# Spin optics for gravitational waves

Pravin Kumar Dahal<EMAIL_ADDRESS>School of Mathematical &
Physical Sciences, Macquarie University

###### Abstract

Abstract: We present the geometric optics expansion for circularly polarized
gravitational waves on a curved spacetime background, to subleading order. We
call spin optics to the subleading order geometric optics expansion, which
involves modifying the standard eikonal function by including a specially
chosen helicity-dependent correction. We show that the techniques developed
for the propagation of electromagnetic waves can also be applied to
gravitational waves in the limit of spin optics. However, one needs to account
for the difference in the photon and graviton helicity, which we do here.

Geometric optics; spin Hall effect; Gravitational Faraday rotation; Spin
optics.

##  I Introduction

The geometric optics approximation can be applied to study high-frequency
gravitational waves. It states that in the infinitely large frequency limit,
gravitational waves in the fixed curved background behave as null ray
trajectories, similar to electromagnetic waves [1, 2, 3]. The geometric optics
expansion reduces the problem of solving the linearized Einstein field
equations to solving the ray and transport equations along these rays. In the
geometric optics limit, the backreaction onto the ray equations from the
polarization vector is absent. Because of this, the laws of geometric optics
are no longer valid at large but finite frequencies.

The backreaction from helicity might cause ray trajectories to deviate from
geodesics appreciably. This effect, in which the propagation of gravitational
waves in the subleading order in curved spacetime (orbital motion) depends on
the helicity (spin), is also called the gravitational spin Hall effect [4, 5,
6, 7]. Thus, the spin-orbit interaction modifies the gravitational wave
propagation from the original ray trajectory as determined by geometric
optics. Here, we present the covariant formulation of the Wentzel-Kramers-
Brillouin (WKB) analysis for polarized gravitational waves in the subleading
order geometric optics expansion in wavelength. We call the “spin optics”
approximation to the WKB analysis up to subleading order in the geometric
optics approximation. The procedure here will be similar to that developed for
electromagnetic waves [5, 8].

The spin Hall effect results from the interaction of the polarization/spin
with the orbital motion of the rays [9, 10, 11]. The effective ray equations
describing this effect for gravitational waves are similar to those for light.
The spin Hall effect for light (also called the optical Magnus effect [12]) is
observed when electromagnetic waves travel in an inhomogeneous medium [13,
14]. Spin-orbit coupling results from the interaction of the polarization
degrees of freedom with the gradient of the refractive index of the medium. As
a result, transverse deflection of electromagnetic waves in a direction
perpendicular to the refractive index gradient occurs. The spin Hall effect
for light can be explained in terms of Berry curvature [16, 15] and provides a
correction to the geometric optics approximation, which scales approximately
as the inverse of the frequency (in the subleading order). One can reinterpret
the spin Hall effect from condensed matter physics in the context of general
relativity, where spacetime curvature itself plays the role of an
inhomogeneous medium [4, 17, 8, 18, 19, 20, 21]. In this setting, the
gravitational spin Hall effect arises from the interaction of polarization
with the spacetime curvature itself, resulting in a spin-dependent correction
of the particle dynamics.

A higher-order geometric optics expansion is not the only approach for
calculating the spin Hall effect for gravitational waves. Effects of spin on
the trajectory of massive particles have been worked out by [22, 23, 24]. The
Mathisson-Papapetrou-Dixon equations have been adapted to the massless case by
the work of [25, 26], and their equations can be used to study the
polarization-dependent correction on the trajectory of massless particles (see
[17] and the reference therein for details).

This article is the study of the propagation of gravitational waves through
spacetime curvature. Here we are not concerned about the two other subscopes
of gravitational wave physics, its generation and detection. We also provide
applications depicting how gravitational wave propagation evolves.
Gravitational waves emitted by the source propagate over cosmological
distances before reaching the observer. Such waves might encounter
inhomogeneities in the form of spacetime curvature during their propagation.
If the length scale of these inhomogeneities, which acts as a lensing object,
is much larger than the characteristic wavelength of the gravitational waves,
then the geometric approximation is valid. This is the standard approach to
studying gravitational wave propagation. Geometric optics approximation
implies that we treat gravitational wave as a particle propagating in the
geodesic trajectory and its polarization tensor parallel transported along the
geodesic. However, the subleading order correction from the geometric optics
for gravitational waves might be necessary when studying gravitational lensing
[27, 28], which necessitates calculating the higher-order geometric optics
correction for gravitational waves. In realistic situations, there are
gravitational wave detectors that can detect waves with a wavelength of
astrophysical lengths. This implies that the waves under consideration will
have wavelengths comparable to or even larger than the Schwarzschild radius of
astrophysical objects, and the standard lensing phenomena will likely include
wave effects that must be appropriately taken into account. Spin-orbit
coupling plays a role when analyzing the propagation of spinning particles in
an inhomogeneous medium when the wavelength of a particle is small but not
negligible compared to the inhomogeneity scale of the medium.

This article is organized as follows: In Sec. II we linearize the Einstein
field equations with some simplifying assumptions, giving rise to the
equations governing the propagation of gravitational waves. Then in Sec. III,
we make a WKB approximation of the metric perturbation and substitute it back
into linearized Einstein equations to obtain the gravitational wave solution
in the high-frequency limit. In the leading order approximation in $1/\omega$,
the results are consistent with the geometric optics approximation. This
gravitational wave solution in the geometric optics limit can be represented
by a set of null tetrads satisfying some orthonormal and completeness
conditions. The scalar product of Fermi transported null tetrads satisfies
those conditions everywhere along the null trajectory of gravitational waves,
provided that they are satisfied initially. Next, we cast the gravitational
wave equations in the form of Maxwell’s equations for electromagnetic waves.
In doing so, we can describe the circular polarization of the gravitational
waves in terms of the self-dual and anti-self-dual solutions of some
bivector/field tensor. After laying the foundations for the generalization of
geometric optics, we will write the equations for the trajectory and
polarization of gravitational waves in the subleading order approximation. In
Sec. IV, we calculate the effect of spin-orbit coupling in a number of simple
cases: 1) gravitational lensing of gravitational waves in Schwarzschild
spacetime and 2) the propagation of gravitational waves in an expanding
universe. Finally, we present some concluding remarks and discussions in Sec.
V.

Throughout this article, we consider a Lorentzian manifold with the metric
$g_{\mu\nu}$ of the signature $\left(-,+,+,+\right)$. Similarly, we write the
complex conjugate of $z$ as $\tilde{z}$, adopt the Einstein summation
convention and denote the covariant derivative by a semicolon (;) (except in
Sec. II and the related appendix, where we use $\nabla$ for convenience). We
use natural units where $G=c=1$. $\lambda$ denotes the affine parameter along
gravitational wave trajectories and $\dot{x}=dx/d\lambda$. The sign convention
used here is adopted by [1].

##  II Linearization of Einstein field equations

We begin by considering the Einstein field equations with vanishing
cosmological constant

$R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta}=0,$ (1)

where $R_{\alpha\beta}$ is the Ricci tensor, and $R$ denotes the Ricci scalar.
Here, we describe the propagation of gravitational waves, treating it as a
small metric perturbation around the fixed background solution of the vacuum
Einstein equations. We assume that the fixed background, given by the metric
$g_{\alpha\beta}$, describes the gravitational field of arbitrary strength and
its perturbation is small, that is, $h_{\alpha\beta}\ll g_{\alpha\beta}$.

Let $g_{\alpha\beta}$ be the solution of Einstein equations in a vacuum

$R_{\alpha\beta}=0.$ (2)

Consider another metric $\tilde{g}_{\alpha\beta}$, which is the result of a
small perturbation $h_{\alpha\beta}$ on $g_{\alpha\beta}$

$\tilde{g}_{\alpha\beta}=g_{\alpha\beta}+h_{\alpha\beta}.$ (3)

Substituting this into Eq. (2) gives, up to first order in $h_{\alpha\beta}$
and its derivatives (see [4])

$\Big{(}\delta^{\gamma}_{\alpha}\delta^{\delta}_{\beta}\nabla_{\mu}\nabla^{\mu}-g_{\alpha\beta}g^{\gamma\delta}\nabla_{\mu}\nabla^{\mu}+g^{\gamma\delta}\nabla_{\alpha}\nabla_{\beta}+g_{\alpha\beta}\nabla^{\gamma}\nabla^{\delta}\\\
-\delta^{\delta}_{\beta}\nabla^{\gamma}\nabla_{\alpha}-\delta^{\delta}_{\alpha}\nabla^{\gamma}\nabla_{\beta}\Big{)}h_{\gamma\delta}=0.$
(4)

Now, taking the trace of this equation yields

$\nabla^{\alpha}\nabla^{\beta}h_{\alpha\beta}-\nabla^{\alpha}\nabla_{\alpha}h^{\mu}_{\mu}=0.$
(5)

Using the available gauge freedom, we can further reduce Einstein’s field
equations to a hyperbolic system. Let us choose the Lorentz gauge condition
such that

$\nabla^{\alpha}h_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\nabla^{\alpha}h^{\mu}_{\mu}=0.$
(6)

Substituting this back into Eq. (5) gives

$\nabla^{\alpha}\nabla_{\alpha}h^{\mu}_{\mu}=0.$ (7)

Again substituting Eqs. (6) and (7) into Einstein field Eq. (4) gives (see
Appendix A)

$\nabla^{\mu}\nabla_{\mu}h_{\alpha\beta}+2R_{\alpha\mu\beta\nu}h^{\mu\nu}=0,$
(8)

where we have used the following relation expressing the commutation of
covariant derivatives in terms of the Riemann curvature tensor
$R_{\alpha\mu\beta\nu}$

$\nabla_{\beta}\nabla_{\alpha}h^{\mu}_{\nu}-\nabla_{\alpha}\nabla_{\beta}h^{\mu}_{\nu}=R^{\sigma}_{\nu\alpha\beta}h^{\mu}_{\sigma}-R^{\mu}_{\sigma\alpha\beta}h^{\sigma}_{\nu}.$
(9)

###  II.1 Initial conditions

Here, we assume that the gradient of the trace of the perturbation tensor
$h_{\alpha\beta}$ vanishes initially. Then, Eq. (7) ensures that
$\nabla^{\alpha}h^{\mu}_{\mu}$ vanishes everywhere along the trajectory. Thus,
the field equations and the Lorentz gauge condition for gravitational waves
reduce to

$\displaystyle\nabla^{\mu}\nabla_{\mu}h_{\alpha\beta}+2R_{\alpha\mu\beta\nu}h^{\mu\nu}=0,$
(10) $\displaystyle\nabla^{\alpha}h_{\alpha\beta}=0,$ (11)

respectively. Note the striking similarity of these equations with Maxwell’s
equations and the Lorentz gauge condition for electromagnetic radiation. This
will be the case for their solutions as well, as shown below. We consider
circularly polarized gravitational waves for our calculations.

##  III Formulation of spin optics

The spin optics approximation is valid when the typical wavelength of the
waves is small (but cannot be neglected) compared to the length scale of the
variation of its amplitudes and wavelength and the radius of curvature of the
spacetime on which it propagates. In that limit, waves can be approximated
locally as rays propagating on an approximately flat spacetime. The spin
optics approximation can be expressed mathematically by using the WKB ansatz

$h^{\alpha\beta}=a^{\alpha\beta}e^{i\omega{\cal S}},$ (12)

where $a^{\alpha\beta}$ is the slowly varying complex amplitude, and
$\omega{\cal S}$ is the rapidly varying real phase. Here, $\omega$ is the
characteristic frequency of the problem. In later calculations, we write the
wave vector $l_{\alpha}={\cal S}_{;\alpha}$, where semicolon ; denotes the
covariant derivative. The wave vector and the polarization tensor can be
expanded in terms of $\omega$ as

$\displaystyle
l^{\alpha}=l^{\alpha}_{0}+\frac{l^{\alpha}_{1}}{\omega}+\frac{l^{\alpha}_{2}}{\omega^{2}}+...,$
(13) $\displaystyle
a^{\alpha\beta}=a^{\alpha\beta}_{0}+\frac{a^{\alpha\beta}_{1}}{\omega}+\frac{a^{\alpha\beta}_{2}}{\omega^{2}}+...,$
(14)

Below in Sec. III.1, we will use the Fermi propagated null tetrad. Two of its
components represent solutions for the trajectory and polarization of
gravitational waves. This Fermi propagation reduces the freedom in the
transformation property of the null tetrad $m^{\alpha}\to e^{i{\cal
S}_{1}(\lambda)}m^{\alpha}$ by the condition

$\frac{d{\cal S}_{1}(\lambda)}{d\lambda}=0$ (15)

This is why we cannot absorb higher-order phase factors such as ${\cal
S}_{1}(\lambda)$ into the complex amplitude $a^{\alpha\beta}_{0}$ by such
transformations. We can recover the equations of geometric optics by
substituting this perturbation metric onto the source free wave equation (Eq.
(10)) and the Lorentz gauge condition. Let us first start from the Lorentz
condition, which can be written as

$l_{0}^{\alpha}a_{0\alpha\beta}+\frac{1}{\omega}\left(l_{0}^{\alpha}a_{1\alpha\beta}+l_{1}^{\alpha}a_{0\alpha\beta}-ia^{\alpha}_{0~{}\beta;\alpha}\right)=0,$
(16)

up to subleading order in $\omega$. We next substitute the perturbation tensor
into the source-free wave equation, which again, up to subleading order in
$\omega$ gives

$t^{\alpha\beta}=\vcentcolon
a_{0}^{\alpha\beta}l_{0\mu}l_{0}^{\mu}+\frac{1}{\omega}\bigg{(}a_{1}^{\alpha\beta}l_{0\mu}l_{0}^{\mu}+2a_{0}^{\alpha\beta}l_{1\mu}l_{0}^{\mu}\\\
-i\left(a_{0}^{\alpha\beta}l^{\mu}_{0;\mu}+2a^{\alpha\beta}_{0~{}~{};\mu}l_{0}^{\mu}\right)\bigg{)}=0.$
(17)

Let us now calculate the identically vanishing quantity
$\tilde{a}_{0\alpha\beta}t^{\alpha\beta}+a_{0\alpha\beta}\tilde{t}^{\alpha\beta}$,
which gives the dispersion relation

$l_{0\beta}l_{0}^{\beta}+\frac{2}{\omega}\left(l_{1\beta}-b_{\beta}\right)l_{0}^{\beta}=0.$
(18)

In the above expression, we have used
$\tilde{a}_{0}^{\alpha\beta}a_{0\alpha\beta}=a^{2}$ and substituted

$\frac{i}{2a^{2}}\left(\tilde{a}^{\alpha\beta}a_{\alpha\beta;\mu}-a^{\alpha\beta}\tilde{a}_{\alpha\beta;\mu}\right)\vcentcolon=b_{\mu}.$
(19)

###  III.1 Introduction of null tetrad

In the geometric optics limit, the wave vector $l_{0}^{\alpha}$ and the
polarization vector $m_{0}^{\alpha}$ satisfying the wave equations can be
identified with two components of a null tetrad (see Appendix B). The
remaining two components would then be a nonunique auxiliary null vector
$n_{0}^{\alpha}$ and a complex conjugate $\tilde{m}_{0}^{\alpha}$. A set of
null tetrads
$e^{\alpha}_{i}=\left(l_{0}^{\alpha},n_{0}^{\alpha},m_{0}^{\alpha},\tilde{m}_{0}^{\alpha}\right)$,
where $i=\\{l_{0},n_{0},m_{0},\tilde{m}_{0}\\}$ represents the tetrad indices,
would satisfy the following orthogonality and completeness relationships

$\displaystyle
l_{0}^{\alpha}m_{0\alpha}=l_{0}^{\alpha}l_{0\alpha}=l_{0}^{\alpha}\tilde{m}_{0\alpha}=0,\qquad
m_{0}^{\alpha}\tilde{m}_{0\alpha}=1,$ (20) $\displaystyle
m_{0}^{\alpha}m_{0\alpha}=\tilde{m}_{0}^{\alpha}\tilde{m}_{0\alpha}=0,$ (21)
$\displaystyle
n_{0}^{\alpha}m_{0\alpha}=n_{0}^{\alpha}n_{0\alpha}=n_{0}^{\alpha}\tilde{m}_{0\alpha}=0,\qquad
n_{0}^{\alpha}l_{0\alpha}=-1.$ (22)

Here Eqs. (20) follows from the geometric optics approximation (see Eqs.
(100)) and Eqs. (21) and (22) are purely by choice of $n_{0\alpha}$ and
$m_{0\alpha}$. In Sec. III.2, we will show that choosing $m_{0\alpha}$ to
satisfy Eqs. (21) is equivalent to choosing the polarization state. In
particular, these relations hold for circularly polarized waves. Moreover,
these components of the null tetrad also satisfy

$\displaystyle l^{\alpha}_{0;\beta}l_{0}^{\beta}=0,\qquad
m^{\alpha}_{0;\beta}l_{0}^{\beta}=0=\tilde{m}^{\alpha}_{0;\beta}l_{0}^{\beta},$
(23) $\displaystyle n^{\alpha}_{0;\beta}l_{0}^{\beta}=0,$ (24)

where Eqs. (23) are again from the geometric optics approximation (see Eq.
(110)), and $l_{0\alpha;\beta}=l_{0\beta;\alpha}$ is used in obtaining the
first relation. Eq. (24) is obtained by choice of $n_{0}^{\alpha}$. To show
that this choice is indeed possible, we introduce the Fermi derivative
operator ${\cal D}_{l}$ along the ray $l^{\alpha}$, which gives the following
relation when applied to the tensor $A^{\alpha}$ [8]

${\cal
D}_{l}A^{\alpha}=l_{0}^{\beta}A^{\alpha}_{;\beta}-w_{\beta}A^{\beta}n^{\alpha}+A^{\beta}n_{\beta}w^{\alpha},$
(25)

where $w^{\alpha}=l_{0}^{\beta}l^{\alpha}_{;\beta}$. $w^{\alpha}$ is an
identically vanishing quantity in geometric optics. As
$l^{\alpha}l_{\alpha}=0$, we have ${\cal D}_{l}l^{\alpha}=0$. A vector
$A^{\alpha}$ is Fermi propagated if its Fermi derivative ${\cal
D}_{l}A^{\alpha}=0$, and it is easy to see that the scalar product of any two
Fermi propagated vectors is constant. From this, we can conclude that if the
tetrad $\left(l^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$
satisfies the orthogonality and completeness relations similar to the one
given in Eqs. (20)-(22) at some point on the ray, and if they are Fermi
propagated, they satisfy those relations everywhere on the ray. Therefore, we
Fermi propagate the frame of a null tetrad
$\left(l^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ so that
they satisfy the orthogonality and completeness relations analogous to those
in Eqs. (20)-(22) along the ray. The components of this tetrad also obey

$\displaystyle l_{0}^{\beta}n^{\alpha}_{;\beta}=w^{\beta}n_{\beta}n^{\alpha},$
(26) $\displaystyle
l_{0}^{\beta}m^{\alpha}_{;\beta}=w^{\beta}m_{\beta}n^{\alpha},$ (27)
$\displaystyle
l_{0}^{\beta}\tilde{m}^{\alpha}_{;\beta}=w^{\beta}\tilde{m}_{\beta}n^{\alpha}.$
(28)

We can now use the freedom in the choice of null tetrad

$l^{\alpha}\to Al^{\alpha},n^{\alpha}\to A^{-1}n^{\alpha},$ (29)

to fix $w^{\beta}n_{\beta}=0$, where $A$ is a real function. This condition
fixes the parameter $\lambda$ along the ray up to its possible rescaling
$\lambda\to A^{-1}\lambda$, and we call such a choice a canonical
parametrization [8]. Therefore, in the canonical parametrization, we have

$l_{0}^{\beta}n^{\alpha}_{;\beta}=0,\qquad
l_{0}^{\beta}m^{\alpha}_{;\beta}=w^{\beta}m_{\beta}n^{\alpha},\qquad
l_{0}^{\beta}\tilde{m}^{\alpha}_{;\beta}=w^{\beta}\tilde{m}_{\beta}n^{\alpha}.$
(30)

From these results, we can see that a null tetrad
$\left(l_{0}^{\alpha},n_{0}^{\alpha},m_{0}^{\alpha},\tilde{m}_{0}^{\alpha}\right)$
satisfying evolution Eqs. (23)-(24) in the geometric optics limit naturally
generalizes to a null tetrad
$\left(l^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ satisfying
evolution Eqs. (30).

###  III.2 Introduction of polarization basis

In the geometric optics limit, circularly polarized gravitational waves have
amplitude $a_{0\alpha\beta}=a_{0}m_{0\alpha}m_{0\beta}$ whose evolution is
governed by the equation $m^{\alpha}_{0;\beta}l_{0}^{\beta}=0$ (see Appendix
B). This means that the polarization is parallel propagated along the
trajectory in geometric optics. The same line of arguments used from Eqs.
(100)-(105) can be used to show that, in the spin optics approximation, the
polarization tensor can be written in the form

$a_{\alpha\beta}=am_{\alpha}m_{\beta},$ (31)

provided that an arbitrary vector field $\xi^{\alpha}$ in Eq. (102) satisfies

$\xi_{\alpha}l^{\alpha}l_{\beta}-\frac{i}{\omega}\left(\xi_{\alpha;\beta}+\xi_{\beta;\alpha}\right)l^{\alpha}-\frac{i}{\omega}\left(\xi_{\beta}l^{\alpha}+\xi^{\alpha}l_{\beta}\right)_{;\alpha}=0.$
(32)

We can keep track of the evolution of this polarization tensor by defining an
antisymmetric field analogous to electromagnetism,

$F_{\beta\gamma}\vcentcolon=A_{\gamma;\beta}-A_{\beta;\gamma},$ (33)

where the vector $A_{\gamma}$ is given as

$A_{\gamma}\vcentcolon=am_{\gamma}e^{i{\cal S}/2}.$ (34)

The geometric optics approximation up to subleading order in $1/\omega$
assumes that the typical length scale of variation of amplitude, polarization,
and wavelength of gravitational waves is negligible compared to the radius of
curvature of the spacetime through which the waves propagate. In this
approximation, the curvature terms appearing in the linearized gravitational
wave equation Eq. (10) can be safely neglected, as seen from Eq. (17). In such
a regime, the antisymmetric field $F_{\beta\gamma}$ satisfies

$\displaystyle
F_{\alpha\beta;\gamma}+F_{\gamma\alpha;\beta}+F_{\beta\gamma;\alpha}=0,$ (35)
$\displaystyle F^{\beta\gamma}_{~{}~{}~{}~{};\gamma}=0.$ (36)

Let us define the complex version of the tensor $F^{\alpha\beta}$ as

${\cal F}^{\pm}=F\pm iF^{*},$ (37)

where $F^{*}=\epsilon_{\alpha\beta\mu\nu}F^{\mu\nu}/2$ is the Hodge dual of
$F^{\alpha\beta}$. Here $\epsilon_{\alpha\beta\mu\nu}$ is the Levi-Civita
tensor in four dimensions, and its components in the tetrad basis are
$il\wedge n\wedge m\wedge\tilde{m}$. As $(F^{*})^{*}=-F$, we have $\left({\cal
F}^{\pm}\right)^{*}=\mp i{\cal F}^{\pm}$, a property by virtue of which we
call ${\cal F}^{\pm}$ self-/anti-self-dual antisymmetric field for $+/-$ sign.
Now, the field ${\cal F}^{+}_{\alpha\beta}$ can be expanded by substituting
Eq. (34) into Eq. (33)

${\cal F}^{+}_{\alpha\beta}=\frac{i\omega}{2}{\cal Z}_{\alpha\beta}e^{i{\cal
S}/2},$ (38)

where

${\cal
Z}_{\alpha\beta}=a\bigg{(}l_{\beta}m_{\gamma}-l_{\gamma}m_{\beta}-\frac{2i}{\omega}\Big{(}\frac{a_{;\beta}}{a}m_{\gamma}-\frac{a_{;\gamma}}{a}m_{\beta}\\\
+m_{\gamma;\beta}-m_{\beta;\gamma}\Big{)}\bigg{)}.$ (39)

For this self-dual field ${\cal F}^{+}_{\alpha\beta}$, using the property that
contraction of a self-dual field with an anti-self-dual field vanishes, we
obtain

$\displaystyle{\cal Z}_{\alpha\beta}m^{\alpha}n^{\beta}=0,$ (40)
$\displaystyle{\cal
Z}_{\alpha\beta}\left(\tilde{m}^{\alpha}m^{\beta}-l^{\alpha}n^{\beta}\right)=0,$
(41) $\displaystyle{\cal Z}_{\alpha\beta}l^{\alpha}\tilde{m}^{\beta}=0,$ (42)

By substituting the value of ${\cal Z}_{\alpha\beta}$ from Eq. (39), we can
see that Eq. (42) is identically satisfied in the limit of geometric optics.
However, Eqs. (40) and (41) gives

$m_{0}^{\alpha}m_{0\alpha}=0=l_{0}^{\alpha}m_{0\alpha}$ (43)

These are the orthogonality relations given in Eqs. (20) and (21).

###  III.3 Equations of spin optics

####  III.3.1 Defining the Hamiltonian

We have shown in Sec. III.1 that in the geometric optics limit, light travels
along the null geodesic

$l_{0}^{\alpha}l_{0\alpha}=0,\qquad l^{\alpha}_{0;\beta}l_{0}^{\beta}=0.$ (44)

Let us take $x^{\alpha}(\lambda)$ to be the integral curve of $l^{\alpha}$.
Then

$l_{0}^{\alpha}=\frac{dx_{0}^{\alpha}}{d\lambda}\equiv\dot{x}_{0}^{\alpha}.$
(45)

Eq. (44) can be recovered from Hamilton’s equations of motion if we define the
Hamiltonian as

$H=\frac{1}{2}g^{\alpha\beta}l_{0\alpha}l_{0\beta},$ (46)

Hamilton’s equations of motion are

$\frac{dx_{0}^{\alpha}}{d\lambda}=\frac{\partial H}{\partial
l_{0\alpha}}=g^{\alpha\beta}l_{0\beta},$ (47)

and

$\frac{dl_{0\alpha}}{d\lambda}=\frac{\partial H}{\partial
x_{0}^{\alpha}}=\frac{1}{2}\dot{x}_{0}^{\mu}\dot{x}_{0}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial x_{0}^{\alpha}},$ (48)

where we have used Eq. (47) and the relation

$\frac{\partial g_{\alpha\beta}}{\partial
x_{0\mu}}=-g_{\nu\alpha}g_{\rho\beta}\frac{\partial g^{\nu\rho}}{\partial
x_{0\mu}}.$ (49)

in obtaining this. Further simplification of this equation gives

$\frac{d\left(g_{\alpha\beta}\dot{x}_{0}^{\beta}\right)}{d\lambda}-\frac{1}{2}\dot{x}_{0}^{\mu}\dot{x}_{0}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial x_{0\alpha}}=\frac{D^{2}x_{0}^{\alpha}}{D\lambda^{2}}=0,$
(50)

where $D/D\lambda$ denotes the covariant derivative along the curve
$x^{\alpha}(\lambda)$. The two Eqs. (47) and (50) are identical to Eq. (44),
thereby demonstrating that the Hamiltonian defined in Eq. (46) correctly
reproduces the geometric optics equations. The action corresponding to this
Hamiltonian is

${\cal A}_{0}=\frac{1}{2}\int\dot{x}_{0}^{\alpha}\dot{x}_{0\alpha}d\lambda.$
(51)

This action corresponds to the dispersion relation in the limit of geometric
optics, and this dispersion relation is Eq. (18) in the leading order
approximation. Thus, because of the subleading order terms of the dispersion
relation Eq. (18), the natural generalization to spin optics would give an
action of the form

${\cal
A}=\frac{1}{2}\int\dot{x}^{\alpha}\dot{x}_{\alpha}d\lambda+\frac{1}{\omega}\int
b_{\alpha}\dot{x}^{\alpha}d\lambda.$ (52)

This action is analogous to that of electromagnetic waves considered in [36]
and [8], where the spin Hall effect of light is derived. The variation of the
first term, which is the optical path length, gives

$\frac{1}{2}\delta\int\dot{x}^{\alpha}\dot{x}_{\alpha}d\lambda=\int\dot{x}_{\alpha}\frac{D\delta
x^{\alpha}}{D\lambda}d\lambda=-\int\frac{D^{2}x_{\alpha}}{D\lambda^{2}}\delta
x^{\alpha}d\lambda.$ (53)

Similarly the variation of the second term (which resembles the Berry
connection in optics) gives

$\frac{1}{\omega}\delta\int
b_{\alpha}\dot{x}^{\alpha}d\lambda=\frac{1}{\omega}\int\delta
b_{\alpha}\dot{x}^{\alpha}d\lambda+\frac{1}{\omega}\int
b_{\alpha}\frac{D\delta x^{\alpha}}{D\lambda}d\lambda\\\ =\frac{1}{\omega}\int
b_{\alpha;\beta}\dot{x}^{\alpha}\delta x^{\beta}d\lambda-\frac{1}{\omega}\int
b_{\beta;\alpha}\delta x^{\beta}\dot{x}^{\alpha}d\lambda.$ (54)

The application of the variational principle $\delta{\cal A}=0$ yields

$\frac{D^{2}x_{\beta}}{D\lambda^{2}}+\frac{1}{\omega}\left(b_{\beta;\alpha}-b_{\alpha;\beta}\right)\dot{x}^{\alpha}=0.$
(55)

We can write the corresponding Hamiltonian as

$H=\dot{x}^{\alpha}l_{\alpha}-{\cal L},$ (56)

where ${\cal
L}=\dot{x}^{\alpha}\dot{x}_{\alpha}/2+b_{\alpha}\dot{x}^{\alpha}/\omega$ is
the Lagrangian. We substitute the canonical momentum

$l_{\alpha}=\frac{\partial{\cal
L}}{\partial\dot{x}^{\alpha}}=\dot{x}_{\alpha}+\frac{b_{\alpha}}{\omega},$
(57)

into Eq. (56) to get the Hamiltonian

$H=\frac{1}{2\omega^{2}}\left(\omega
l_{0\alpha}+l_{1\alpha}-b_{\alpha}\right)\left(\omega
l_{0}^{\alpha}+l_{1}^{\alpha}-b^{\alpha}\right).$ (58)

####  III.3.2 Solving Hamilton’s equations of motion

Now, we substitute Eq. (57) into the dispersion relation of Eq. (18) to obtain

$\left(l_{0\beta}+\frac{1}{\omega}(l_{1\beta}-b_{\beta})\right)\left(l_{0}^{\beta}+\frac{1}{\omega}(l_{1}^{\beta}-b^{\beta})\right)=\dot{x}_{\beta}\dot{x}^{\beta}=0.$
(59)

Thus, in the leading order approximation in $1/\omega$,
$\dot{x}^{\alpha}=l_{0}^{\alpha}$ gives the tangent vector. This equation
implies that the gravitational wave trajectory in the spin optics
approximation is still null. It is not geodesic however, as seen from the fact
that

$\frac{D\dot{x}^{\alpha}}{D\lambda}=\frac{1}{\omega}\left(l^{\alpha}_{1;\beta}-b^{\alpha}_{;\beta}\right)l_{0}^{\beta}.$
(60)

We use the Hamiltonian of Eq. (58) to calculate this quantity. This
Hamiltonian has been solved by [8] to obtain the following equation for right-
hand circularly polarized rays:

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=\frac{1}{\omega}k^{\alpha}_{\beta}l_{0}^{\beta},$
(61)

where $k_{\alpha\beta}=b_{\beta;\alpha}-b_{\alpha;\beta}$ and
$b_{\alpha}=2i\tilde{m}_{0}^{\beta}m_{0\beta;\alpha}$. This is our Eq. (55),
which has been derived by using the variational principle. Comparing this with
Eq. (60), we get $l_{1\alpha;\beta}=b_{\beta;\alpha}$. Further simplifying
$k_{\alpha\beta}$, one gets

$k_{\alpha\beta}=-2iR_{\alpha\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}+2i\left(\tilde{m}^{\mu}_{;\alpha}m_{\mu;\beta}-\tilde{m}^{\mu}_{;\beta}m_{\mu;\alpha}\right).$
(62)

Substituting this back into Eq. (60) gives

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=-\frac{2i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}l_{0}^{\beta}\approx-\frac{2i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}l_{0}^{\beta}m_{0}^{\mu}\tilde{m}_{0}^{\nu}.$
(63)

We thus obtain a nongeodesic trajectory in the spin optics approximation.

####  III.3.3 Polarization equation

We can substitute $w^{\alpha}=l_{0}^{\beta}l^{\alpha}_{;\beta}$ from Eq. (60)
into the Eqs. (30), which gives the evolution of the polarization vector

$\displaystyle l_{0}^{\beta}n^{\mu}_{;\beta}=$ $\displaystyle 0,$ (64)
$\displaystyle l_{0}^{\beta}m^{\mu}_{;\beta}=$
$\displaystyle\frac{1}{\omega}\left(l^{\beta}_{1;\alpha}-b^{\beta}_{;\alpha}\right)l_{0}^{\alpha}m_{\beta}n^{\mu}$
$\displaystyle=$
$\displaystyle\frac{2i}{\omega}R_{\alpha\beta\gamma\delta}l_{0}^{\alpha}m_{0}^{\beta}m_{0}^{\gamma}\tilde{m}_{0}^{\delta}n_{0}^{\mu},$
(65) $\displaystyle l_{0}^{\beta}\tilde{m}^{\mu}_{;\beta}=$
$\displaystyle\frac{1}{\omega}\left(l^{\beta}_{1;\alpha}-b^{\beta}_{;\alpha}\right)l_{0}^{\alpha}\tilde{m}_{\beta}n^{\mu}$
$\displaystyle=$
$\displaystyle-\frac{2i}{\omega}R_{\alpha\beta\gamma\delta}l_{0}^{\alpha}\tilde{m}_{0}^{\beta}\tilde{m}_{0}^{\gamma}m_{0}^{\delta}n_{0}^{\mu}.$
(66)

These equations ensure that the set of tetrads
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$
satisfy the normalization and orthogonality relations in Eqs. (20)-(22)
throughout the ray up to subleading order in $1/\omega$. To see this, let us
first simplify Eq. (65) as

$l_{0}^{\beta}m^{\mu}_{;\beta}\approx\frac{1}{\omega}l_{0}^{\beta}m^{\mu}_{1;\beta}\approx\frac{1}{\omega}l_{0}^{\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)_{;\beta}m_{0\alpha}n_{0}^{\mu}.$
(67)

As the covariant derivatives of $m_{0\alpha}$ and $n_{0}^{\mu}$ are zero, we
can write

$m_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)m_{0\alpha}n_{0}^{\mu}.$
(68)

Simplification of Eqs. (64) and (66) in a similar manner gives

$n_{1}^{\mu}=0,\qquad\tilde{m}_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)\tilde{m}_{0\alpha}n_{0}^{\mu},$
(69)

respectively. One can also write the subleading order of the component of
tetrad $\dot{x}^{\alpha}$ differently by noting that
$w^{\alpha}l_{0\alpha}=0=w^{\alpha}n_{0\alpha}$. This is how we have
constructed the Fermi propagated tetrad in Sec. III.1. Therefore, $w^{\alpha}$
can be written in the form

$\displaystyle
w^{\alpha}\equiv\frac{1}{\omega}l_{0}^{\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)_{;\beta}=-\tilde{\kappa}m_{0}^{\alpha}-\kappa\tilde{m}_{0}^{\alpha},$
(70)
$\displaystyle\kappa=-\frac{1}{\omega}m_{0}^{\alpha}l_{0}^{\beta}\left(l_{1\alpha}-b_{\alpha}\right)_{;\beta}.$

From this, one can write

$l_{1}^{\alpha}-b^{\alpha}=\tilde{m}_{0}^{\beta}\left(l_{1\beta}-b_{\beta}\right)m_{0}^{\alpha}+m_{0}^{\beta}\left(l_{1\beta}-b_{\beta}\right)\tilde{m}_{0}^{\alpha}.$
(71)

It is easy to see that Eq. (68), in conjunction with the Eqs. (69) and (71)
taken as the subleading order correction of the tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$
satisfies the scalar products of Eqs. (20)-(22). The leading order terms of
this tetrad are
$\left(l_{0}^{\alpha},n_{0}^{\alpha},m_{0}^{\alpha},\tilde{m}_{0}^{\alpha}\right)$
of Sec. III.1. Moreover, the subleading order terms of the Lorentz gauge
condition of Eq. (16) gives

$\frac{a_{;\alpha}}{a}m_{0}^{\alpha}=-m^{\alpha}_{0;\alpha}-ib_{\alpha}m_{0}^{\alpha}+m_{0}^{\alpha}\tilde{m}_{0\beta;\alpha}m_{0}^{\beta}\\\
=-m^{\alpha}_{0;\alpha}-\frac{i}{2}b_{\alpha}m_{0}^{\alpha}.$ (72)

All of the polarization Eqs. (40)-(42) are not satisfied in the limit of spin
optics, thereby implying that the field, with the subleading order solution
given by Eq. (68), Eqs. (69) and (71), is not self-dual (see Appendix C).
However, a self-dual solution of these field equations should exist in the
subleading order geometric optics approximation. We can find this self-dual
solution by first writing the Fermi-like derivative operator as follows:

${\cal
D}^{\prime}_{l}A^{\alpha}=l_{0}^{\beta}A^{\alpha}_{;\beta}-w_{\beta}A^{\beta}n^{\alpha}+A^{\beta}n_{\beta}w^{\alpha}\\\
-\frac{2i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}A^{\beta}m^{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}A^{\beta}\tilde{m}^{\alpha}\right),$
(73)

where $\tilde{\lambda}=m_{0}^{\alpha}n_{0\alpha;\beta}m_{0}^{\beta}$. The
vanishing of this derivative ${\cal D}^{\prime}_{l}A^{\alpha}=0$ implies

$l_{0}^{\beta}A^{\alpha}_{;\beta}=w_{\beta}A^{\beta}n^{\alpha}-A^{\beta}n_{\beta}w^{\alpha}\\\
+\frac{2i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}A^{\beta}m^{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}A^{\beta}\tilde{m}^{\alpha}\right),$
(74)

and in that case, the scalar product of any two components of a tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ are
constant except that of $m^{\alpha}$ with itself and that of
$\tilde{m}^{\alpha}$ with itself. This can be seen by calculating

$\left(u^{\alpha}v_{\alpha}\right)_{;\beta}l_{0}^{\beta}=u^{\alpha}v_{\alpha_{;}\beta}l_{0}^{\beta}+v^{\alpha}u_{\alpha_{;}\beta}l_{0}^{\beta}\\\
=\frac{4i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}u^{\beta}m^{\alpha}v_{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}u^{\beta}\tilde{m}^{\alpha}v_{\alpha}\right).$
(75)

This scalar product is non-zero only if $u^{\alpha}=v^{\alpha}=m^{\alpha}$ or
$u^{\alpha}=v^{\alpha}=\tilde{m}^{\alpha}$. A tetrad with a vanishing Fermi-
like derivative satisfies these conditions as well as the following
orthogonality and completeness relations everywhere on the ray if they satisfy
them at some point on the ray:

$\displaystyle\dot{x}^{\alpha}m_{\alpha}=\dot{x}^{\alpha}\dot{x}_{\alpha}=\dot{x}^{\alpha}\tilde{m}_{\alpha}=0,\qquad
m^{\alpha}\tilde{m}_{\alpha}=1,$ (76) $\displaystyle
n^{\alpha}m_{\alpha}=n^{\alpha}n_{\alpha}=n^{\alpha}\tilde{m}_{\alpha}=0,\qquad
n^{\alpha}l_{\alpha}=-1,$ (77)

However, in general, $m^{\alpha}m_{\alpha}\neq 0$ and
$\tilde{m}^{\alpha}\tilde{m}_{\alpha}\neq 0$ along the circularly polarized
ray in the subleading order approximation. That is, the polarization vectors
are not null like they are in the geometric optics approximation. Moreover,
the tetrad evolves as

$\displaystyle l_{0}^{\beta}n^{\alpha}_{;\beta}=0,$ (78) $\displaystyle
l_{0}^{\beta}m^{\alpha}_{;\beta}=w^{\beta}m_{\beta}n^{\alpha}-\frac{2i}{\omega}\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}^{\alpha},$
(79) $\displaystyle
l_{0}^{\beta}\tilde{m}^{\alpha}_{;\beta}=w^{\beta}\tilde{m}_{\beta}n^{\alpha}+\frac{2i}{\omega}\lambda_{;\mu}l^{\mu}m^{\alpha}.$
(80)

The following simplification procedure is used to obtain Eqs. (68) and (69)
gives

$\displaystyle
m_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)m_{0\alpha}n_{0}^{\mu}-2i\tilde{\lambda}\tilde{m}_{0}^{\mu},\qquad
n_{1}^{\mu}=0,$ (81)
$\displaystyle\tilde{m}_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)\tilde{m}_{0\alpha}n_{0}^{\mu}+2i\lambda
m_{0}^{\mu}.$ (82)

These components of tetrads constitute a solution of the linearized Einstein
field equations in Lorentz gauge and are right-hand circularly polarized, as
they also satisfy Eqs. (40)-(42). Therefore, they are the solutions for the
propagation of circularly polarized gravitational waves in curved spacetime in
the spin optics approximation.

##  IV Applications

###  IV.1 Gravitational lensing of gravitational waves

As a first application of the spin-optics, we consider a localized object
acting as a lens for gravitational waves of frequency $\omega$. The point-like
lensing object is described by a Schwarzschild geometry of mass $M$. The
geometric optics expansion for gravitational waves is valid only when their
wavelength is much smaller than the Schwarzschild radius of the lensing body
($1/\omega\ll M$) [29, 30]. Detectors like pulsar timing arrays (PTAs) [31]
and the Laser Interferometer Space Antenna (LISA) [32] can collectively detect
gravitational waves across a wide range of wavelengths, ranging from $10^{8}m$
to $10^{17}m$ (the corresponding frequency range of $10^{-9}-1Hz$). Thus, the
lensing of real gravitational waves by the astrophysical objects in the wide
mass range, including the galaxies, is likely to contain effects not seen in
the geometric optics approximation [27].

Here, we calculate the effect of spin-orbital coupling on gravitational wave
lensing. In the geometric optics regime, the angle of deflection of
gravitational waves by gravitating objects is the same as that of
electromagnetic waves, and the maximum angle of deflection is given by

$\Delta\phi\approx\frac{4M}{R_{0}},$ (83)

where $M$ is the mass and $R_{0}$ is the radius of a gravitating object.

For the Schwarzschild geometry, the general vector tangent to the congruence
of null geodesics can be written as

$l_{0\mu}=\left(-E,\frac{1}{1-\frac{2M}{r}}\sqrt{E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)},0,L\sin\theta\right),$
(84)

where $E$ and $L$ are the constants of motion whose values are determined by
the initial conditions of the geodesic trajectory. The three other null fields
$n^{\mu}$, $m^{\mu}$ and $\tilde{m}^{\mu}$, along with $l^{\mu}$, form a set
of null tetrads and satisfy the orthogonality and completeness relations given
as

$\begin{split}&n_{0\mu}=\frac{1-\frac{2M}{r}}{2\left(E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)\right)}\left(-E,-\frac{1}{1-\frac{2M}{r}}\sqrt{E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)},0,L\sin\theta\right),\\\
&m_{0\mu}=\frac{r}{\sqrt{2}}\left(-\frac{iL\left(1-\frac{2M}{r}\right)}{r^{2}\sqrt{E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)}},0,1,\frac{iE\sin\theta}{\sqrt{E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)}}\right),\qquad\tilde{m}_{0\mu}=\left(m_{0\mu}\right)^{*}.\end{split}$
(85)

We now apply the following transformation relations

$\displaystyle l_{0}^{\mu}\to l_{0}^{\mu},\qquad m_{0}^{\mu}\to
m_{0}^{\mu}+al_{0}^{\mu},$ $\displaystyle n_{0}^{\mu}\to
n_{0}^{\mu}+\tilde{a}m_{0}^{\mu}+a\tilde{m}_{0}^{\mu}+a\tilde{a}l_{0}^{\mu},$
(86)

to the null tetrad such that the resulting tetrad will be Fermi propagated
along the rays. Here, the value of $a$ can be evaluated in such a way that the
transformed vector $m_{0}^{\mu}+al_{0}^{\mu}$ has vanishing covariant
derivative to leading order:

$\left(m_{0}^{\mu}+al_{0}^{\mu}\right)_{;\nu}l_{0}^{\nu}=m^{\mu}_{0;\nu}l_{0}^{\nu}+a_{,\nu}l_{0}^{\mu}l_{0}^{\nu}=0.$
(87)

In a spherically symmetric spacetime, $a$ can only be a function of $r$,
thereby giving

$a=\int{\frac{m^{\mu}_{0;\nu}l_{0}^{\nu}n_{0\mu}}{l_{0}^{r}}}dr\\\
=\int{\frac{iEL(r-3M)}{\sqrt{2}r^{3}\left(E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)\right)^{3/2}}}dr.$
(88)

These transformed null tetrads satisfy Eqs. (20)-(24). These null trajectories
$l_{0}^{\mu}$ and polarization vectors $m_{0}^{\mu}$ constitute solutions of
the linearized Einstein equations in the geometric optics limit.

Now, to obtain the first-order correction to the null trajectory, we
substitute these values into the propagation Eqs. (63), and we get

$\displaystyle\frac{D^{2}x^{2}}{D\lambda^{2}}=\frac{6iLM}{\omega}\left(\frac{\sqrt{2}La}{r^{6}}+\frac{iE}{r^{5}\sqrt{E^{2}-\frac{L^{2}}{r^{2}}\left(1-\frac{2M}{r}\right)}}\right),$
$\displaystyle\frac{D^{2}x^{\mu}}{D\lambda^{2}}=0\qquad\text{for}\qquad\mu\neq
2.$ (89)

Integrating this equation up to second-order in $M/r$ gives

$\dot{x}^{2}=\frac{1}{\omega}\left(\frac{C_{1}}{r}+\frac{2LM}{Er^{4}}+\frac{3LM^{2}}{2Er^{5}}\right),$
(90)

where $C_{1}$ is an integration constant. The first term, which gives a
divergent result upon integration, can be made to vanish by choosing
$C_{1}=0$. We can see that $\dot{x}^{2}\neq 0$ for $L\neq 0$, and in that
case, the gravitational wave travels along a null nongeodesic trajectory even
in the Schwarzschild spacetime, due to the gravitational spin Hall effect.

We can now calculate the angle of deflection of gravitational waves in the
$\theta$-direction while passing near a lensing object of mass $M$, using the
relation (here, $x_{2}=\theta$ and $x_{3}=\phi$)

$d\theta=\frac{\dot{\theta}}{\dot{r}}dr=\frac{2LM}{\omega
Er^{3}\sqrt{E^{2}r^{2}-L^{2}}}dr,$ (91)

up to leading order in $M/r$. A photon starting from infinity and approaching
the lensing object within a closest distance of $R_{0}$ has $E=1$ and

$L=\frac{R_{0}}{1-\frac{2M}{R_{0}}}.$ (92)

We can now integrate to obtain the total deflection angle up to leading order
in $M/r$,

$\theta=2\left(\theta_{\infty}-\theta_{0}\right)=\frac{\pi M}{\omega
R_{0}^{2}}.$ (93)

This gives an additional deflection resulting from the spin-orbit coupling,
which agrees with the result obtained in Ref. [20], up to a numerical
prefactor. For a lensing object of mass $M\ll R_{0}$, this quantity may be too
small to detect by LIGO [34]. This deflection, however, is twice the
deflection of electromagnetic waves due to the spin-orbit correction [29].
Moreover, although $\omega$ is only the expansion, it multiplies with the
phase ${\cal S}$ in Eq. (12) and thus flipping the sign of $\omega$ from the
positive to the negative sign corresponds to the flip in the helicity of
waves. As the subleading order corrections are proportional to $\omega$, waves
of opposite helicities give the more pronounced spin Hall effect.

###  IV.2 Propagation of a binary’s gravitational waves through an expanding
universe

If we model the large-scale structure of the expanding universe by the line
element

$ds^{2}=R_{0}t^{2}\left(-dt^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right),$ (94)

then no spin Hall effect is observed for gravitational waves sourced by a
binary system. This is because, in this spacetime, the null rays along which
gravitational waves propagate in the geometric optics regime are the curves of
constant $\theta$, $\phi$ and $t-r$. A set of null tetrads for such spacetime
can be written as

$\begin{split}&l_{0\mu}=R_{0}^{2}\left(-1,1,0,0\right),\\\
&n_{0\mu}=\frac{t^{4}}{2}\left(-1,-1,0,0\right),\\\
&m_{0\mu}=\frac{R_{0}t^{2}r}{\sqrt{2}}\left(0,0,1,i\sin\theta\right),\quad\tilde{m}_{0\mu}=\left(m_{0\mu}\right)^{*}.\end{split}$
(95)

This is indeed a Fermi propagated null tetrad along the rays that satisfy Eqs.
(20)-(24). Therefore, these null trajectories $l_{0}^{\mu}$ and polarization
vectors $m_{0}^{\mu}$ constitute the gravitational wave solutions of the
linearized Einstein equations in the limit of geometric optics.

Using the propagation Eqs. (63), we obtain no first-order correction of the
null trajectory, thereby implying no spin-orbit coupling for the propagation
of gravitational waves in such a spacetime. Thus, the propagation of
gravitational waves from a binary does not deviate from the null geodesic
trajectories in such spacetime. The absence of terms with a subleading order
correction in the geometric optics approximation in such spacetime could be
attributed to the fact that waves propagate in the curves of constant $\theta$
and $\phi$. The absence of orbital motion/angular momentum of gravitational
waves propagating in such spacetime causes spin-orbit coupling to vanish.

##  V Conclusion and discussions

We have presented calculations that account for the spin Hall effect of
gravitational waves on curved backgrounds and applied the results to some
simple scenarios. The equations of spin optics developed here for
gravitational waves are shown to have the exact same form as their
electromagnetic counterparts. The only difference is a factor of two, which
correctly accounts for the difference in helicity between gravitational and
electromagnetic waves. This factor of two causes the deviation of
gravitational waves from the null geodesic trajectory to be twice that of
electromagnetic waves, due to spin-orbit coupling. It should also be noted
that this deviation is frequency-dependent.

In leading-order geometric optics, the trajectory of light/gravitational waves
is geodesic. However, this is no longer true when the subleading order
correction is considered. Once we include the subleading order correction to
the geometric optics, the diffraction effects start to appear and the
propagation properties of waves changes. Wave-like effects differ increasingly
from the geometric optics as the wavelength increases relative to the length
scale of the inhomogeneity. The study of this effect, resulting from the
propagation in the Universe with the structure, also helps in disentangling
the effects coming from the intrinsic properties of the source or the nature
of gravity in the strong field regime. As gravitons/photons still travel at
the speed of light, it takes longer for them to reach the observer from the
source [29]. Due to their helicity difference, it takes an even longer time
for gravitons than for photons. This frequency-dependent time delay for
circularly polarized waves might be observed in cosmological events like
neutron star mergers [33]. We can also notice from Sec. III.2 that
electromagnetic and gravitational waves in the weak field limit (up to the
subleading order geometric optics approximation) follow the same set of
equations except for a difference in the factor of two that accounts for their
helicity difference. Therefore, the procedure for obtaining the subleading
order correction described here can be applied to massless particles of
arbitrary spin (obviously, by taking into account the difference in the spin
of these particles).

There are different approaches to studying massless particles’ motion with
spins in curved spacetimes [17]. The similarity of our geometric optics
approach to the quantum mechanical approach used by [35] to obtain the
gravitational spin Hall effect is evident from the similarity of our
Hamiltonian Eq. (58) with theirs. This gives a similar equation for the null
trajectory deviating from the geodesic in the subleading order in $1/\omega$,
where the term proportional to $b_{\beta}$ occurring in our Hamiltonian should
be interpreted as a Berry connection. Similarly, the equivalence between the
quantum mechanical approach, for example, of Gosselin and the approach using
Souriau-Saturnini equations [25, 26] was demonstrated by Ref. [17] (at least
for the Schwarzschild spacetime). For the Schwarzschild spacetime, the
trajectory equations from all of these approaches closely resemble those used
in optics to derive the spin Hall effect, where spacetime is treated as an
effective medium with perfect impedance matching. This resemblance is
particularly encouraging, as the spin Hall effect has been verified
experimentally in optics. In future work, we hope to rigorously establish the
equivalence of our spin optics approach with these different approaches.

Acknowledgements: We would like to thank Ioannis Soranidis and Phil Simovic
for helpful comments. PKD is supported by an International Macquarie
University Research Excellence Scholarship.

##  Appendix A Deriving gravitational wave equations

For simplification, we first write Eq. (4) as

$\nabla_{\mu}\nabla^{\mu}h_{\alpha\beta}-g_{\alpha\beta}\nabla_{\mu}\nabla^{\mu}h^{\nu}_{\nu}+\nabla_{\alpha}\nabla_{\beta}h^{\mu}_{\mu}\\\
+g_{\alpha\beta}\nabla^{\gamma}\nabla^{\delta}h_{\gamma\delta}-\nabla^{\gamma}\nabla_{\alpha}h_{\gamma\beta}-\nabla^{\gamma}\nabla_{\beta}h_{\gamma\alpha}=0.$
(96)

Next, we use Eq. (5) to reduce this equation to

$\nabla_{\mu}\nabla^{\mu}h_{\alpha\beta}+\nabla_{\alpha}\nabla_{\beta}h^{\mu}_{\mu}-\nabla^{\gamma}\nabla_{\alpha}h_{\gamma\beta}\\\
-\nabla^{\gamma}\nabla_{\beta}h_{\gamma\alpha}=0.$ (97)

Further simplification can be achieved by using Eq. (9) for the commutation of
covariant derivatives

$\displaystyle\nabla_{\gamma}\nabla_{\alpha}h^{\gamma}_{\beta}=$
$\displaystyle\nabla_{\alpha}\nabla_{\gamma}h^{\gamma}_{\beta}-R^{\gamma}_{\sigma\alpha\gamma}h^{\sigma}_{\beta}+R^{\sigma}_{\beta\alpha\gamma}h^{\gamma}_{\sigma}$
$\displaystyle=$
$\displaystyle\nabla_{\alpha}\nabla_{\gamma}h^{\gamma}_{\beta}+R^{\sigma}_{\beta\alpha\gamma}h^{\gamma}_{\sigma},$
(98)

where we have used the relation
$R^{\gamma}_{\sigma\alpha\gamma}=-R_{\sigma\alpha}=0$, which follows from Eq.
(2). Substituting this into Eq. (97) gives

$\nabla_{\mu}\nabla^{\mu}h_{\alpha\beta}+\nabla_{\alpha}\nabla_{\beta}h^{\mu}_{\mu}-\nabla_{\alpha}\nabla_{\gamma}h^{\gamma}_{\beta}-\nabla_{\beta}\nabla_{\gamma}h^{\gamma}_{\alpha}\\\
-2R_{\sigma\beta\alpha\gamma}h^{\gamma\sigma}=0$ (99)

Further use of Eq. (6) gives the linearized gravitational wave Eq. (8).

##  Appendix B Reproducing the equations of geometric optics

From Eqs. (16)-(18), we can recover the results of geometric optics by
substituting $a_{1}^{\alpha\beta}=0=l_{1}^{\beta}$. In the leading order
approximation in $\omega$, the Lorentz condition Eq. (16) and the wave
equation Eq. (17) gives

$l_{0}^{\alpha}a_{0\alpha\beta}=0=l_{0}^{\alpha}l_{0\alpha}.$ (100)

This equation allows one to write
$a_{0\alpha\beta}=a_{0\alpha}\left(c_{1}l_{0\beta}+c_{2}m_{0\beta}\right)$,
where $c_{1}$ and $c_{2}$ are arbitrary constants, $a_{0\alpha}$ is an
arbitrary vector and $m_{0\beta}$ is a complex vector satisfying
$m_{0\beta}\tilde{m}_{0}^{\beta}=1$ and $m_{0\beta}l_{0}^{\beta}=0$. As the
polarization tensor $a_{0\alpha\beta}$ is symmetric in its indices, we should
also have $a_{0\alpha}=c_{1}l_{0\alpha}+c_{2}m_{0\alpha}$. Now, using the
property $\tilde{a}_{0}^{\alpha\beta}a_{0\alpha\beta}=a^{2}$ we obtain
$c_{2}^{2}=ae^{i\phi}$. We can further reduce the expression for
$a_{0\alpha\beta}$ by considering the gauge transformation

$h^{\prime}_{\alpha\beta}=h_{\alpha\beta}-\Xi_{\alpha;\beta}-\Xi_{\beta;\alpha},$
(101)

under which the linearized Einstein field equations presented above are
invariant. Here,

$\Xi_{\alpha}=g_{\alpha\beta}\Xi^{\beta}=\frac{1}{\omega}\xi_{\alpha}e^{i{\cal
S}},$ (102)

are small arbitrary functions. This gauge transformation preserves the form of
WKB ansatz given in Eq. (12) if

$a^{\prime}_{\alpha\beta}=a_{\alpha\beta}-\frac{1}{\omega}\left(\xi_{\alpha;\beta}+\xi_{\beta;\alpha}\right)-i\left(\xi_{\beta}l_{\alpha}+\xi_{\alpha}l_{\beta}\right),$
(103)

is satisfied. The change in amplitude in the leading order approximation can
be written as

$a^{\prime}_{0\alpha\beta}=a_{0\alpha\beta}-i\left(\xi_{0\beta}l_{0\alpha}+\xi_{0\alpha}l_{0\beta}\right).$
(104)

As $l_{0\alpha}l^{0\alpha}=0$, the Lorentz condition of Eq. (16) holds for
arbitrary $\xi_{0\alpha}$ satisfying $\xi_{0\alpha}l_{0}^{\alpha}=0$. This
freedom in the choice of $\xi_{0\alpha}$ can be used to choose $c_{1}=0$.
Moreover, an arbitrary constant parameter $\phi$ from $c_{2}^{2}$ can be
absorbed in a redefinition of the phase function ${\cal S}$. We finally obtain

$a_{0\alpha\beta}=c_{2}^{2}m_{0\alpha}m_{0\beta}=am_{0\alpha}m_{0\beta}.$
(105)

Note that it is equally legitimate to write
$a_{0\alpha\beta}=a_{0\alpha}\left(c_{2}m_{0\beta}+c_{3}\tilde{m}_{0\beta}\right)$,
where $\tilde{m}_{0\beta}$ is the complex conjugate of $m_{0\beta}$ and
$c_{3}$ is another arbitrary constant. However, as in electromagnetism, if the
polarization tensor with $m_{0\beta}$ represents right-hand circularly
polarized waves, then the polarization tensor with $\tilde{m}_{0\beta}$
represents left-hand circularly polarized waves.

We calculate $\tilde{a}_{0\alpha\beta}t^{\alpha\beta}$ from Eq. (17) while
considering only the terms relevant to the geometric optics approximation,
that is, we take $a_{1}^{\alpha\beta}=0$ and $l_{1\mu}=0$ to obtain

$l^{\mu}_{0;\mu}+\frac{2}{a^{2}}\tilde{a}_{0\alpha\beta}a^{\alpha\beta}_{0~{}~{};\mu}l_{0}^{\mu}=0.$
(106)

Adding this equation to its complex conjugate gives

$l^{\mu}_{0;\mu}+\frac{1}{a^{2}}\left(\tilde{a}_{0\alpha\beta}a_{0}^{\alpha\beta}\right)_{;\mu}l_{0}^{\mu}=\frac{1}{a^{2}}\left(l_{0}^{\mu}{\cal
I}_{0}\right)_{;\mu}=0.$ (107)

where

${\cal
I}_{0}=\vcentcolon\tilde{a}_{0\alpha\beta}a_{0}^{\alpha\beta}-\frac{1}{2}\tilde{a}^{\mu}_{0\mu}a^{\nu}_{0\nu}$
(108)

is the intensity of the wave (note that $a^{\mu}_{0\mu}=0$). These constitute
a complete set of equations for gravitational waves in curved spacetime in the
limit of geometric optics.

We substitute the value of the polarization amplitude from Eq. (105) to Eq.
(106), which upon further simplification gives

$l^{\beta}_{0;\beta}+2\frac{a_{;\beta}}{a}l_{0}^{\beta}+4\tilde{m}_{0\alpha}m^{\alpha}_{0;\beta}l_{0}^{\beta}=0.$
(109)

Since the term $\tilde{m}_{0\alpha}m^{\alpha}_{0;\beta}l_{0}^{\beta}$ is
purely imaginary and the remaining terms
$l^{\beta}_{0;\beta}+2l_{0}^{\beta}a_{;\beta}/a$ are purely real, they should
be separately zero, thereby giving

$l^{\beta}_{0;\beta}+2\frac{a_{;\beta}}{a}l_{0}^{\beta}=0,\qquad
m^{\alpha}_{0;\beta}l_{0}^{\beta}=0.$ (110)

These relations will be used below in Appendix C.

##  Appendix C Checking self-duality in the spin optics approximation

To see that the subleading order solution of the tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ given
by Eq. (68), Eqs. (69) and (71) is not self-dual, we calculate each of these
equations separately, starting with Eq. (42)

${\cal
Z}_{\alpha\beta}l^{\alpha}\tilde{m}^{\beta}=\frac{2ia}{\omega}\left(-\frac{a_{;\alpha}}{a}l_{0}^{\alpha}+m_{0\alpha;\beta}l_{0}^{\alpha}\tilde{m}_{0}^{\beta}\right)\\\
=\frac{2ia}{\omega}\left(\frac{1}{2}l^{\alpha}_{0;\alpha}-m_{0}^{\alpha}l_{0\alpha;\beta}\tilde{m}_{0}^{\beta}\right)=0,$
(111)

where we have used Eq. (110) to arrive at this identity. Similarly, Eq.(41)
gives

${\cal
Z}_{\alpha\beta}\left(\tilde{m}^{\alpha}m^{\beta}-l^{\alpha}n^{\beta}\right)=\frac{2ia}{\omega}\left(\frac{a_{;\alpha}}{a}m_{0}^{\alpha}-m_{0\alpha;\beta}l_{0}^{\alpha}n_{0}^{\beta}\right)\\\
=\frac{2ia}{\omega}\left(-m^{\alpha}_{0;\alpha}-\frac{i}{2}b_{\alpha}m_{0}^{\alpha}-m_{0\alpha;\beta}l_{0}^{\alpha}n_{0}^{\beta}\right)=0,$
(112)

where we have used Eq. (72) to obtain this identity. Finally, Eq.(40) gives

${\cal
Z}_{\alpha\beta}m^{\alpha}n^{\beta}=\frac{2ia}{\omega}\left(-m_{0\alpha;\beta}m_{0}^{\beta}n_{0}^{\alpha}\right)\\\
=\frac{2ia}{\omega}\left(m_{0}^{\alpha}n_{0\alpha;\beta}m_{0}^{\beta}\right)\equiv\frac{2ia}{\omega}\tilde{\lambda},$
(113)

where $\tilde{\lambda}$ is one of the Newman-Penrose scalars. Thus, Eq.(40) is
not satisfied unless $\tilde{\lambda}=0$. Thus, the tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$,
satisfying the scalar products of Eqs. (20)-(22) does not give the self-dual
solution of the field Eqs. (35) and (36).

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