# Spinoptics in a curved spacetime

Valeri P. Frolov<EMAIL_ADDRESS>Theoretical Physics Institute,
Department of Physics, University of Alberta,
Edmonton, Alberta, T6G 2E1, Canada

###### Abstract

In this paper we study propagation of the high frequency electromagnetic waves
in a curved spacetime. We discuss a so call spinoptics approach which
generalizes a well known geometric optics approximation and allows one to take
into account spin-dependent corrections. A new element proposed in this work
is the use of effective action which allows one to derive spinoptics
equations. This method is simpler and more transparent than the earlier
proposed methods. It is explicitly covariant and can be applied to an
arbitrary spacetime background. We also demonstrate how the initial value
problem for the high frequency electromagnetiv wave can be soled in the
developed spinoptics approximation.

Alberta Thy 4-24

##  I Introduction

Most information in the cosmology and astrophysics is obtained by observation
of electromagnetic radiation. In this paper we consider propagation of a high
frequency electromagnetic wave in a curved spacetime and discuss how its
polarization affects the wave propagation. Before study of the problem in the
curved background let us remind basic well known properties of Maxwell field
$F_{\mu\nu}$ in a flat spacetime. A general solution of source-free Maxwell
equations can be presented as a superposition of the plane monochromatic
waves, characterized by a 4D wave vector ${\boldsymbol{k}}$. For each
${\boldsymbol{k}}$ there exist two linearly independent solutions with
different polarizations. To identify the polarization state one can consider a
following gedanken experiment. Consider an inertial observer with 4D velocity
${\boldsymbol{U}}$. Two vectors ${\boldsymbol{U}}$ and ${\boldsymbol{k}}$
uniquely determine a 2D plane $\Pi$ orthogonal to both vectors. The electric
field $E^{\mu}=F^{\mu\nu}U_{\nu}$ in the frame of the observer
${\boldsymbol{U}}$ lies in a plane $\Pi$. In a general case its ”end point”
moves along an ellipse. One can present such a wave as a superposition, of
right- and left-circular polarized waves. In ”quantum” language, these states
are characterized by their helicity. It is easy to check that if the
monochromatic plane wave is right- or left-polarized for the observer
${\boldsymbol{U}}$ then it has the same property for any other inertial
observer. This is in an agreement with Lorentz invariance of the helicity.

For electromagnetic waves in a media the left and right circularly polarized
waves propagate at slightly different speed. Since a linear polarization state
is a superposition of two equal-amplitude right- and left- circularly
polarized components, the direction of the vector of the linear polarization
slowly rotates (Faraday effect).

In many practically important situations the wavelength of the electromagnetic
radiation is much shorter compared to the size of structures with which it
interacts. In such cases the geometric optics (high frequency) approximation
is very useful. In this approach one searches for approximate solutions of the
Maxwell equation with the following form of the 4D vector potential

$A_{\mu}=am_{\mu}\exp(i\omega S)\,.$ (1)

Here $\omega$ is a large frequency, and amplitude $a$ and phase function $S$
are real functions which slowly change at the characteristic wavelength scale
$\lambda=2\pi/\omega$. For linear polarization ${\boldsymbol{m}}$ is a real
vector, while for the circular polarization ${\boldsymbol{m}}$ is a complex
null vector. In this approximation finding a solution of the Maxwell equations
reduces to study of null rays propagation.

The geometric optics approach has been applied for study solutions of the
Maxwell equations in a curved spacetime. It is nicely described in the MWT
book Misner  _et al._ (1974), where many useful references can be found (for a
more recent review see e.g. Dolan (2018)). The main results can be formulated
as follows: (1) $k_{\mu}=S_{,\mu}$ is a parallel propagated null vector. Its
integral lines are geodesic null rays. (2) For linear polarization the vector
of polarization $m^{\mu}$ is parallel propagated along null rays. (3) Scalar
amplitude $a$ obeys the conservation law
$\nabla_{k}(a^{2}{\boldsymbol{k}})=0$.

Long time ago it was demonstrated that in the general case the polarization
vector of the electromagnetic radiation propagating in a curved spacetime can
rotate Skrotskii (1957); Plebanski (1960). This so called gravitational
Faraday effect was discussed and studied for special cases in a number of
publication Connors  _et al._ (1980); Connors and Stark (1977); Piran and
Safier (1985); Ishihara  _et al._ (1988); Nouri-Zonoz (1999); Sereno (2004);
Li  _et al._ (2022a). This effect can be explained by assuming that waves with
different circular polarization in an external gravitational field propagate
in a slightly different way. This is a result of the interaction of the spin
of photons with the spacetime curvature. To study this effect one should
slightly modify (”upgrade”) the geometric optics approximation.

The main idea of such spinoptics (or gravitational spin-Hall) approximation is
the following. One starts with a standard ansatz similar to (1) for the vector
potential with a complex null vector corresponding to the circular polarized
state. The helicity dependent contribution appears only in the subleading
terms of the $1/\omega$ expansion. To take it into account, one should include
these helicity dependent terms in the eikonal equation. At short distances
this modifies the rays trajectory only slowly. However, for a long distance
such a spin effect is ”enhanced”. For two opposite circular polarizations the
ray trajectories in an external gravitational field are slightly different.
They are still null, but in a general case they are not geodesic. As a result
a linearly polarized beam of radiation, which it is a superposition of
opposite circular polarization states, splits and becomes a superposition of
two spatially separated circular polarized beams.

Spinoptics in the special types of the gravitational fields (Kerr black holes
and cosmology) was discussed in many publications (see e.g. Frolov and Shoom
(2011, 2012); Yoo (2012); Shoom (2021); Oancea and Harko (2024); Shoom (2024)
and references therein). In papers Oancea  _et al._ (2019, 2020); Frolov
(2020); Dahal (2023) a general formulation of the spinoptics equations in an
arbitrary gravitational field was proposed. Spinoptics approximation for high
frequency gravitational waves propagating in a curved spacetime background was
developed in Yamamoto (2018); Andersson  _et al._ (2021); Dahal (2022); Li
_et al._ (2022b); Kubota  _et al._ (2024). In the present paper we propose a
different derivation of the spinoptics equations for the high frequency
electromagnetic waves in a curved geometry. Instead of study the high-
frequency expansion of the Maxwell equations, we use an approach which is
similar to the one developed earlier in the paper Stachel and Plebanski
(1977). Namely, by substituting into the action for the electromagnetic field
the ansatz, similar to (1), we derive an effective action. We demonstrate,
that such an effective action calculated in the high-frequency approximation,
which includes first order in $1/\omega$ frequency helicity sensitive terms,
allows one to obtain the spinoptics equations.

In this paper we use units in which $G=c=1$, and the signature conventions of
the book Misner  _et al._ (1974).

##  II Effective action

Let us formulate a problem. We study approximate high frequency solutions of
the source-free Maxwell equations in a curved spacetime $\mathcal{M}$ with
metric $g_{\mu\nu}$

$F^{\mu\nu}_{\,\,;\nu}=0\,.$ (2)

For this purpose we proceed as follows. Consider a complex Maxwell field
$F_{\mu\nu}$ the action for which is of the form

$W=\dfrac{1}{8}\int\mathcal{F}^{2}\,\sqrt{-g}\,d^{4}x,\hskip
14.22636pt\mathcal{F}^{2}=F_{\mu\nu}\bar{F}^{\mu\nu}\,.$ (3)

To develop the spinoptics approach we write a complex field potential
$A_{\mu}$ in the form

$A_{\mu}=aM_{\mu}e^{i\omega S}\,.$ (4)

Here amplitude $a$ and phase $S$ are real scalar functions. $M^{\mu}$ is a
complex null vector, satisfying the following conditions

$M_{\mu}{M}^{\mu}=\bar{M}_{\mu}\bar{M}^{\mu}=0,\hskip
5.69046ptM_{\mu}\bar{M}^{\mu}=1,\hskip 5.69046ptM^{\mu}S_{,\mu}=0\,.$ (5)

As in the standard geometric optics a surface $S=$const defines a wavefront
and the last condition in (5) means that the complex null vectors
${\boldsymbol{M}}$ and $\bar{{\boldsymbol{M}}}$ are tangent to these surfaces.
Simple calculations give the following expression for the complex field
strength

$\begin{split}&F_{\mu\nu}=A_{\nu;\mu}-A_{\mu;\nu}=\Big{[}i\omega
a(S_{,\mu}M_{\nu}-S_{,\nu}M_{\mu})\\\
&+[(aM_{\nu})_{;\mu}-(aM_{\mu})_{;\nu}]\Big{]}\exp(i\omega S)\,.\end{split}$
(6)

Substituting this expression in $\mathcal{F}^{2}$ and keeping terms of the
second and first order in $\omega$ one obtains

$\begin{split}&\mathcal{F}^{2}=4\omega^{2}a^{2}\left[\dfrac{1}{2}(\nabla
S)^{2}-\dfrac{1}{\omega}B^{\mu}S_{,\mu}\right]\,,\\\
&B_{\mu}=i\bar{M}^{\nu}M_{\nu;\mu}\,.\end{split}$ (7)

It is easy to check that the vector $B^{\mu}$ is real.

By substituting the expression (7) into the action (3) one obtains a
functional which depends on three variables: the amplitude $a$, polarization
vector $\bf{M}$ and the phase function $S$. We write
$W=\dfrac{1}{2}\omega^{2}I$, use $I$ is the effective action for the
spinoptics approximation. It differs from $W$ by a constant factor. This does
not affect the equations of motion.

Since in the derivation of $\mathcal{F}^{2}$ we used relations (5), one should
include the corresponding constraints in the obtained effective action. Let us
denote

$\begin{split}&\Lambda=\dfrac{1}{2}\bar{\lambda}_{1}M_{\mu}M^{\mu}+\dfrac{1}{2}\lambda_{1}\bar{M}_{\mu}\bar{M}^{\mu}+\lambda_{2}({M}_{\mu}\bar{M}^{\mu}-1)\\\
&+\bar{\lambda}_{3}M^{\mu}S_{,\mu}+\lambda_{3}\bar{M}^{\mu}S_{,\mu}\,.\end{split}$
(8)

Here $\lambda_{1}$ and $\lambda_{3}$ are complex Lagrange multipliers, and
$\lambda_{2}$ is a real one. We include these constraints into the effective
action $I$ and write in in the form

$I=\int d^{4}x\,\sqrt{-g}\,\Big{[}a^{2}[\dfrac{1}{2}(\nabla
S)^{2}-\dfrac{1}{\omega}B^{\mu}S_{,\mu}]+\Lambda\Big{]}\,.$ (9)

##  III Equations

Let us note that constraint $\Lambda$ does not depend on the amplitude $a$.
Variation of $I$ with respect to $a$ gives

$\dfrac{\delta I}{\delta a}=2a[\dfrac{1}{2}(\nabla
S)^{2}-\dfrac{1}{\omega}B^{\mu}S_{,\mu}]\,.$ (10)

Since $a\neq 0$ we write the equation obtained by varying $a$ in the form

$H=\dfrac{1}{2}(\nabla S)^{2}-\dfrac{1}{\omega}B^{\mu}S_{,\mu}=0\,.$ (11)

Let us denote

$J^{\mu}=a^{2}(S^{,\mu}-\dfrac{1}{\omega}B^{\mu})+\bar{\lambda}_{3}M^{\mu}+\lambda_{3}\bar{M}^{\mu}\,.$
(12)

Then the variation of the action $I$ over $S$ gives

$J^{\mu}_{\ ;\mu}=0\,.$ (13)

Variation of $I$ with respect to $\bar{M}^{\mu}$ gives

$-\dfrac{a^{2}i}{\omega}S^{,\nu}M_{\mu;\nu}+\lambda_{1}\bar{M}_{\mu}+\lambda_{2}M_{\mu}+\lambda_{3}S_{,\mu}=0\,.$
(14)

By multiplying this relation by $M^{\mu}$ and using (5) one gets
$\lambda_{1}=0$.

Let us note that the relations (5) are invariant under the rotation in the 2D
plane spaned by the complex null vectors ${\boldsymbol{M}}$ and
$\bar{{\boldsymbol{M}}}$

$M^{\mu}\to e^{i\psi}M^{\mu},\hskip 14.22636pt\bar{M}^{\mu}\to
e^{i\psi}\bar{M}^{\mu}\,.$ (15)

Under this transformation the vector ${\boldsymbol{B}}$ changes as follows
$B_{\mu}\to B_{\mu}-\psi_{,\mu}$. We use this freedom to put

$B^{\mu}S_{,\mu}=0\,.$ (16)

Then by multiplying (14) by $\bar{M}^{\mu}$ and using (5) and (16) one gets
$\lambda_{2}=0$. Thus one can write (14) in the following form

$i\dfrac{a^{2}}{\omega}S^{,\nu}M_{\mu;\nu}=\lambda_{3}S_{,\mu}\,.$ (17)

Following the prescription of Stachel and Plebanski (1977) one should identify
(11) with a Hamilton-Jacobi equation for a particle with 4-momentum
$p_{\mu}=S_{,\mu}$

$H(p,x)=\dfrac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}-\dfrac{1}{\omega}B^{\mu}p_{\mu}\,.$
(18)

This Hamiltonian determines a class of mechanical trajectories associated with
the high frequency electromagnetic waves. The corresponding Hamiltonian
equations are

$\begin{split}&\dfrac{dx^{\mu}}{ds}=\dfrac{\partial H}{\partial
p_{\mu}}=p_{\mu}-\dfrac{1}{\omega}B_{\mu}\,,\\\
&\dfrac{dp_{\mu}}{ds}=-\dfrac{\partial H}{\partial x^{\mu}}\,.\end{split}$
(19)

Let us note, that equations (19) are similar to the equations of motion for a
charges particle with the charge $1/\omega$ in the magnetic field with the
potential $B_{\mu}$.

##  IV Null ray congruence and complex null tetrad associated with it

In order to formulate spinoptics equations we use a special complex null
frame. We denote by $l^{\mu}$ a tangent vector to the ray

$l^{\mu}=\dfrac{dx^{\mu}}{ds}=p^{\mu}-\dfrac{1}{\omega}B^{\mu}\,.$ (20)

Then the second equation in (19) can be written as follows

$w^{\mu}\equiv
l^{\nu}l^{\mu}_{\,\,;\nu}=\dfrac{1}{\omega}K^{\mu}_{\,\,\nu}l^{\nu},\hskip
14.22636ptK_{\mu\nu}=B_{\nu;\mu}-B_{\mu;\nu}\,.$ (21)

This equation implies that

$l^{\mu}w_{\mu}=0,\hskip 14.22636ptl^{\mu}({\boldsymbol{l}}^{2})_{,\mu}=0\,.$
(22)

The latter equation shows that if ${\boldsymbol{l}}^{2}=0$ at some moment of
time, then ${\boldsymbol{l}}^{2}$ vanishes along all the trajectory and the
ray is null.

Let us consider a congruence of null rays and introduce a complex null tetrad
$(l^{\mu},m^{\mu},\bar{m}^{\mu},n^{\mu})$ associated with it. These vectors
are normalized as follows

$({\boldsymbol{m}},\bar{{\boldsymbol{m}}})=-({\boldsymbol{l}},{\boldsymbol{n}})=1\,,$
(23)

all other scalar products are zero. For a given congruence ${\boldsymbol{l}}$
there exists the following freedom in the definition of such a null complex
tetrad associated it and preserving these normalization conditions

  1. 1.

${\boldsymbol{l}}\to\gamma{\boldsymbol{l}},\hskip
14.22636pt{\boldsymbol{n}}\to\gamma^{-1}{\boldsymbol{n}}$ .

  2. 2.

${\boldsymbol{m}}\to e^{i\phi}{\boldsymbol{m}}$, $\bar{{\boldsymbol{m}}}\to
e^{-i\phi}\bar{{\boldsymbol{m}}}$ .

  3. 3.

${\boldsymbol{l}}\to{\boldsymbol{l}}$,
${\boldsymbol{m}}\to{\boldsymbol{m}}+\alpha{\boldsymbol{l}}$,
$\bar{{\boldsymbol{m}}}\to\bar{{\boldsymbol{m}}}+\bar{\alpha}{\boldsymbol{l}}$
,
${\boldsymbol{n}}\to{\boldsymbol{n}}+\bar{\alpha}{\boldsymbol{m}}+\alpha\bar{{\boldsymbol{m}}}+\alpha\bar{\alpha}{\boldsymbol{l}}$
.

  4. 4.

${\boldsymbol{m}}\to\bar{{\boldsymbol{m}}}$,
$\bar{{\boldsymbol{m}}}\to{\boldsymbol{m}}$.

One can use the normalization conditions to show that

$e^{\mu\nu\alpha\beta}l_{\mu}n_{\nu}m_{\alpha}\bar{m}_{\beta}=is,\hskip
14.22636pts=\pm 1\,.$ (24)

We call a complex null tetrad right-handed if the helicity parameter $s=1$,
and left-handed if $s=-1$. It is easy to see that the transformation 4 changes
the helicity to the opposite one.

Let us note that under the transformation 1 the acceleration vector
${\boldsymbol{w}}$ changes as follows

$w^{\mu}\to\gamma^{2}w^{\mu}+\gamma l^{\nu}\gamma_{,\nu}l^{\mu}\,.$ (25)

Since $({\boldsymbol{w}},{\boldsymbol{l}})=0$, (25) implies that one can
always find $\gamma$ such that the vector ${\boldsymbol{w}}$ takes the form

$w^{\mu}=\dfrac{1}{\omega}\kappa^{\mu},\hskip
14.22636pt\kappa^{\mu}=K^{\mu}_{\
\nu}l^{\nu}=\bar{\kappa}m^{\mu}+{\kappa}\bar{m}^{\mu}\,.$ (26)

The complex function $\kappa$ in (26) is of the order of 1.

In the limit $\omega\to\infty$ equation (21) reduces to a geodesic equation.
We restrict the choice of the complex null tetrad by requiring that in the
same limit its other vectors are also parallel propagated along the null ray.
Thus one has

$Dm^{\mu}=\dfrac{1}{\omega}Z^{\mu},\hskip
14.22636ptDn^{\mu}=\dfrac{1}{\omega}N^{\mu}\,,$ (27)

where complex vector $Z^{\mu}$ and real vector $N^{\mu}$ are of the order of
$\omega^{0}$. The normalization conditions imply

$\begin{split}&({\boldsymbol{l}},{\boldsymbol{\kappa}})=({\boldsymbol{n}},{\boldsymbol{N}})=({\boldsymbol{m}},{\boldsymbol{Z}})=(\bar{{\boldsymbol{m}}},\bar{{\boldsymbol{Z}}})=0\,,\\\
&({\boldsymbol{l}},{\boldsymbol{N}})=-({\boldsymbol{n}},{\boldsymbol{\kappa}})=0\,,\
\
({\boldsymbol{l}},{\boldsymbol{Z}})=-({\boldsymbol{m}},{\boldsymbol{\kappa}})=-\kappa\,,\\\
&({\boldsymbol{m}},{\boldsymbol{N}})=-({\boldsymbol{n}},{\boldsymbol{Z}}),\hskip
14.22636pt(\bar{{\boldsymbol{m}}},{\boldsymbol{Z}})=-({\boldsymbol{m}},\bar{{\boldsymbol{Z}}})\,.\end{split}$
(28)

The vectors $N^{\mu}$ and $Z^{\mu}$ in the complex null frame have the form

$\begin{split}&N^{\alpha}=N_{l}l^{\alpha}+N_{n}n^{\alpha}+N_{m}m^{\alpha}+\bar{N}_{{m}}\bar{m}^{\alpha}\,,\\\
&Z^{\alpha}=Z_{l}l^{\alpha}+Z_{n}n^{\alpha}+Z_{m}m^{\alpha}+Z_{\bar{m}}\bar{m}^{\alpha}\,.\end{split}$
(29)

Relations (28) impose restrictions on the coefficients in (29). Using them one
gets

$\begin{split}&N^{\alpha}=\bar{Z}_{l}m^{\alpha}+Z_{l}\bar{m}^{\alpha}\,,\\\
&Z^{\alpha}=Z_{l}l^{\alpha}+\kappa n^{\alpha}+iZm^{\alpha}\,.\end{split}$ (30)

Here $Z$ is a real function and $Z_{l}$ is a complex one. One still has
freedom to use transformations 2 and 3, provided that
$\phi=\dfrac{1}{\omega}\phi_{1}$ and $\alpha=\dfrac{1}{\omega}\alpha_{1}$,
where $\phi_{1}$ and $\alpha_{1}$ are of the order of 1. Under these
transformations one has

$\begin{split}&Dn^{\mu}\to
Dn^{\mu}+\dfrac{1}{\omega}[D\bar{\alpha}_{1}m^{\mu}+D{\alpha}_{1}\bar{m}^{\mu}]\,,\\\
&Dm^{\mu}\to
Dm^{\mu}+\dfrac{1}{\omega}[iD\phi_{1}m^{\mu}+D\alpha_{1}l^{\mu}]\,,\end{split}$
(31)

By choosing $D\phi_{1}=-Z$ and $D\alpha_{1}=-Z_{l}$ one can simplify
expressions in (30) and present them in the form

$N^{\mu}=0,\hskip 14.22636ptZ^{\mu}=\kappa n^{\mu}\,.$ (32)

The above results means that for large $\omega$ one can choose the complex
null tetrad
$({\boldsymbol{l}},{\boldsymbol{n}},{\boldsymbol{m}},\bar{{\boldsymbol{m}}})$
associated with the null congruence ${\boldsymbol{l}}$ so that the following
set of equations is valid

$Dl^{\mu}=\dfrac{1}{\omega}(\bar{\kappa}m^{\mu}+{\kappa}\bar{m}^{\mu}),\hskip
5.69046ptDn^{\mu}=0,\hskip 5.69046ptDm^{\mu}=\dfrac{1}{\omega}\kappa
n^{\mu}\,.$ (33)

##  V Spinoptics equations

Our goal now is write a set of the obtained earlier equations (19), (12),
(13), and (17) in terms of equations for the congruence of null rays and a
complex null tetrads associated with it. We write the complex null vector
${\boldsymbol{M}}$ in the form

$M^{\alpha}=m^{\alpha}+\dfrac{1}{\omega}\mu^{\alpha}\,.$ (34)

Normalization conditions (5) imply

$({\boldsymbol{\mu}},{\boldsymbol{m}})=0,\hskip
14.22636pt(\bar{{\boldsymbol{\mu}}},{\boldsymbol{m}})+({\boldsymbol{\mu}},\bar{{\boldsymbol{m}}})=0\,.$
(35)

Using the definition of ${\boldsymbol{B}}$ one gets

$\begin{split}&B^{\alpha}=b^{\alpha}+\dfrac{1}{\omega}\beta^{\alpha},\hskip
14.22636ptb_{\alpha}=i\bar{m}^{\nu}m_{\nu;\alpha}\,,\\\
&\beta_{\alpha}=i(\bar{\mu}^{\nu}m_{\nu;\alpha}+\bar{m}^{\nu}\mu_{\nu;\alpha})\,.\end{split}$
(36)

Since the difference between $B_{\mu}$ and $b_{\mu}$ is of the order of
$O(\dfrac{1}{\omega})$, one can simplify equations (20) and (26) by keeping in
them terms up to the order $O(\dfrac{1}{\omega})$. One has 111 The leading
order contribution to the field strength (6) is $F^{(0)}_{\mu\nu}=-i\omega
a(l_{\mu}m_{\nu}-l_{\nu}m_{\mu})\exp(i\omega S)$. It is easy to check that
${}^{{}^{*}\\!\\!}F^{(0)}_{\mu\nu}\equiv\dfrac{1}{2}e_{\mu\nu\alpha\beta}F^{(0)\alpha\beta}=isF^{(0)}_{\mu\nu}$.
That is in the leading order ${\boldsymbol{F}}$ is self-dual (for $s=1$) or
anti self-dual for $s=-1$.

$\begin{split}&l^{\mu}=p^{\mu}-\dfrac{1}{\omega}b^{\mu}\,,\\\
&\kappa^{\mu}=k^{\mu}_{\ \nu}l^{\nu},\hskip
14.22636ptk_{\mu\nu}=b_{\nu;\mu}-b_{\mu;\nu}\,.\end{split}$ (37)

Condition (16) implies that

$l^{\mu}b_{\mu}=O(\dfrac{1}{\omega})\,.$ (38)

The relations in the second line of (37) written in the explicit form give

$\begin{split}&\kappa_{\mu}=i(m_{\alpha;\nu\mu}-m_{\alpha;\mu\nu})\bar{m}^{\alpha}l^{\nu}\\\
&+i(\bar{m}_{\alpha;\mu}Dm^{\alpha}-{m}_{\alpha;\mu}D\bar{m}^{\alpha})\end{split}$
(39)

Equations (33) imply that $Dm^{\alpha}$ and $D\bar{m}^{\alpha}$ are of the
order of $O(\dfrac{1}{\omega})$, and hence in our approximation the terms in
the second line of (39) can be omitted. Using a formula for the commutator of
the covariant derivatives one gets

$\kappa_{\mu}=iR_{\mu\nu\alpha\beta}m^{\alpha}\bar{m}^{\beta}l^{\nu}\,.$ (40)

Relation $M^{\mu}S_{,\mu}=0$ in the leading order is satisfied, while in a
subleading order it gives

$l^{\alpha}\mu_{\alpha}=-m^{\alpha}b_{\alpha}\,.$ (41)

Relations (35) and (41) do not specify uniquely ${\boldsymbol{\mu}}$. In
particular, they remain valid under the transformation

$\mu^{\alpha}\to\mu^{\alpha}+\nu l^{\alpha}\,.$ (42)

”Dynamical” equation (17) gives

$\dfrac{ia^{2}}{\omega^{2}}[D\mu^{\alpha}+\kappa
n^{\alpha}+b^{\beta}m^{\alpha}_{\
;\beta}]=\lambda_{3}(l^{\alpha}+\dfrac{1}{\omega}b^{\alpha})\,.$ (43)

This relation shows that $\lambda_{3}$ is of the order $O(1/\omega^{2})$. This
implies that the terms with the Lagrange multiplier $\lambda_{3}$ in equation
(12) can be omitted and (13) takes the form

$(a^{2}l^{\mu})_{;\mu}=0\,.$ (44)

This relation has a simple interpretation: a number of photons in a beam
formed by null rays is conserved.

We denote $\lambda_{3}=(ia^{2}/2\omega^{2})\lambda$. Then in the leading order
equation (43) takes the form

$D\mu^{\alpha}+\kappa n^{\alpha}+b^{\beta}m^{\alpha}_{\ ;\beta}=\lambda
l^{\alpha}\,.$ (45)

Using the ambiguity (42) one can put $\lambda=0$.

###  V.1 Helicity dependence

Let us note that the vector $b^{\mu}$, and other quantities which are defined
in terms of it, are helicity sensitive. To specify a complex null tetrad we
impose a condition that it is right-handed. To get the results for the
opposite helicity one should make a substitution

$b_{\mu}\to sb_{\mu},\hskip 5.69046ptk_{\mu\nu}\to sk_{\mu\nu},\hskip
5.69046pt\kappa_{\mu}\to s\kappa_{\mu},\hskip 5.69046pt\kappa\to s\kappa\,.$
(46)

Then the basic spinoptics equations take the form

$\begin{split}&Dl^{\mu}=isR^{\mu}_{\
\nu\alpha\beta}m^{\alpha}\bar{m}^{\beta}l^{\nu}\,,\\\ &Dn^{\mu}=0,\hskip
5.69046ptDm^{\mu}=\dfrac{s}{\omega}\kappa n^{\mu}\,,\\\
&D\mu^{\alpha}=-s(\kappa n^{\alpha}+b^{\beta}m^{\alpha}_{\ ;\beta})\,,\\\
&(a^{2}l^{\mu})_{;\mu}=0\,.\end{split}$ (47)

##  VI Initial conditions

We specify the initial conditions as follows. Consider a 3D spacelike surface
$\Sigma$ imbedded in $\mathcal{M}$. We denote by $x^{\mu}$ ($\mu=0,1,2,3)$)
coordinates in the bulk spacetime $\mathcal{M}$ and by $y^{i}$ ($i=1,2,3$)
coordinates in $\Sigma$. Then $\Sigma$ is defined by the following equations
$x^{\mu}=x^{\mu}(y^{i})$. The induced metric on $\Sigma$ is

$h_{ij}=\dfrac{\partial x^{\mu}}{\partial y^{i}}\dfrac{\partial
x^{\nu}}{\partial y^{j}}g_{\mu\nu}\,.$ (48)

We denote by ${\boldsymbol{t}}$ a future directed unit vector orthogonal to
$\Sigma$. We denote also by $S_{0}$ the value of the phase function on
$\Sigma$

$S_{0}(y^{i})=S(x^{\mu}(y^{i}))\,.$ (49)

Equation $S_{0}(y)=$const defines a position of the 2D surface of equal phase
(wave surface) at the initial moment of time.

3D vector $S_{0,i}$ is orthogonal to the wave surface and is tangent to
$\Sigma$. It determines a direction of propagation of the wave. Denote
$h^{ij}S_{0,i}S_{0,j}=q^{2}$. Then a 3D unit vector in the direction of the
wave propagation is $\hat{e}_{i}=S_{0,i}/q$. Its 4D form is defined by the
conditions

$t^{\mu}e_{\mu}=0,\hskip 14.22636pt\dfrac{\partial x^{\mu}}{\partial
y^{i}}{e}_{\mu}=\hat{e}_{i}\,.$ (50)

We define two future directed null vectors on $\Sigma$

$l_{0}^{\mu}=\dfrac{1}{\sqrt{2}}(t^{\mu}+e^{\mu}),\hskip
14.22636ptn_{0}^{\mu}=\dfrac{1}{\sqrt{2}}(t^{\mu}-e^{\mu})\,.$ (51)

We denote by ${\boldsymbol{m}}_{0}$ the complex null vector at the initial
time and choose it to be tangent to $\Sigma$ and orthogonal to $l_{0}^{\mu}$
and $n_{0}^{\mu}$. We also specify the slowly changing function of the
amplitude of the field on $\Sigma$: $a=a_{0}(y^{i})$.

To obtain a solution of the Maxwell equations with given initial conditions in
the spinoptics approximation one proceeds as follows. One searches for a
solution in the form

$A_{\mu}=a\Big{(}m_{\mu}+\dfrac{1}{\omega}\mu_{\mu}\Big{)}\exp(i\omega S)\,.$
(52)

Using the initial conditions
$({\boldsymbol{l}}_{0},{\boldsymbol{n}}_{0},{\boldsymbol{m}}_{0},\bar{{\boldsymbol{m}}}_{0})$
on $\Sigma$ one should solve the ”dynamical equations” in the first two lines
of (47) The null rays are defined by the equation $dx^{\mu}/d\sigma=l^{\mu}$.
We assume that the surface $\Sigma$ is chosen so that only one ray passes
through each point in its vicinity. A null ray is defined by coordinates
$y^{i}$ of the point on $\Sigma$ where the ray intersects it. We choose
$\sigma=0$ at $\Sigma$. Then $(y^{i},\sigma)$ specify a point on the ray.
Using relation $S_{,\mu}=l_{\mu}+(s/\omega)b_{\mu}$ one can show in the
spinoptics approximation the phase function $S$ remains constant on a given
null ray up to the order of $O(1/\omega^{2})$. Really, for two points 1 and 2
on the ray one has

$\Delta S=\int_{1}^{2}dx^{\mu}S_{,\mu}=\int_{1}^{2}d\sigma
l^{\mu}(l_{\mu}+\dfrac{s}{\omega}b_{\mu})=O(1/\omega^{2})\,.$ (53)

To get this result we use relation (38). This means that the value of $S$ at
point $x^{\mu}$ is equal to the value of the phase function $S_{0}$ a the
point where a null ray passing through $x^{\mu}$ intersects $\Sigma$. The last
equation in (47) determines the amplitude $a$ of the field.

Let us summarize. In this paper we studied propagation of the high-frequency
electromagnetic waves in a curved space. We started with with the ansatz (4)
which (locally) reproduces the complex vector potential for a monochromatic
circularly polarized plane wave. After substitution of this ansatz into the
action for the complex electromagnetic field and keeping the leading and first
order subleading terms in its high frequency expansion, we derived an
effective action. The variation of this effective action produces the
spinoptics equations. We demonstrated that the helicity dependent contribution
enters in the subleading terms of these equations. We also demonstrated how
using the spinoptics equations one can find a solution for an initial value
problem for the high frequency electromagnetic ways. Let us emphasise that the
effective action method developed in this paper is simpler and more
transparent, then earlier discussed derivation of the spinoptics equations
based on the direct study of high frequency solutions of the Maxwell
equations. The developed effective action approach allows a natural
generalization for study spinoptics of gravitational waves propagating in a
curved spacetime.

## Acknowledgments

The author thanks the Natural Sciences and Engineering Research Council of
Canada and the Killam Trust for their financial support.

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