimensional coordinate on the lens plane and
$\bm{\eta}$ is the 2-dimensional coordinate of the source object on the source
plane. $d_{S}$, $d_{L}$, and $d_{LS}$ are the angular diameter distances from
the observer to the source, to the lens, and from the lens to the source,
respectively. In (48), $\theta_{E}$ is the Einstein deflection angle, and
$z_{L}$ is the cosmological redshift of the lens object.
$\tau(\bm{\xi},\bm{\eta})$ is the time delay of a photon or graviton emitted
from $\bm{\eta}$ passing through $\bm{\xi}$, and finally arriving at detector.
The exact definition of time delay is given by Schneider  _et al._ (1993); Guo
and Lu (2020); Dai  _et al._ (2018); Takahashi and Nakamura (2003)

$\tau(\bm{\xi},\bm{\eta})\equiv\frac{d_{L}d_{S}}{d_{LS}}\theta_{E}^{2}\left[\frac{1}{2}(\bm{\xi}-\bm{\eta})^{2}-\phi(\bm{\xi})\right].$
(49)

$\phi(\bm{\xi})$ is called the lens potential and can be generally determined
from the surface mass density of the lens object. The nonspinning point-like
mass assumption is not necessary. The readers can find more details in
Schneider  _et al._ (1993). For example, the Schwarzschild lens is considered
in Takahashi and Nakamura (2003) and the Kerr lens in Baraldo  _et al._
(1999).

In the geometrical optics limit, the $j$-th ($j=1,2$ for point-mass lens)
image position $\bm{\xi}_{j}$ is determined by the lens equation,
$\bm{\nabla}_{\bm{\xi}}\tau(\bm{\xi},\bm{\eta})=0$. Therefore, the
transmission factor $F(f)$ is contributed by each image and is expressed as

$F(f)=\sum_{j}F_{j}(f)=\sum_{j}\sqrt{|\mu_{j}|}e^{2\pi if\tau_{j}-i\pi
n_{j}},\quad\tau_{j}\equiv\tau(\bm{\xi}_{j},\bm{\eta})$ (50)

whereas the Morse indices are $n_{j}=0,1/2,1$ when $\bm{\xi}_{j}$ is a
minimum, saddle and maximum of the time-delay function
$\tau(\bm{\xi},\bm{\eta})$, respectively. The magnifications $\mu_{j}$ is
defined as
$\mu_{j}\equiv[1/\det(\partial\bm{\xi}/\partial\bm{\eta})]_{\bm{\xi}=\bm{\xi}_{j}}$.
Specially, the point-mass lens forms two images in the geometrical optics
limit. For a point-mass lens, when $|\tau_{1}-\tau_{2}|$ is large enough, the
amplitude and phase of $F_{j}(f)$, i.e., $\sqrt{|\mu_{j}|}$ and $2\pi
if\tau_{j}-\pi n_{j}$, describe the amplitude enhancement and phase shift of
lensed GW relative to that of the unlensed one. This framework describes
amplitude enhancement, phase shift, interference, and diffraction. However,
the polarization effect of the lensed GW is not included. If taking the GFR
effect into consideration, transformation (47) becomes

$\displaystyle\tilde{h}^{+}_{j,{\rm len}}(f)$
$\displaystyle=F_{j}(f)\left[\tilde{h}^{+}_{j,{\rm unlen}}(f)\cos
2\chi_{j}-\tilde{h}^{\times}_{j,{\rm unlen}}(f)\sin 2\chi_{j}\right],$ (51)
$\displaystyle\tilde{h}^{\times}_{j,{\rm len}}(f)$
$\displaystyle=F_{j}(f)\left[\tilde{h}^{+}_{j,{\rm unlen}}(f)\sin
2\chi_{j}+\tilde{h}^{\times}_{j,{\rm unlen}}(f)\cos 2\chi_{j}\right],$

or equivalently

$\tilde{h}^{L,R}_{j,{\rm len}}(f)=F_{j}(f)\ \tilde{h}^{L,R}_{j,{\rm
unlen}}(f)\exp\left(-i\cdot 2\chi_{j}\right),$ (52)

in terms of left-hand and right-hand polarization modes. We abbreviate the GFR
angle of $j$-th image as $\chi_{j}$. The left-hand and right-hand polarization
modes are defined as $h^{L,R}\equiv(1/\sqrt{2})(h^{+}\pm ih^{\times})$
Maggiore (2008). As shown in (51) or (52), the lensing effects can be divided
into two parts. Firstly, the unlensed frequency-domain GW waveform will be
multiplied by a complex factor $F(f)$, which contains amplitude enhancement
and phase shift effects. Secondly, the polarization plane rotates by an angle
$\chi_{g}$ caused by the GFR effect. Our work gives the theoretical prediction
of the GFR angle (46).

The unlensed signal received by detectors can be written as a linear
combination of $+$ and $\times$ polarization modes,

$\tilde{h}_{\rm unlen}=F_{+}(\alpha,\lambda,\psi)\tilde{h}_{\rm
unlen}^{+}+F_{\times}(\alpha,\lambda,\psi)\tilde{h}^{\times}_{\rm unlen}.$
(53)

$F_{+,\times}(\alpha,\lambda,\psi)$ is the pattern function of GW detectors
Maggiore (2008) for $+$ and $\times$ modes. $(\alpha,\lambda)$ are the right
ascension and declination of GW sources, and $\psi$ is the so-called
polarization angle. The polarization angle is defined in terms of the
direction of orbital angular moment of binary $\hat{\mathbf{L}}$ and GW
propagation $\hat{\mathbf{N}}$ Jaranowski  _et al._ (1998). However, this
definition is inconvenient for lensing system. Instead, we define the
polarization angle using the basis vectors
$(\hat{\mathbf{u}},\hat{\mathbf{v}})$ shown in Fig. 1. Thus, the lensed GW
signal is given by

$\displaystyle\tilde{h}_{j,{\rm len}}$ $\displaystyle=\tilde{h}_{j,{\rm
len}}^{+}F_{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},\psi)+\tilde{h}_{j,{\rm
len}}^{\times}F_{\times}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},\psi)$ (54)
$\displaystyle=F_{j}(f)\Big{\\{}(\tilde{h}_{j,{\rm unlen}}^{+}\cos
2\chi_{j}-\tilde{h}_{j,{\rm unlen}}^{\times}\sin
2\chi_{j})[F_{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\cos 2\psi-
F_{\times}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\sin 2\psi]$
$\displaystyle\qquad+(\tilde{h}_{j,{\rm unlen}}^{+}\sin
2\chi_{j}+\tilde{h}_{j,{\rm unlen}}^{\times}\cos
2\chi_{j})[F_{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\sin
2\psi+F_{\times}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\cos
2\psi]\Big{\\}}$ $\displaystyle=F_{j}(f)\Big{\\{}\tilde{h}_{j,{\rm
unlen}}^{+}[F_{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\cos
2(\psi-\chi_{j})+F_{\times}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\sin
2(\psi-\chi_{j})]$ $\displaystyle\qquad+\tilde{h}_{j,{\rm
unlen}}^{\times}[F_{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\sin
2(\psi-\chi_{j})F_{\times}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},0)\cos
2(\psi-\chi_{j})]\Big{\\}}$ $\displaystyle=F_{j}(f)\left[\tilde{h}_{j,{\rm
unlen}}^{+}F^{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},\psi-\chi_{j})+\tilde{h}_{j,{\rm
unlen}}^{\times}F^{+}(\alpha^{\prime}_{j},\lambda^{\prime}_{j},\psi-\chi_{j})\right].$

In the second step, we use the properties of pattern functions,
$F_{+}(\alpha,\lambda,\psi)=F_{+}(\alpha,\lambda,0)\cos 2\psi-
F_{\times}(\alpha,\lambda,0)\sin 2\psi$ and
$F_{\times}(\alpha,\lambda,\psi)=F_{+}(\alpha,\lambda,0)\sin
2\psi+F_{\times}(\alpha,\lambda,0)\cos 2\psi$. The massive lens object
deflects the path of gravitons, changing the direction of the incoming GW
signal from $(\alpha,\lambda)$ to $(\alpha^{\prime}_{j},\lambda^{\prime}_{j})$
on the sky plane. From (54) we can find that the effect of GFR is equivalent
to a change in the polarization angle $\psi$ in the pattern function,
$\psi\rightarrow\psi-\chi_{g}$. Additionally, the total amplitude of the
lensed GW signal at detectors is proportional to $|F_{j}|/D_{L}$, where
$D_{L}$ is the luminosity distance of the GW source. The total phase is
$\phi_{c}+\arg(F_{j})$, where $\phi_{c}$ is the coalescence phase of the
binary system. Therefore, lensing amplification and the GW source distance are
degenerate, the phase shift and initial phase $\phi_{c}$ are degenerate
Ezquiaga  _et al._ (2021). Similarly, the GFR angle and initial polarization
angle $\psi$ are degenerate.

We need to find a way to break the degeneracy between the initial polarization
angle and the GFR angle. In the geometrical optics limit, the lens focuses the
original GW from several paths toward an observer, forming multiple images
Baraldo  _et al._ (1999); Sereno and De Luca (2006). Every path’s pericenter
radius $r_{m}$ is usually different. The observed GFR effects from each image
are distinguished. A Kerr lens forms two images on the lens plane in the
geometrical optical approximation. The observed values of the polarization
angle of the lensed GW from each image will be $\psi_{j}=\psi-\chi_{j}$. These
two GFR angles are different because of different graviton paths. Therefore,
the difference in GFR angles is obtained,

$\Delta\chi_{g}=\chi_{2}-\chi_{1}=\psi_{2}-\psi_{1}.$ (55)

From the GFR angle (46), we find that $\Delta\chi_{g}$ is a third-order
quantity, $\sim\mathcal{O}(\epsilon^{3})$. The first term in (46),
$\tilde{a}\mathcal{W}\cos\theta_{d}=a(1/r_{s}+1/r_{d})\cos\theta_{d}$, is
independent of the pericenter radius $r_{m}^{(0)}$. The second-order
corrections cancel each other out in $\Delta\chi_{g}$, and all of the three
three-order terms survive. Therefore, by observing the images of lensed GW
rays, one can obtain $\Delta\chi_{g}$ and estimate the spin of the lens black
hole accordingly. However, such a method brings more difficulties. Firstly, it
requires sophisticated modeling of gravitational lensing. Secondly, it makes
the leading-order term vanish, and one has to compare the tiny difference
between multiple images, which requires highly accurate measurement. We
consider a supermassive BH and a GW source system similar to that in the
previous section. The GW signals form two images on the lens plane. The impact
distances of these two graviton orbits are assumed to be $10^{2}{\rm AU}$ and
$1.1\times 10^{2}{\rm AU}$. Generally, the difference between the two impact
distances should not be too large because the GW rays with a significant
impact distance will be too dim to be detected. Only considering the first
three-order term, the subleading terms of the GFR angle of the two images are
about $0.2\ {\rm deg}$ and $0.15\ {\rm deg}$, respectively. As a result, the
GFR angle difference is $0.05\ {\rm deg}$, which is beyond the detectability
of the third-generation GW detector or networks.

There are other potential methods to address this problem, such as considering
the motion of lens objects or background GW sources. One can consider that the
initial polarization angle of GWs does not change with the motion of the
source. However, the propagation path of the GW and, thus, the GFR angle
changes, when the lens object or GW source moves. Therefore, it is possible to
detect the polarization angle changing with time from the lensed GW signals to
obtain characteristics of the GFR effect. However, this method needs to
consider the moving GW source and lens object, and solving the geodesic
equation will become more challenging.

The actual measurement of the GFR angle helps reveal the relevant
characteristics of spinning objects. Breaking the degeneracy is of great
significance for the observation of GFR. More works are needed in this field
for in-depth discussion.

##  VII Conclusion and Discussion

The gravitationally-lensed GWs can be successfully described by a transmission
factor, evaluated from the Kirchhoff integral Takahashi and Nakamura (2003);
Meena and Bagla (2019); Born and Wolf (1959); Baraldo  _et al._ (1999); Guo
and Lu (2020). This framework includes the effects of amplitude enhancement,
phase shift, interference, and diffraction, but the GFR effect is absent. In
this work, by applying the Walker-Penrose theorem, we derive an explicit
result of the GFR angle of GWs lensed by a Kerr BH (46) in the geometric
optical approximation and WDL assumption.

The GFR effect describes the mixture of original GW polarization modes, is an
important correction to the polarization angle of gravitationally-lensed GW
signals. It is a frame-dragging effect sourced by the angular momentum of
spinning lens objects. The GFR of the lensed GW was not considered in previous
works because of its weak effect. However, in this work, we find that the GFR
angle is a second-order correction of the polarization angle rather than a
third-order correction, as predicted in previous work. The GFR angle of the GW
signal lensed by a central supermassive BH in a galaxy has been estimated in
Section V ($\sim 0.5\ {\rm deg}$). On the other hand, Fisher estimation for
the standard deviation of polarization angle $\Delta\psi$ of the stellar-mass
binary is about $0.6\ {\rm deg}$ for HLV/ET/B-DECIGO network Grimm and Harms
(2020). This kind of system has been predicted by Graham  _et al._ (2020), and
its GFR angle $\chi_{g}$ is close to the detection capability of the third-
generation GW detector network. However, we have to break the degeneracy
between the GFR and polarization angles.

As well known, the amplitude enhancement factor of lensed GWs and the
luminosity distance of the GW source are degenerate, the phase shift and the
coalescence phase of compact binary coalescence of lensed GWs are degenerate
Ezquiaga  _et al._ (2021). Similarly, the GFR angle and initial polarization
angle $\psi$ are also degenerate. Compared with an unlensed signal, only the
polarization angle changes from $\psi$ to $\psi-\chi_{g}$ for lensed signals.
Theoretically, one can break the degeneracy between the GFR and the initial
polarization angle from the multiple lensed images, although higher-accuracy
observations is required.

The GFR effect is sourced from the angular momentum of spinning lens objects
and changes the initial polarization state of lensed GW signals. As a result,
this study may help future research measure the spin of BHs. Faraday rotation
of EMW depends on both the magnetic field and the rotating gravitational field
during propagation. However, the GFR of GW is only affected by the
gravitational field. Therefore, this effect is suitable for measuring the BH
or galaxy spin. Similarly, lensed EMWs and GWs from a binary system with an
electromagnetic counterpart provide a potential method to measure the magnetic
field along the line of sight. It is anticipated that many lensed GW signals
will be detected in the future Li  _et al._ (2018); Piórkowska  _et al._
(2013); Hannuksela  _et al._ (2019); Lo and Hernandez (2021), which will help
measure the spin of more supermassive BHs and galaxies statistically and test
the Universe’s chiral symmetry Yu  _et al._ (2020).

_Acknowledgments._ We would like to thank Youjun Lu, Yefei Yuan, Xilong Fan,
Shaoqi Hou and Aoxiang Jiang for helpful discussions and comments. This work
is supported by the National Key R&D Program of China Grant No.
2021YFC2203100, NSFC No. 12273035, 11773028, 11633001, 11653002, 11903030,
11873006 the Fundamental Research Funds for the Central Universities under
Grant No. WK2030000036 and WK3440000004, Key Research Program of the Chinese
Academy of Sciences, Grant No. XDPB15, and the science research grants from
the China Manned Space Project with No.CMS-CSST-2021-B01, CMS-CSST-2021-B11
and CMS-CSST-2021-A12.

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