# Covariant formulation of spin optics for electromagnetic waves

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

###### Abstract

We develop geometric optics expansion up to the subleading order for
circularly polarized electromagnetic waves on curved spacetime. This
subleading order geometric optics expansion, in which the conventional eikonal
function is modified by inserting a carefully chosen helicity-dependent
correction, is called spin optics. We derive the propagation and polarization
equations in the spin optics approximation as electromagnetic waves travel in
curved spacetime. Polarization-dependent deviation of the light ray trajectory
from the geodesic, describing the gravitational spin Hall effect, is observed.
We also establish an analogy with the related phenomena (optical Magnus
effect) of condensed matter physics.

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

##  I Introduction

Geometric optics is valid in the infinitely large frequency limit Misner et
al. (1973); Dolan (2017, 2018). In this approximation, electromagnetic waves
propagating on the fixed curved background follow a null ray trajectory. At
large but finite frequencies, the geometric optics approximation no longer
remains valid, as backreaction from the helicity may cause ray trajectory to
depart from the geodesics by a significant amount. The gravitational spin Hall
effect refers to this helicity dependent phenomenon of the propagation of a
light ray in curved spacetime in the subleading order Frolov & Shoom (2011);
Yamamoto (2018); Yoo (2012); Shoom (2021). This is similar to the
gravitational Faraday effect (or the spin Hall effect for gravitational
waves), which is observed for high-frequency polarized gravitational waves
propagating in curved spacetimes p35; Hou et al. (2019); Andersson et al.
(2021).

The spin Hall effect is due to the interaction of polarization/spin with the
orbital motion of the rays Mashhoon (1974, 1975); Ling et al. (2017). Related
phenomena are observed in different areas of physics, including optics, where
the polarization-dependent deflection of light was predicted Dooghin et al.
(1992); Onoda et al. (2004); Liberman & Zel’dovich (1992) and verified
experimentally for both electrons Wunderlich et al. (2005); Kato et al. (2004)
and light Hosten & Kwiat (2008); Bliokh et al. (2008) (also see Ref. Aiello &
Woerdman (2008) for the explanation of these results in terms of classical
optics). When an electromagnetic wave propagates in an inhomogeneous medium,
the spin Hall effect of light (also known as the optical Magnus effect) is
observed. The coupling of spin with the orbital motion comes from the
interaction of the polarization of waves with the refractive index gradient of
the medium. As a result, electromagnetic waves are deflected transversally in
a direction perpendicular to the refractive index gradient. The spin Hall
effect can be expressed in terms of Berry curvature Bliokh et al. (2015);
Aiello et al. (2015); Sundaram & Niu (1999) and provides correction to the
geometric optics, which is approximately proportional to the frequency
inverse, in the subleading order. This phenomenon of the optical Magnus effect
can be extended to general relativity, where spacetime curvature itself acts
as an inhomogeneous medium Oancea et al. (2019, 2020); Frolov (2020); Frolov &
Shoom (2011); Yamamoto (2018); Harte (2019); Gosselin et al. (2007). This
analogy comes from the fact that the equations for electromagnetic waves
propagating in some optical medium and curved spacetime are identical. Thus,
the gravitational spin Hall effect is caused by the interaction of
polarisation with spacetime curvature, requiring a spin-dependent correction
to particle dynamics.

Various approaches can be found in the literature for the calculation of the
gravitational spin Hall effect, and they can be broadly classified into three
categories: 1) use of the Souriau-Saturnini equations Saturnini (1976);
Souriau (1974), which is a modification of the Mathisson-Papapertou-Dixon
equation Mathisson (2010); Papapetrou (1951); Corinaldesi & Papapetrou (1951);
Dixon (1964) for the motion of massive spinning particles to the massless
ones, 2) application of the methods of quantum mechanics, such as using the
Foldy-Wouthyusyen transformation on the Bargmann-Wigner equations Bérard &
Mohrbach (2006); Gosselin et al. (2007, 2007), and 3) spin optics Frolov &
Shoom (2011); Dolan (2017, 2018); Yoo (2012). See the review by Oancea et al.
(2019) and the references therein for the detailed discussions of each of
these approaches and their comparisons. Here, we focus on spin optics (or
modified geometric optics), where the eikonal function is modified by
including the spin-dependent term. This correction is of the order
$\omega^{-1}$ and thus suppressed in the high-frequency approximation.
However, such correction becomes essential for the propagation of polarized
rays of high but finite frequencies at large distances (where the ray
trajectory could be modified).

Self-dual and anti-self-dual solutions of the Maxwell equations represent
electromagnetic fields with right- and left-hand circular polarization,
respectively. The use of the Wentzel-Kramers-Brillouin (WKB) ansatz for each
of these solutions does not require the same eikonal functions for both types
of waves. We instead include the helicity-dependent first-order high-frequency
correction in the eikonal function in the spin optics approximation. The
development of spin optics for general spacetime has been demonstrated in
recent papers by Refs. Oancea et al. (2020) and Frolov (2020). Their results
are slightly different from each other and our paper. We will compare these
results.

Here, we formulate the theory of spin optics and develop self-consistent ray
and transport equations for the propagation of electromagnetic waves in curved
spacetime. Electromagnetic waves from astrophysical sources might propagate
over cosmological distances before reaching the observer. As a result, they
might encounter inhomogeneities in the form of spacetime curvature. These
inhomogeneities act as a lensing object, and if their length scale is much
larger than the characteristic wavelength of the electromagnetic waves, then
the geometric optics approximation is valid. However, there might be a
situation, where the characteristic wavelength is small but not negligible
compared to the length scale of the variation of spacetime curvature. In such
a situation, the necessity of the subleading order correction to the geometric
optics arises while studying the gravitational lensing of electromagnetic
waves. The standard lensing phenomena likely include wave effects, changing
the propagation properties of waves Takahashi (2017). In Sec. II, we write the
Maxwell equations in curved spacetime and obtain the equation for
electromagnetic waves. To solve the electromagnetic wave equation, we present
the WKB ansatz for the vector potential in Sec. III. Next, we show that we can
reduce the geometric optics solution to a set of Fermi transported null
tetrads and then generalize this (requirement) to the subleading order. We
impose more constraints on the subleading order equations considering the
solutions to be self-dual and anti-self-dual. These requirements and
considerations are sufficient to obtain the propagation and polarization
equations up to the subleading order. We do this in Sec. IV. We also obtain
the stress-energy tensor of the electromagnetic field up to the subleading
order approximation in Sec. V to see the energy flow as the wave propagates.
We then compare our results with the literature using the WKB approximation,
the same formalism as used here, in Sec. VI. Finally, we discuss our results
in Sec. VII.

In this article, we consider the metric $g_{\mu\nu}$ of signature
$\left(-,+,+,+\right)$ in a Lorentzian manifold $M$. The phase space is the
cotangent bundle $T^{*}M$ and its points are written as $(x,p)$. Similarly, we
write $\tilde{z}$ for the complex conjugate of $z$ and adopt the Einstein
summation convention. We use the system of units with $G=c=1$. A semicolon (;)
denote the covariant derivative, $\lambda$ denotes the parameter of
electromagnetic wave curves, and $\dot{x}=dx/d\lambda$. We use the sign
convention for the curvature adopted in Misner et al. (1973).

##  II Maxwell equations in curved spacetimes

In flat spacetime, the Maxwell equations of electrodynamics are

$\displaystyle\frac{\partial}{\partial x^{\alpha}}F^{\alpha\beta}=-J^{\beta},$
(1) $\displaystyle\frac{\partial}{\partial
x^{\alpha}}F^{\beta\gamma}+\frac{\partial}{\partial
x^{\beta}}F^{\gamma\alpha}+\frac{\partial}{\partial
x^{\gamma}}F^{\alpha\beta}=0,$ (2)

where $F^{\alpha\beta}$ is the antisymmetric field tensor with components

$F^{jk}=e^{jkl}B_{l},\qquad F^{0j}=E_{j}.$ (3)

Here, $e^{jkl}$ is the Levi-Civita symbol in three dimensions, and
$J^{\alpha}$ is the conserved four-current. The principle of general
covariance implies that these equations hold in a general curved spacetime if
covariant derivatives replace the partial derivatives occurring in the
equations. Thus, the Maxwell equations in curved spacetime are

$\displaystyle F^{\alpha\beta}_{~{}~{}~{}~{};\alpha}=-J^{\beta},$ (4)
$\displaystyle
F_{\alpha\beta;\gamma}+F_{\gamma\alpha;\beta}+F_{\beta\gamma;\alpha}=0.$ (5)

Eq.(5) allows us to represent the electromagnetic field tensor
$F^{\alpha\beta}$ in terms of the vector potential $A^{\alpha}$ as

$F_{\alpha\beta}=A_{\beta;\alpha}-A_{\alpha;\beta}.$ (6)

We can use available gauge freedom in the Maxwell equations such that the
vector $A^{\alpha}$ satisfies the Lorenz gauge condition,
$A^{\alpha}_{;\alpha}=0$. We substitute this into Eq. (4) to obtain the wave
equation

$-A^{\alpha;\beta}_{~{}~{}~{}~{};\beta}+R^{\alpha}_{\beta}A^{\beta}=J^{\alpha},$
(7)

where $R^{\alpha}_{\beta}$ denotes the Ricci tensor.

##  III WKB approximation

Spin optics approximation is valid when the typical wavelength of the wave is
very small (but nonnegligible) in comparison with the length scale over which
its amplitude and wavelength vary and the radius of curvature of the spacetime
on which it propagates. We can locally approximate the wave as a ray
propagating on approximately flat spacetime in such a case. Mathematically, we
formulate the spin optics approximation using the WKB ansatz as

$A^{\alpha}=a^{\alpha}e^{i\omega{\cal S}},$ (8)

where $a^{\alpha}$ is the complex amplitude that varies slowly, and
$\omega{\cal S}$ is the real phase that changes rapidly. Here, $\omega$ is the
characteristic frequency of the problem. The wave vector is the phase
gradient, that is, $l_{\alpha}={\cal S}_{;\alpha}$. Let us write the square
amplitude $a=\left(\tilde{a}^{\alpha}a_{\alpha}\right)^{1/2}$ and the
polarization vector $m^{\alpha}=a^{\alpha}/a$. We can expand the wave vector
and polarization vector in powers of $1/\omega$ as

$\displaystyle l^{\alpha}=$ $\displaystyle
l^{\alpha}_{0}+\frac{l^{\alpha}_{1}}{\omega}+\frac{l^{\alpha}_{2}}{\omega^{2}}+...,$
(9) $\displaystyle m^{\alpha}=$ $\displaystyle
m^{\alpha}_{0}+\frac{m^{\alpha}_{1}}{\omega}+\frac{m^{\alpha}_{2}}{\omega^{2}}+....$
(10)

We should mention here that we cannot absorb higher order phase factors like
${\cal S}_{1}(\lambda)$ into the complex amplitude $m^{\alpha}_{0}$ by
transformation $m^{\alpha}\to e^{i{\cal S}_{1}(\lambda)/\omega}m^{\alpha}$.
This is because we will use the Fermi propagated null tetrad in Sec. IV.1 and
two of its components, $l^{\alpha}$ and $m^{\alpha}$, give the trajectory and
polarization of the waves, respectively. Fermi propagation reduces the freedom
in the transformation of the null tetrad $m^{\alpha}\to e^{i{\cal
S}_{1}(\lambda)/\omega}m^{\alpha}$ by the condition

$\frac{d{\cal S}_{1}(\lambda)}{d\lambda}=0.$ (11)

This is the reason why we should expand both the wave vector and polarization
vector separately in powers of $\omega$. Let us substitute this vector
potential onto the source free wave equation (Eq. (7) with $J^{\alpha}=0$) and
the Lorenz gauge condition. We start from the Lorenz condition, which up to
the subleading order in $\omega$ can be written as

$l_{0}^{\alpha}m_{0\alpha}+\frac{1}{\omega}\left(l_{0}^{\alpha}m_{1\alpha}+l_{1}^{\alpha}m_{0\alpha}-i\left(\frac{a_{;\alpha}}{a}m_{0}^{\alpha}+m^{\alpha}_{0;\alpha}\right)\right)=0.$
(12)

Again, we substitute the vector potential into the source-free wave equation,
which gives

$j^{\alpha}=\vcentcolon
m_{0}^{\alpha}l_{0\beta}l_{0}^{\beta}+\frac{1}{\omega}\bigg{(}m_{1}^{\alpha}l_{0\beta}l_{0}^{\beta}+2m_{0}^{\alpha}l_{1\beta}l_{0}^{\beta}\\\
-i\left(m_{0}^{\alpha}l^{\beta}_{0;\beta}+2m^{\alpha}_{0;\beta}l_{0}^{\beta}+2\frac{a_{;\beta}}{a}m_{0}^{\alpha}l_{0}^{\beta}\right)\bigg{)}=0,$
(13)

up to the subleading order in $\omega$. We now calculate
$\tilde{m}_{0\alpha}j^{\alpha}+m_{0\alpha}\tilde{j}^{\alpha}$, which is the
identically vanishing quantity

$l_{0\beta}l_{0}^{\beta}+\frac{2}{\omega}\left(l_{1\beta}-b_{\beta}\right)l_{0}^{\beta}=0.$
(14)

This is the generalization of the dispersion relation of geometric optics.
Here, we have used $\tilde{m}_{0}^{\alpha}m_{0\alpha}=1$ and substituted

$\frac{i}{2}\left(\tilde{m}^{\alpha}m_{\alpha;\beta}-m^{\alpha}\tilde{m}_{\alpha;\beta}\right)=i\tilde{m}^{\alpha}m_{\alpha;\beta}\vcentcolon=b_{\beta}.$
(15)

##  IV Spin optics approximation

###  IV.1 Propagation and polarization vector as tetrad components

Let us construct a set of null tetrads
$\left(l_{0}^{\alpha},n_{0}^{\alpha},m_{0}^{\alpha},\tilde{m}_{0}^{\alpha}\right)$
satisfying the following orthogonality and completeness relationships

$\displaystyle
l_{0}^{\alpha}m_{0\alpha}=l_{0}^{\alpha}l_{0\alpha}=l_{0}^{\alpha}\tilde{m}_{0\alpha}=0,\qquad
m_{0}^{\alpha}\tilde{m}_{0\alpha}=1,$ (16) $\displaystyle
m_{0}^{\alpha}m_{0\alpha}=\tilde{m}_{0}^{\alpha}\tilde{m}_{0\alpha}=0,$ (17)
$\displaystyle
n_{0}^{\alpha}m_{0\alpha}=n_{0}^{\alpha}n_{0\alpha}=n_{0}^{\alpha}\tilde{m}_{0\alpha}=0,\qquad
n_{0}^{\alpha}l_{0\alpha}=-1.$ (18)

Comparison of Eqs. (16) with the equations of geometric optics (see Eqs.
(103)) shows that two components of the tetrad $l_{0}^{\alpha}$ and
$m_{0}^{\alpha}$ could be identified with the wave vector and polarization
vector, respectively. Auxiliary null vectors $n_{0\alpha}$ and $m_{0\alpha}$
are not unique and can be chosen in such a way that they satisfy Eqs. (17) and
(18). We will discuss below in Sec. IV.2 that the validity of Eqs. (17)
determines the state of polarization; specifically, circularly polarized waves
satisfy these relations. Moreover, these components of the null tetrad also
satisfy

$\displaystyle l^{\alpha}_{0;\beta}l_{0}^{\beta}=0,\qquad
m^{\alpha}_{0;\beta}l_{0}^{\beta}=0=\tilde{m}^{\alpha}_{0;\beta}l_{0}^{\beta},$
(19) $\displaystyle n^{\alpha}_{0;\beta}l_{0}^{\beta}=0,$ (20)

where Eqs. (19) again follows from geometric optics (to obtain the first
relation, we have applied $l_{0\alpha;\beta}=l_{0\beta;\alpha}$). We can
choose $n_{0}^{\alpha}$, such that it satisfies Eq. (20). We can show that
this choice is indeed possible by introducing the Fermi derivative operator
${\cal D}_{l}$ along the ray $l^{\alpha}$, which applied to the vector
$A^{\alpha}$ gives Frolov & Shoom (2011)

${\cal
D}_{l}A^{\alpha}=l_{0}^{\gamma}A^{\alpha}_{;\gamma}-w_{\gamma}A^{\gamma}n^{\alpha}+A^{\gamma}n_{\gamma}w^{\alpha},$
(21)

where $w^{\alpha}=l_{0}^{\gamma}l^{\alpha}_{;\gamma}$ vanishes identically in
geometric optics. We have ${\cal D}_{l}l^{\alpha}=0$ as
$l^{\gamma}l_{\gamma}=0$. If the Fermi derivative ${\cal D}_{l}A^{\alpha}$ of
a vector $A^{\alpha}$ is zero, then it is said to be Fermi propagated, and we
can easily prove that the scalar product of any two Fermi propagated vectors
is constant. We can apply this to the set of tetrads
$\left(l^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$: they
satisfy the orthogonality and completeness relations similar to the one given
in Eqs. (16)-(18) everywhere on the ray if they satisfy those relations the at
some point on the ray and if they all are Fermi propagated. As Fermi
propagation preserves the scalar product, the set of null tetrad
$\left(l^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ satisfies
the orthogonality and completeness relations like those in Eqs. (16)-(18)
along the ray and obeys

$\displaystyle l_{0}^{\beta}n^{\alpha}_{;\beta}=w^{\beta}n_{\beta}n^{\alpha},$
(22) $\displaystyle
l_{0}^{\beta}m^{\alpha}_{;\beta}=w^{\beta}m_{\beta}n^{\alpha},$ (23)
$\displaystyle
l_{0}^{\beta}\tilde{m}^{\alpha}_{;\beta}=w^{\beta}\tilde{m}_{\beta}n^{\alpha}.$
(24)

Next, let us take advantage of flexibility in selecting a null tetrad

$l^{\alpha}\to Fl^{\alpha},n^{\alpha}\to F^{-1}n^{\alpha},$ (25)

to fix $w^{\beta}n_{\beta}=0$, where $F$ is some real function. This condition
defines the parameter $\lambda$ along the ray up to some rescaling $\lambda\to
F^{-1}\lambda$, and such a choice is called a canonical parametrization Frolov
(2020). Therefore, in the canonical parametrization, we have

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

These relations generalize Eqs. (16)-(20) of geometric optics.

###  IV.2 Self-dual and anti-self-dual solutions of Maxwell equations

One can define the complex version of the electromagnetic field tensor
$F^{\alpha\beta}$ as

${\cal F}^{s}=F+isF^{*},$ (27)

where $s=\pm 1$ and $F^{*}=\epsilon_{\alpha\beta\mu\nu}F^{\mu\nu}/2$ is the
Hodge dual of $F^{\alpha\beta}$. Here, $\epsilon_{\alpha\beta\mu\nu}$ is the
Levi-Civita symbol in four dimensions, and its components in the tetrad basis
are $il\wedge n\wedge m\wedge\tilde{m}$. As $(F^{*})^{*}=-F$, we have
$\left({\cal F}^{s}\right)^{*}=-is{\cal F}^{s}$, a feature due to which we
call ${\cal F}_{\alpha\beta}^{s}$ self- (or anti-self-) dual antisymmetric
field for $s=+1(\mathrm{or}-1)$. A general self-dual antisymmetric field can
be expanded in terms of the self-dual basis

$\left(\mathbf{U},\mathbf{V},\mathbf{W}\right)=\left(\tilde{m}\wedge n,l\wedge
m,m\wedge\tilde{m}-l\wedge n\right),$ (28)

as

${\cal F}^{+1}=\Phi_{0}\mathbf{U}+\Phi_{1}\mathbf{W}+\Phi_{2}\mathbf{V}.$ (29)

In the limit of geometric optics, $\Phi_{0}=\Phi_{2}=0$. Now, the field ${\cal
F}^{+1}_{\alpha\beta}$ corresponding to the potential of Eq. (8) is

${\cal F}^{+1}_{\alpha\beta}=i\omega{\cal Z}_{\alpha\beta}e^{i{\cal S}},$ (30)

where

${\cal
Z}_{\alpha\beta}=l_{\alpha}a_{\beta}-l_{\beta}a_{\alpha}-\frac{i}{\omega}\left(a_{\beta;\alpha}-a_{\alpha;\beta}\right).$
(31)

Using the condition that the contraction of the self-dual field with the anti-
self-dual field vanishes, we obtain

$\displaystyle{\cal Z}_{\alpha\beta}m^{\alpha}n^{\beta}=0,$ (32)
$\displaystyle{\cal
Z}_{\alpha\beta}\left(\tilde{m}^{\alpha}m^{\beta}-l^{\alpha}n^{\beta}\right)=0,$
(33) $\displaystyle{\cal Z}_{\alpha\beta}l^{\alpha}\tilde{m}^{\beta}=0.$ (34)

An anti-self-dual solution can be found by the complex conjugation of the
amplitude ${\cal Z}_{\alpha\beta}$ of the self-dual field. Eq. (34) is
satisfied identically in the geometric optics approximation, which can be
verified easily by substituting the value of ${\cal Z}_{\alpha\beta}$ from Eq.
(31). However, Eqs. (32) and (33) gives

$m_{0}^{\alpha}m_{0\alpha}=0=l_{0}^{\alpha}m_{0\alpha}.$ (35)

These relations are presented in Eqs. (16) and (17) as the orthogonality
conditions.

###  IV.3 Equations of spin optics

####  IV.3.1 Generalization of the Hamilton-Jacobi equation

The dispersion equation of geometric optics (see Eq. (103)) could be written
as

$\frac{1}{2}g_{\alpha\beta}l_{0}^{\alpha}l_{0}^{\beta}=0,$ (36)

which is the Hamilton-Jacobi equation for the leading order phase function
${\cal S}_{0}$ defined as ${\cal S}_{0;\alpha}=l_{0\alpha}$. To solve this
Hamilton-Jacobi equation, we define a Hamiltonian function on the cotangent
bundle $T^{*}M$ as

$H(x,l)=\frac{1}{2}g^{\alpha\beta}l_{0\alpha}l_{0\beta},$ (37)

where $x^{\alpha}(\lambda)$ is the integral curve of $l^{\alpha}$. We obtain
the following Hamilton’s equations of motion

$\frac{dx_{0}^{\mu}}{d\lambda}=\frac{\partial H}{\partial
l_{0\mu}}=g^{\mu\nu}l_{0\nu},$ (38)

and

$\frac{dl_{0\alpha}}{d\lambda}=\frac{\partial H}{\partial
x_{0}^{\alpha}}=-\frac{1}{2}\dot{x}_{0}^{\mu}\dot{x}_{0}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial x_{0}^{\alpha}},$ (39)

where Eq. (38) and the relation

$\frac{\partial g_{\alpha\beta}}{\partial
x_{0\mu}}=-g_{\nu\alpha}g_{\rho\beta}\frac{\partial g^{\nu\rho}}{\partial
x_{0\mu}},$ (40)

are used in obtaining this. Given a solution of Hamilton’s equations of
motion, the corresponding solution of the Hamilton-Jacobi equation (36) is
obtained from Goldstein et al. (2002)

${\cal
S}_{0}(x,l)=\int_{\lambda}(\dot{x}_{0}^{\alpha}l_{0\alpha}-H(x,l))d\lambda.$
(41)

The Euler-Lagrange equation for this action, whose Lagrangian is the Legendre
transformation of the Hamiltonian in Eq. (37), is the geodesic equation.
Therefore, Eq. (39) describes null geodesics, and further simplification of
this equation gives

$\frac{d\left(g_{\alpha\beta}\dot{x}_{0}^{\beta}\right)}{d\lambda}-\frac{1}{2}\dot{x}_{0}^{\mu}\dot{x}_{0}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial x_{0\alpha}}=\frac{D^{2}x_{0}^{\alpha}}{D\lambda^{2}}=0,$
(42)

where $D/D\lambda$ denotes the covariant derivative along the curve
$x^{\alpha}(\lambda)$. The two Eqs. (38) and (42) obtained by solving
Hamilton’s equations of motion constitute the two equations of geometric
optics. We have thus shown that the Hamiltonian defined in Eq. (37) correctly
reproduces the equations of geometric optics. Substituting Eqs. (37) and (38)
into the action of Eq. (41) gives

${\cal S}_{0}=\frac{1}{2}\int\dot{x}_{0}^{\alpha}\dot{x}_{0\alpha}d\lambda.$
(43)

Now, to evaluate the trajectory equation up to the subleading order, we
consider the generalized dispersion Eq. (14) and write it as

$\frac{1}{2}g_{\alpha\beta}l_{0}^{\alpha}l_{0}^{\beta}+\frac{1}{\omega}g_{\alpha\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)l_{0}^{\beta}=0,$
(44)

which is the Hamilton-Jacobi equation for the subleading order phase function
${\cal S}$ defined as ${\cal S}_{;\alpha}=l_{0\alpha}+l_{1\alpha}/\omega$. The
corresponding Hamiltonian function on the cotangent bundle $T^{*}M$ is

$H(x,l)=\frac{1}{2}g^{\alpha\beta}l_{0\alpha}l_{0\beta}+\frac{1}{\omega}g^{\alpha\beta}\left(l_{1\alpha}-b_{\alpha}\right)l_{0\beta}\\\
=\frac{1}{2\omega^{2}}g^{\alpha\beta}\left(\omega
l_{0\alpha}+l_{1\alpha}-b_{\alpha}\right)\left(\omega
l_{0\beta}+l_{1\beta}-b_{\beta}\right).$ (45)

Hamilton’s equations of motion are

$\frac{dx^{\alpha}}{d\lambda}=\frac{\partial H}{\partial
l_{\alpha}}=g^{\alpha\beta}\left(l_{\beta}-\frac{b_{\beta}}{\omega}\right),$
(46)

and

$\frac{dl_{\alpha}}{d\lambda}=-\frac{\partial H}{\partial
x^{\alpha}}=\frac{1}{2}\dot{x}^{\mu}\dot{x}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial
x^{\alpha}}+\frac{1}{\omega}g^{\mu\nu}\dot{x}_{\nu}\frac{\partial
b_{\mu}}{\partial x^{\alpha}},$ (47)

where we have used Eqs. (40) and (46) in obtaining this. Thus, the
corresponding solution of the Hamilton-Jacobi equation (44) is obtained from
Goldstein et al. (2002)

${\cal S}(x,l)=\int_{\lambda}(\dot{x}^{\alpha}l_{\alpha}-H(x,l))d\lambda\\\
=\frac{1}{2}\int\dot{x}^{\alpha}\dot{x}_{\alpha}d\lambda+\frac{1}{\omega}\int
b_{\alpha}\dot{x}^{\alpha}d\lambda={\cal S}_{0}-{\cal S}_{B},$ (48)

where Eqs. (45) and (46) are used for simplification. In addition to the
scalar phase, photons acquire a polarization-dependent phase ${\cal S}_{B}$.
This is the Berry geometric phase acquired by the circularly polarized modes
propagating in curved spacetime Berry (1987); Zel’dovich & Kundikova (1995);
Vinitskiĭ (1990). This form of action was considered by Refs. Frolov (2020)
and Duval et al. (2006) to derive the spin Hall effect of light. The first
term is the optical path length and its variation yields

$\frac{1}{2}\delta\int\dot{x}^{\mu}\dot{x}_{\mu}d\lambda=\int\dot{x}_{\mu}\frac{D\delta
x^{\mu}}{D\lambda}d\lambda\\\ =-\int\frac{D^{2}x_{\mu}}{D\lambda^{2}}\delta
x^{\mu}d\lambda.$ (49)

Similarly, the second term resembles the Berry connection of optics (see Sec.
IV.3.2), and its variation gives

$\frac{1}{\omega}\delta\int
b_{\alpha}\dot{x}^{\alpha}d\lambda=\frac{1}{\omega}\int\delta
b_{\alpha}\dot{x}^{\alpha}d\lambda+\frac{1}{\omega}\int
b_{\alpha}\frac{D\delta x^{\alpha}}{D\lambda}d\lambda\\\ =\frac{1}{\omega}\int
b_{\alpha;\beta}\dot{x}^{\alpha}\delta x^{\beta}d\lambda-\frac{1}{\omega}\int
b_{\beta;\alpha}\delta x^{\beta}\dot{x}^{\alpha}d\lambda.$ (50)

Thus, the variational principle $\delta{\cal S}=0$ gives

$\frac{D^{2}x_{\mu}}{D\lambda^{2}}+\frac{1}{\omega}\left(b_{\mu;\nu}-b_{\nu;\mu}\right)\dot{x}^{\nu}=0.$
(51)

One can also show that simplification of Eq. (47) gives this exact same
equation Frolov (2020), as should be the case. In the subleading order
geometric optics approximation, the action acquires a new topological term
depending on the wave polarization (because the action for the left-handed
circularly polarized waves would be the same except for the replacement
$\omega\to-\omega$). This term gives the deflection of the ray trajectory in
the transverse direction. The resulting phenomenon is called the spin Hall
effect because of the coupling of spin with the curved ray trajectory. As
could be seen from Eq. (46), the topological term makes velocity and momentum
noncollinear, which is also the characteristic of waves travelling in
anisotropic media (see, for example, Ziman (1979)).

####  IV.3.2 Propagation equation in the subleading order

To calculate the subleading order terms in the geometric optics approximation,
we take the dispersion relation of Eq. (14)

$\left(l_{0\mu}+\frac{1}{\omega}(l_{1\mu}-b_{\mu})\right)\left(l_{0}^{\mu}+\frac{1}{\omega}(l_{1}^{\mu}-b^{\mu})\right)=\dot{x}_{\mu}\dot{x}^{\mu}=0,$
(52)

where we have used Eq. (46). We can see that in the leading order in
$1/\omega$, $\dot{x}^{\mu}=l_{0}^{\mu}$ is the tangent vector. This equation
suggests that the electromagnetic wave trajectory is still null in the spin
optics approximation. However, it is not a geodesic, as evidenced from the
equation

$\frac{D\dot{x}^{\mu}}{D\lambda}=\frac{1}{\omega}\left(l^{\mu}_{1;\nu}-b^{\mu}_{;\nu}\right)l_{0}^{\nu}.$
(53)

Comparing this with Eq. (51), we get $l_{1\alpha;\beta}=b_{\beta;\alpha}$.
Further simplification of
$b_{\beta;\alpha}-b_{\alpha;\beta}\vcentcolon=k_{\alpha\beta}$ gives

$k_{\alpha\beta}=-iR_{\alpha\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}+i\left(\tilde{m}^{\nu}_{;\alpha}m_{\nu;\beta}-\tilde{m}^{\nu}_{;\beta}m_{\nu;\alpha}\right).$
(54)

We substitute this back into Eq. (53) to obtain

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}l_{0}^{\beta}\approx-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}l_{0}^{\beta}m_{0}^{\mu}\tilde{m}_{0}^{\nu}.$
(55)

Thus, in the spin optics approximation, light travels in a null but
nongeodesic trajectory. Let us interpret this result by comparing it with the
related phenomena in condensed matter physics. The Lagrangian corresponding to
the action of Eq. (48) could be written as

${\cal L}={\cal L}_{0}+{\cal L}_{1};\qquad{\cal
L}_{0}=\frac{1}{2}\dot{x}^{\alpha}\dot{x}_{\alpha},\\\ {\cal
L}_{1}=\frac{1}{\omega}b_{\alpha}\dot{x}^{\alpha}=-\frac{1}{\omega}\mathscr{A}_{\alpha}\dot{x}^{\alpha},$
(56)

where ${\cal L}_{0}$ is the Lagrangian corresponding to the leading order
geometric optics and ${\cal L}_{1}$ gives the spin-orbit coupling. Here,
$\mathscr{A}_{\alpha}=-b_{\alpha}=-i\tilde{m}^{\beta}m_{\beta;\alpha}$ could
be identified with the Berry gauge field (this has the same form as the spin-
orbit interaction of light in gradient-index medium and spin-orbit interaction
of electrons occurring in Dirac equation, see, for example, Refs. Duval et
al., 2006; Bliokh et al., 2008; Bérard & Mohrbach, 2006; Spohn, 2000). The
nonrelativistic version of ${\cal L}_{1}$ appears in the theory of spinning
particles in Duval et al. (2006); Y Bliokh (2009) (also see, for example,
Fanelli (1975); Bhandari (1997)). ${\cal L}_{1}$ introduces an additional
polarization-dependent wave phase, which could be explained as the Berry
phase. This Berry phase manifests itself dynamically, thereby inducing an
additional term in the equation of motion of the ray trajectory that describes
the spin Hall effect. Thus, the Berry phase and spin Hall effect together
characterize the spin-orbit interaction of electromagnetic waves.

Berry connection $\mathscr{A}_{\alpha}$ appears in the Lagrangian as an
external vector potential affecting the light trajectory. The curvature
associated with the Berry connection can be defined as

$\frac{\partial\mathscr{A}_{\alpha}}{\partial
x^{\beta}}-\frac{\partial\mathscr{A}_{\beta}}{\partial
x^{\alpha}}=b_{\beta;\alpha}-b_{\alpha;\beta}\equiv k_{\alpha\beta}.$ (57)

This quantity is known by the name of Berry curvature and plays the role of a
field strength tensor corresponding to the vector potential
$\mathscr{A}_{\alpha}$.

####  IV.3.3 Polarization equation in the subleading order

The polarization vector depends only on the direction of the ray trajectory
(or the momentum of photons). The momentum of free particles propagating in
curved spacetime is a function of position only. Therefore, the Berry
connection determines the evolution of the wave polarization in curved
spacetime. Let us substitute $w^{\alpha}=l_{0}^{\beta}l^{\alpha}_{;\beta}$
from Eq. (53) into the Eqs. (26) to obtain equations for the evolution of the
polarization vector

$\displaystyle l_{0}^{\beta}n^{\mu}_{;\beta}=$ $\displaystyle 0,$ (58)
$\displaystyle l_{0}^{\beta}m^{\mu}_{;\beta}=$
$\displaystyle\frac{1}{\omega}\left(l^{\beta}_{1;\alpha}-b^{\beta}_{;\alpha}\right)l_{0}^{\alpha}m_{\beta}n^{\mu}$
$\displaystyle=$
$\displaystyle\frac{i}{\omega}R_{\alpha\beta\gamma\delta}l_{0}^{\alpha}m_{0}^{\beta}m_{0}^{\gamma}\tilde{m}_{0}^{\delta}n_{0}^{\mu},$
(59) $\displaystyle l_{0}^{\beta}\tilde{m}^{\mu}_{;\beta}=$
$\displaystyle\frac{1}{\omega}\left(l^{\beta}_{1;\alpha}-b^{\beta}_{;\alpha}\right)l_{0}^{\alpha}\tilde{m}_{\beta}n^{\mu}$
$\displaystyle=$
$\displaystyle-\frac{i}{\omega}R_{\alpha\beta\gamma\delta}l_{0}^{\alpha}\tilde{m}_{0}^{\beta}\tilde{m}_{0}^{\gamma}m_{0}^{\delta}n_{0}^{\mu}.$
(60)

These equations assure that, up to the subleading order in $1/\omega$, the set
of tetrads
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$
satisfies the normalization and orthogonality relations in Eqs. (16)-(18)
throughout the ray. To verify this, we first simplify Eq. (59) as

$l_{0}^{\beta}m^{\mu}_{;\beta}\approx\frac{1}{\omega}l_{0}^{\beta}m^{\mu}_{1;\beta}\approx\frac{1}{\omega}l_{0}^{\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)_{;\beta}m_{0\alpha}n_{0}^{\mu}.$
(61)

We can use the fact that the covariant derivatives of $m_{0\alpha}$ and
$n_{0}^{\mu}$ are zero to write

$m_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)m_{0\alpha}n_{0}^{\mu}.$
(62)

We simplify Eqs. (58) and (60) in a similar way to obtain

$n_{1}^{\mu}=0,\qquad\tilde{m}_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)\tilde{m}_{0\alpha}n_{0}^{\mu},$
(63)

respectively. Noting that the Fermi propagated tetrad is constructed in Sec.
IV.1 in such a way that it satisfies
$w^{\alpha}l_{0\alpha}=0=w^{\alpha}n_{0\alpha}$, we can also write the
subleading order of its component $\dot{x}^{\alpha}$ differently. Therefore,
we can write $w^{\alpha}$ as

$w^{\alpha}\equiv\frac{1}{\omega}l_{0}^{\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)_{;\beta}=-\tilde{\kappa}m_{0}^{\alpha}-\kappa\tilde{m}_{0}^{\alpha},\\\
\kappa=-\frac{1}{\omega}m_{0}^{\alpha}l_{0}^{\beta}\left(l_{1\alpha}-b_{\alpha}\right)_{;\beta}.$
(64)

This allows one to write

$\left(l_{1}^{\alpha}-b^{\alpha}\right)=\tilde{m}_{0}^{\beta}\left(l_{1\beta}-b_{\beta}\right)m_{0}^{\alpha}+m_{0}^{\beta}\left(l_{1\beta}-b_{\beta}\right)\tilde{m}_{0}^{\alpha}.$
(65)

We can easily see that the subleading order correction of the tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$,
explicitly presented in Eqs. (62), (63) and (65) satisfy the scalar products
of Eqs. (16), (17) and (18). The leading order terms of this tetrad are
obviously the tetrad
$\left(l_{0}^{\alpha},n_{0}^{\alpha},m_{0}^{\alpha},\tilde{m}_{0}^{\alpha}\right)$
of Sec. IV.1. Moreover, the subleading order terms of the Lorenz condition Eq.
(12) gives

$\frac{a_{;\mu}}{a}m_{0}^{\mu}=-m^{\mu}_{0;\mu}-ib_{\mu}m_{0}^{\mu}.$ (66)

The field is not self-dual in the subleading order in $1/\omega$, since all
the polarization Eqs. (32)-(34) are not satisfied in the limit of spin optics
(see Appendix B).

####  IV.3.4 Self-dual solution up to the subleading order

The tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$,
satisfying the scalar products of Eqs. (16), (17) and (18) is not the self-
dual solution of the Maxwell equations. However, a self-dual solution of the
Maxwell equations in Eqs. (4) and (5) should exist for $J^{\alpha}=0$ in the
subleading order geometric optics approximation. One can obtain this self-dual
solution by first introducing the Fermi-like derivative operator

${\cal
D}^{\prime}_{l}A^{\alpha}=l_{0}^{\beta}A^{\alpha}_{;\beta}-w_{\beta}A^{\beta}n^{\alpha}+A^{\beta}n_{\beta}w^{\alpha}\\\
-\frac{i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}A^{\beta}m^{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}A^{\beta}\tilde{m}^{\alpha}\right).$
(67)

The vanishing of the Fermi-like derivative ${\cal D}^{\prime}_{l}A^{\alpha}=0$
gives

$l_{0}^{\beta}A^{\alpha}_{;\beta}=w_{\beta}A^{\beta}n^{\alpha}-A^{\beta}n_{\beta}w^{\alpha}\\\
+\frac{i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}A^{\beta}m^{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}A^{\beta}\tilde{m}^{\alpha}\right),$
(68)

and this implies that the scalar product of any two tetrad components
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$ is
constant except that of $m^{\alpha}$ with itself and of $\tilde{m}^{\alpha}$
with itself. To see this, we calculate

$\left(a^{\alpha}b_{\alpha}\right)_{;\beta}l_{0}^{\beta}=a^{\alpha}b_{\alpha_{;}\beta}l_{0}^{\beta}+b^{\alpha}a_{\alpha_{;}\beta}l_{0}^{\beta}\\\
=\frac{2i}{\omega}\left(\lambda_{;\mu}l^{\mu}m_{\beta}a^{\beta}m^{\alpha}b_{\alpha}-\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}_{\beta}a^{\beta}\tilde{m}^{\alpha}b_{\alpha}\right).$
(69)

This scalar product is nonzero only if $a^{\alpha}=b^{\alpha}=m^{\alpha}$ or
$a^{\alpha}=b^{\alpha}=\tilde{m}^{\alpha}$. For tetrads with vanishing Fermi-
like derivatives, if they satisfy the following orthogonality and completeness
relations at some point on the ray

$\displaystyle\dot{x}^{\alpha}m_{\alpha}=\dot{x}^{\alpha}\dot{x}_{\alpha}=\dot{x}^{\alpha}\tilde{m}_{\alpha}=0,\qquad
m^{\alpha}\tilde{m}_{\alpha}=1,$ (70) $\displaystyle
n^{\alpha}m_{\alpha}=n^{\alpha}n_{\alpha}=n^{\alpha}\tilde{m}_{\alpha}=0,\qquad
n^{\alpha}l_{\alpha}=-1,$ (71)

then they satisfy these relations everywhere on the ray. However,
$m^{\alpha}m_{\alpha}\neq 0$ and $\tilde{m}^{\alpha}\tilde{m}_{\alpha}\neq 0$,
in general, along the circularly polarized ray in the subleading order
approximation. This means that the polarization vectors are no more null as in
the geometric optics limit. In addition to these, the tetrad evolves as

$\displaystyle l_{0}^{\beta}n^{\alpha}_{;\beta}=0,$ (72) $\displaystyle
l_{0}^{\beta}m^{\alpha}_{;\beta}=w^{\beta}m_{\beta}n^{\alpha}-\frac{i}{\omega}\tilde{\lambda}_{;\mu}l^{\mu}\tilde{m}^{\alpha},$
(73) $\displaystyle
l_{0}^{\beta}\tilde{m}^{\alpha}_{;\beta}=w^{\beta}\tilde{m}_{\beta}n^{\alpha}+\frac{i}{\omega}\lambda_{;\mu}l^{\mu}m^{\alpha}.$
(74)

As in Eqs. (62) and (63), we can write

$\displaystyle
m_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)m_{0\alpha}n_{0}^{\mu}-i\tilde{\lambda}\tilde{m}_{0}^{\mu},\qquad
n_{1}^{\mu}=0,$ (75)
$\displaystyle\tilde{m}_{1}^{\mu}=\left(l_{1}^{\alpha}-b^{\alpha}\right)\tilde{m}_{0\alpha}n_{0}^{\mu}+i\lambda
m_{0}^{\mu}.$ (76)

These tetrads components constitute the solution of the Maxwell equation in
the Lorenz gauge, and they are self-dual, as they also satisfy the
polarization Eqs. (32)-(34). Therefore, they provide a solution for the
propagation of right-handed circularly polarized electromagnetic waves in
curved spacetime in the spin optics approximation.

####  IV.3.5 Constructing a gauge independent Hamiltonian

The gauge properties of the Berry connection and Berry curvature are related
to the choice of the comoving frame. One can introduce noncanonical
coordinates, such that the gauge-dependent Berry connection term Littlejohn &
Flynn (1991) can be removed from the Hamiltonian of Eq. (45). We consider the
following transformation relations to the noncanonical coordinates

$\displaystyle X^{\alpha}=$ $\displaystyle x^{\alpha},$ (77) $\displaystyle
L_{\alpha}=$ $\displaystyle
l_{0\alpha}+\frac{l_{1\alpha}-b_{\alpha}}{\omega}.$ (78)

It is shown in Oancea et al. (2020) that such substitutions could be generated
as the linearization of coordinates changes. The Hamiltonian Eq. (45) under
this transformation becomes

$\displaystyle H^{\prime}(X,L)$ $\displaystyle=H(x,l)$ (79)
$\displaystyle=H\left(X^{\alpha},L_{\alpha}-\frac{l_{1\alpha}-b_{\alpha}}{\omega}\right)$
(80) $\displaystyle=H(X,L)-\frac{1}{\omega}\frac{\partial H_{0}}{\partial
L_{\alpha}}(l_{1\alpha}-b_{\alpha})$ (81) $\displaystyle=H_{0}(X,L),$ (82)

where Eq. (45) is used in obtaining the last equality. Therefore, in the new
coordinates $(X^{\alpha},L_{\alpha})$, the Hamiltonian reduces to

$H^{\prime}(X,L)=\frac{1}{2}g^{\alpha\beta}(X)L_{\alpha}L_{\beta}.$ (83)

The corresponding Hamilton’s equations of motion are

$\displaystyle\begin{pmatrix}\dot{X}^{\alpha}\\\
\dot{L}_{\alpha}\end{pmatrix}=T^{\prime}\begin{pmatrix}\frac{\partial
H^{\prime}}{\partial X^{\beta}}\\\ \frac{\partial H^{\prime}}{\partial
L_{\beta}}\end{pmatrix},$ (84)

where $T^{\prime}$ is the Poisson tensor in $(X,L)$ Marsden (2013). The
Poisson tensor could be written as

$\displaystyle T^{\prime}=\begin{pmatrix}0\quad&\delta^{\alpha}_{\beta}\\\
-\delta^{\beta}_{\alpha}\qquad&\frac{k_{\alpha\beta}}{\omega}\end{pmatrix},$
(85)

where $k_{\alpha\beta}$ is defined in Eq. (54) (it is compared with the Berry
curvature term of condensed matter physics in Sec. IV.3.2). Hamilton’s
equations of motion in the new variables are

$\displaystyle\dot{X}^{\alpha}=$ $\displaystyle L^{\alpha},$ (86)
$\displaystyle\dot{L}_{\alpha}=$
$\displaystyle\Gamma^{\mu}_{\nu\alpha}L_{\mu}L^{\nu}+\frac{1}{\omega}k_{\alpha\beta}L^{\beta}.$
(87)

Covariant differentiation of Eq. (86) gives

$\frac{D\dot{X}^{\alpha}}{D\lambda}=\frac{DL^{\alpha}}{D\lambda}=\frac{1}{\omega}k^{\alpha}_{\beta}L^{\beta},$
(88)

which is precisely the trajectory equation given in Eq. (55). This shows that
although the Hamiltonian of Eq. (45) contains the gauge-dependent term, the
trajectory equation obtained from it is gauge invariant. The reason is, as
explained in Eq. (11), $U(1)$ gauge transformation freedom of the polarization
basis $m^{\alpha}$ is constrained by the requirement that it should be Fermi
propagated along the trajectory.

##  V Stress-energy tensor up to the subleading order

To see how energy flows as waves propagate, we calculate the stress-energy
tensor up to the subleading order approximation using the field tensor, given
in Eq. (30). The stress-energy tensor due to the electromagnetic field is
given by the relation

$4\pi
T_{\alpha\beta}=F_{\alpha\gamma}F_{\beta}^{\gamma}-\frac{1}{4}g_{\alpha_{\beta}}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}Re\left[{\cal
F}^{+1}_{\alpha\gamma}\tilde{\cal F}^{+1\gamma}_{\beta}\right],$ (89)

where $Re\left[z\right]$ is the real part of $z$. This equation can be
simplified by substituting ${\cal F}^{+1}_{\alpha\gamma}$ from Eq. (30)

$4\pi T_{\alpha\beta}=\frac{\omega^{2}}{2}Re\left[{\cal
Z}_{\alpha\gamma}\tilde{\cal Z}_{\beta}^{\gamma}\right].$ (90)

As ${\cal Z}_{\alpha\gamma}$ is self-dual, it can be expanded as (see Eq.
(29))

${\cal Z}=\Phi_{0}\mathbf{U}+\Phi_{1}\mathbf{W}+\Phi_{2}\mathbf{V},$ (91)

where

$\displaystyle\Phi_{0}=$ $\displaystyle\frac{1}{2}{\cal
Z}_{\alpha\gamma}\left(l^{\alpha}m^{\gamma}-l^{\gamma}m^{\alpha}\right),$
$\displaystyle\Phi_{2}=$ $\displaystyle\frac{1}{2}{\cal
Z}_{\alpha\gamma}\left(\tilde{m}^{\alpha}n^{\gamma}-\tilde{m}^{\gamma}n^{\alpha}\right),$
$\displaystyle\Phi_{1}=$ $\displaystyle-\frac{1}{4}{\cal
Z}_{\alpha\gamma}\left(m^{\alpha}\tilde{m}^{\gamma}-m^{\gamma}\tilde{m}^{\alpha}-l^{\alpha}n^{\gamma}+l^{\gamma}n^{\alpha}\right).$
(92)

Substituting ${\cal Z}_{\alpha\gamma}$ from Eq. (31), we get

$\displaystyle\Phi_{0}=$
$\displaystyle\frac{ia}{\omega}\sigma,\qquad\Phi_{1}=\frac{ia}{2\omega}\left(\chi+2\tau\right),$
$\displaystyle\Phi_{2}=$ $\displaystyle
a-\frac{ia}{\omega}\left(-\frac{a_{,\alpha}}{a}n^{\alpha}-\tilde{\mu}+\frac{\gamma-\tilde{\gamma}}{2}\right),$
(93)

where $\sigma=-m^{\alpha}l_{\alpha;\beta}m^{\beta}$,
$\chi=\tilde{m}^{\beta}m_{\beta;\alpha}m^{\alpha}$,
$\tau=l^{\beta}m_{\beta;\alpha}n^{\alpha}$,
$\mu=\tilde{m}^{\beta}n_{\beta;\alpha}m^{\alpha}$ and
$\gamma-\tilde{\gamma}=-2\tilde{m}^{\alpha}m_{\alpha;\beta}n^{\beta}$ are
Newman-Penrose scalars. Thus, up to the subleading order in $1/\omega$,

$4\pi
T_{\alpha\beta}=\frac{\omega^{2}}{2}Re\Big{[}\Phi_{0}\tilde{\Phi}_{2}U_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}+\Phi_{1}\tilde{\Phi}_{2}W_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}\\\
+\Phi_{2}\tilde{\Phi}_{0}V_{\alpha\gamma}\tilde{U}_{\beta}^{\gamma}+\Phi_{2}\tilde{\Phi}_{1}V_{\alpha\gamma}\tilde{W}_{\beta}^{\gamma}+\Phi_{2}\tilde{\Phi}_{2}V_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}\Big{]}.$
(94)

However, we have

$\displaystyle
U_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}=\tilde{m}_{\alpha}\tilde{m}_{\beta},\quad
W_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}=-\tilde{m}_{\alpha}l_{\beta}-\tilde{m}_{\beta}l_{\alpha},$
$\displaystyle
V_{\alpha\gamma}\tilde{U}_{\beta}^{\gamma}=m_{\alpha}m_{\beta},\quad
V_{\alpha\gamma}\tilde{W}_{\beta}^{\gamma}=-m_{\alpha}l_{\beta}-l_{\alpha}m_{\beta},$
$\displaystyle
V_{\alpha\gamma}\tilde{V}_{\beta}^{\gamma}=l_{\alpha}l_{\beta}.$ (95)

Collecting these values, we obtain, up to the subleading order in $1/\omega$,

$\displaystyle 4\pi T_{\alpha\beta}$
$\displaystyle=\frac{\omega^{2}a^{2}}{2}l_{\alpha}l_{\beta}+\frac{i\omega
a^{2}}{4}\bigg{(}2\left(\sigma\tilde{m}_{\alpha}\tilde{m}_{\beta}-\tilde{\sigma}m_{\alpha}m_{\beta}\right)$
$\displaystyle-$ $\displaystyle
2\left(\mu-\tilde{\mu}+\gamma-\tilde{\gamma}\right)l_{\alpha}l_{\beta}-\left(\chi+2\tau\right)\left(\tilde{m}_{\alpha}l_{\beta}+\tilde{m}_{\beta}l_{\alpha}\right)$
$\displaystyle+\left(\tilde{\chi}+2\tilde{\tau}\right)\left(m_{\alpha}l_{\beta}+m_{\beta}l_{\alpha}\right)\bigg{)}$
$\displaystyle=\frac{\omega^{2}a^{2}}{2}L_{\alpha}L_{\beta}+\frac{i\omega
a^{2}}{2}\left(\sigma\tilde{m}_{\alpha}\tilde{m}_{\beta}-\tilde{\sigma}m_{\alpha}m_{\beta}\right),$
(96)

where we redefined the wave vector as

$L_{\alpha}=l_{\alpha}+\frac{i}{2\omega}\Big{(}-\left(\mu-\tilde{\mu}+\gamma-\tilde{\gamma}\right)l_{\alpha}+\left(\tilde{\chi}+2\tilde{\tau}\right)m_{\alpha}\\\
-\left(\chi+2\tau\right)\tilde{m}_{\alpha}\Big{)}.$ (97)

The expression shows that the wave carries transverse stress due to the
nonvanishing shear $\sigma$. The significance of this result is that we have
obtained it without using the solution of the Hamilton-Jacobi equation (44).
We could thus interpret this result as an independent verification of the spin
Hall effect as depicted by the fact that the transverse stress of the wave is
frequency-dependent. The energy-momentum tensor is not unique, in the limit of
spin optics, in the sense that the tensor

$\Theta^{\mu\nu}=T^{\mu\nu}+\nabla_{\lambda}A^{\mu\nu\lambda};\qquad
A^{\mu\nu\lambda}=-A^{\mu\lambda\nu}$ (98)

also satisfies the criterion to be the energy-momentum tensor. Thus, it is not
a surprise that with the appropriate choice of $A^{\mu\nu\lambda}$ we could
reduce this energy-momentum tensor into the expression given in Ref. Dolan
(2018), which is derived by a similar mathematical formulation. As expected,
the direction of energy flow is not along the direction of the wave vector
$l^{\alpha}$, a characteristic of waves propagating in anisotropic medium.

##  VI Comparison of results

Ref. Oancea et al. (2020) also developed a covariant formulation of the spin
Hall effect. Our results differ slightly from theirs, primarily owing to the
difference in our eikonal function Eq. (8) with theirs. In their formulation,
the amplitude $a^{\alpha}$ is assumed to be a function of the phase gradient
$l^{\alpha}$, that is, $a^{\alpha}=a^{\alpha}\left(\lambda,l(\lambda)\right)$.
Using the WKB analysis with this form of eikonal function, the following
equation for the ray trajectory was obtained:

$\dot{x}^{\gamma}=\frac{1}{\omega}\left(l^{\gamma}-B^{\gamma}-l_{\mu}\frac{\partial
B^{\mu}}{\partial l_{\gamma}}\right),$ (99)

where

$\displaystyle
B_{\beta}(\lambda,l(\lambda))=\frac{i}{2}\left(\tilde{m}^{\alpha}\overset{h}{\nabla}_{\beta}m_{\alpha}-m^{\alpha}\overset{h}{\nabla}_{\beta}\tilde{m}_{\alpha}\right)=i\tilde{m}^{\alpha}\overset{h}{\nabla}_{\beta}m_{\alpha},$
$\displaystyle\overset{h}{\nabla}_{\beta}m_{\alpha}=\nabla_{\beta}m_{\alpha}+\Gamma^{\mu}_{\beta\nu}l_{\mu}\frac{\partial
m_{\alpha}}{\partial l_{\nu}}.$ (100)

Thus, if we start with the WKB expansion whose amplitude does not depend on
the phase gradient, that is, if we take

$a^{\alpha}(\lambda,l(\lambda))=a^{\alpha}(\lambda).$ (101)

then we could show that the ray trajectory derived by Ref. Oancea et al.
(2020) would be the same as the propagation equation of Eq. (55).

Ref. Frolov (2020) started with the following form of the WKB expansion for
the vector potential:

$A^{\alpha}=a^{\alpha}(\lambda)e^{i{\cal S}},$ (102)

where $l_{\alpha}={\cal S}_{;\alpha}/\omega$ is the wave vector, and $\lambda$
is the usual parameter from above. The only difference in this eikonal
function from the one given in Eq. (8) is: here, the phase gradient is not
expanded in powers of $1/\omega$, that is, $l^{\alpha}=l_{0}^{\alpha}$. Thus,
all of our Eqs. (52)-(66) for spin optics would be the same as Frolov’s if
$l_{1}^{\alpha}=0$. However, as explained above Eq. (11), we could not take
$l_{1}^{\alpha}=0$ as we are using parallel propagated tetrad in the leading
order approximation.

##  VII Discussion and conclusions

We formulated the theory of spin optics using WKB formalism, where we expand
both the amplitude and phase in powers of $1/\omega$, $\omega$ being the
characteristic frequency. This expansion in both the amplitude and phase is
necessary if we want to simplify the problem by imposing gauge conditions (for
example, we needed to use the Fermi propagated tetrad here) that reduce some
of the freedom in the transformation of the null tetrad. This novel
application of the WKB formalism is used to derive the evolution equations for
both the propagation vector and polarization tensor. We are aware of only the
evolution equation for the propagation vector derived in the literature. In
the geometric optics approximation, polarization and propagation vectors are
two of the components of a parallel propagated null tetrad. This requirement
of parallel propagation, which is necessary in the subleading order also,
restricts some of the freedom in the transformation of the null tetrad,
thereby resulting in the observer/ gauge independent spin Hall effect. We have
proved that our result is indeed observer independent in Sec. IV.3.5 and thus
distinguishes itself from the observer/emitter-dependent effects occurring in
the geometric optics regime Dahal & Terno (2021). Such effects are: 1) the
special relativistic effect due to the change in polarization direction
occurring from the Wigner rotation and 2) the general relativistic effect due
to the absence of a global reference direction, making us unable to determine
a unique standard polarization triad in curved spacetime. Moreover, we have
pointed out in Ref. Dahal & Terno (2021) that the initially divergent
trajectories do not reconverge; the separation between the rays of different
frequencies increases with increasing distance. We could draw the trajectories
from numerical calculations, confirming that the initially divergent
trajectories do not reconverge. This is because the gravitational spin Hall
effect results from spin-orbit interaction, and neither spin nor orbital
angular momentum reverses in direction as the particle crosses the distance of
the closest approach, causing the effect of spin-orbit interaction to
increase, not decrease.

One of the ways to understand the nature of light propagation in curved
spacetime is to carry forward an analogy from condensed matter physics, where
the phenomena are well understood and experimentally verified Bliokh et al.
(2008); Dooghin et al. (1992); Volyar (1998); Kundikova (1999). Notably, this
analogy is beneficial when several authors have approached this problem with
the eikonal formalism itself and obtained slightly different results. The
analogy with the optical Magnus effect from condensed matter physics revealed
that the Berry phase and spin Hall effect are closely related to the dynamics
of the intrinsic angular momentum of the wave. In particular, the spin Hall
effect results from the bending of the trajectory of photons with nonzero
spin. Moreover, the transverse deflection of the ray trajectory is
proportional to the curvature of the ray Bliokh et al. (2008)(or equivalently
to the curvature of the spacetime, where the ray is propagating Eq.(55)).

The derivation of the spin Hall effect made here is based on classical
arguments. However, one can make the quantum mechanical interpretation in the
following manner. The geometric optics approximation is valid in the infinite
frequency limit, which implies the evolution of the confined wave packet. The
spin Hall effect arises while extending this approximation to waves of finite
but large frequency. This effect is due to the interference of multiple
partial plane waves constituting the wave packet that propagates in slightly
different directions and thus acquires slightly different geometric phases.
Therefore, because of the transverse gradient of the Berry phase, which is the
phase of the plane wave in the packet, the spin Hall effect is observed.

###### Acknowledgements.

PKD is supported by an International Macquarie University Research Excellence
Scholarship.

##  Appendix A Geometric optics limit

All the results of geometrical optics could be retrieved by taking Eqs.
(12)-(14) and substituting $m_{1}^{\beta}=0=l_{1}^{\beta}$. The Lorenz
condition Eq. (12) and the wave equation Eq. (13) in the leading order
approximation in $\omega$ reduces to

$l_{0}^{\alpha}m_{0\alpha}=0=l_{0}^{\alpha}l_{0\alpha}.$ (103)

Next, we calculate $\tilde{m}_{0\alpha}j^{\alpha}$ from Eq. (13) by taking
$m_{1}^{\alpha}=0=l_{1\mu}$ as they are subleading order terms in $\omega$ and
thus irrelevant in geometric optics approximation, to obtain

$l^{\beta}_{0;\beta}+2\tilde{m}_{0\alpha}m^{\alpha}_{0;\beta}l_{0}^{\beta}+2\frac{a_{;\beta}}{a}l_{0}^{\beta}=0.$
(104)

Since the term $\tilde{m}_{0\alpha}m^{\alpha}_{0;\beta}l_{0}^{\beta}$ is
purely imaginary and the remaining terms

$l^{\beta}_{0;\beta}+2\frac{a_{;\beta}}{a}l_{0}^{\beta},$

are purely real, they should be separately zero, thereby giving

$l^{\beta}_{0;\beta}+2\frac{a_{;\beta}}{a}l_{0}^{\beta}=0,\qquad
m^{\alpha}_{0;\beta}l_{0}^{\beta}=0.$ (105)

In the geometric optics approximation, these are the entire set of equations
for electromagnetic waves in curved spacetime.

##  Appendix B Checking self-duality

To verify that the tetrad
$\left(\dot{x}^{\alpha},n^{\alpha},m^{\alpha},\tilde{m}^{\alpha}\right)$,
satisfying Eqs. (62), (63) and (65) in the subleading order, is not a self-
dual solution, let us first calculate Eq. (34)

${\cal
Z}_{\alpha\beta}l^{\alpha}\tilde{m}^{\beta}=\frac{ia}{\omega}\left(-\frac{a_{;\alpha}}{a}l_{0}^{\alpha}+m_{0\alpha;\beta}l_{0}^{\alpha}\tilde{m}_{0}^{\beta}\right)\\\
=\frac{ia}{\omega}\left(\frac{1}{2}l^{\alpha}_{0;\alpha}-m_{0}^{\alpha}l_{0\alpha;\beta}\tilde{m}_{0}^{\beta}\right)=0,$
(106)

where Eq. (105) is used to obtain this identity. Similarly, Eq.(33) gives

${\cal
Z}_{\alpha\beta}\left(\tilde{m}^{\alpha}m^{\beta}-l^{\alpha}n^{\beta}\right)=\frac{i}{\omega}\left(\frac{a_{;\alpha}}{a}m_{0}^{\alpha}-m_{0\alpha;\beta}l_{0}^{\alpha}n_{0}^{\beta}\right)\\\
=\frac{i}{\omega}\left(-m^{\alpha}_{0;\alpha}-ib_{\alpha}m_{0}^{\alpha}-m_{0\alpha;\beta}l_{0}^{\alpha}n_{0}^{\beta}\right)=0,$
(107)

where Eq. (66) is used to arrive at this identity. Finally, Eq. (32) gives

${\cal
Z}_{\alpha\beta}m^{\alpha}n^{\beta}=\frac{i}{\omega}\left(-m_{0\alpha;\beta}m_{0}^{\beta}n_{0}^{\alpha}\right)\\\
=\frac{i}{\omega}\left(m_{0}^{\alpha}n_{0\alpha;\beta}m_{0}^{\beta}\right)\equiv\frac{i}{\omega}\tilde{\lambda},$
(108)

where $\lambda$ denotes the Newman-Penrose scalar. Hence Eq. (32) is not
satisfied unless $\tilde{\lambda}=0$.

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