# Spin Hall effects and the localization of massless spinning particles

Abraham I. Harte<EMAIL_ADDRESS>Centre for Astrophysics and Relativity,
School of Mathematical Sciences Dublin City University, Glasnevin, Dublin 9,
Ireland  Marius A. Oancea<EMAIL_ADDRESS>Faculty of Physics,
University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria  Erwin
Schrödinger International Institute for Mathematics and Physics, University of
Vienna, Boltzmanngasse 9, 1090 Vienna, Austria  Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476
Potsdam, Germany

###### Abstract

The spin Hall effects of light represent a diverse class of polarization-
dependent physical phenomena involving the dynamics of electromagnetic wave
packets. In a medium with an inhomogeneous refractive index, wave packets can
be effectively described by massless spinning particles following
polarization-dependent trajectories. Similarly, in curved spacetime the
gravitational spin Hall effect of light is represented by polarization-
dependent deviations from null geodesics. In this paper, we analyze the
equations of motion describing the gravitational spin Hall effect of light. We
show that these equations are a special case of the Mathisson-Papapetrou
equations for spinning objects in general relativity. This allows us to use
several known results for the Mathisson-Papapetrou equations, and apply them
to the study of electromagnetic wave packets. We derive conservation laws, we
discuss the limits of validity of the spin Hall equations, and we study how
the energy centroids of wave packets, effectively described as massless
spinning particles, depend on the external choice of a timelike vector field,
representing a family of observers. In flat spacetime, the relativistic Hall
effect and the Wigner(-Souriau) translations are recovered, while our
equations also provide a generalization of these effects in arbitrary
spacetimes. We construct a large class of wave packets that can be described
by the spin Hall equations, but also find its limits by giving examples of
wave packets which are more general and are not described by the spin Hall
equations. Lastly, we examine the assumption that electromagnetic wave packets
are massless. While this is approximately true in many contexts, it is not
exact. We show that failing to carefully account for the limitations of the
massless approximation results in the appearance of unphysical “centroids”
which are nowhere near the wave packet itself.

##  I Introduction

Many observations of the physical world—particularly in astrophysical
contexts—involve measurements of electromagnetic and (more recently)
gravitational radiation. Interpreting this radiation requires a theoretical
model for its propagation. In the case of electromagnetic waves, one might
begin with Maxwell’s equations. In the case of gravitational waves, one might
instead use the Einstein field equation. Regardless, exact solutions are
rarely available and the geometric optics approximation111The geometric optics
approximation, with or without higher-order corrections, is sometimes referred
to as the high-frequency approximation, or as the Wentzel-Kramers-Brillouin
(WKB) approximation. is typically applied in order to make progress. This
assumes that wavelengths are small compared with all other relevant length
scales, and forms the basis for most of the theory of gravitational lensing
[1, 2, 3, 4, 5].

Mathematically, the geometric optics approximation allows the field equations,
which are partial differential equations, to be approximated by a set of
ordinary differential equations. The problem of solving partial differential
equations is thereby reduced to the much simpler problem of solving ordinary
differential equations. More specifically, this process shows that the
amplitudes and polarization states of high-frequency electromagnetic and
gravitational waves propagate along null geodesics. The resulting field acts,
in this approximation, as though it were formed from a collection of
noninteracting massless particles.

It is the purpose of this paper to investigate what happens beyond geometric
optics, when wavelengths are small but not completely ignorable. More
specifically, how do corrections to geometric optics affect propagation
directions? While the equations which govern small corrections to geometric
optics were derived long ago [6, 7] for electromagnetic and gravitational
waves propagating through curved spacetimes, their consequences have not been
thoroughly explored. It is nevertheless known that all reasonable definitions
for the local “propagation direction” agree in geometric optics: The direction
of the electromagnetic momentum density is identical for all observers, and
that coincides with the direction of the local phase gradient, the direction
along which “information” propagates, and the (necessarily degenerate)
principal null direction of the electromagnetic field. Beyond geometric
optics, different notions of propagation direction no longer agree: The
direction of the 4-momentum density can be different for different observers,
there can be two principal null directions, and phase gradients can depend on
a choice of basis [8]. Moreover, amplitude and polarization states no longer
propagate independently along each ray. Instead, there is a transport of
information between neighboring rays as well as along them. This means that
beyond leading order, there is no well-defined direction which can be
associated with “information flow” in a high-frequency field.

This complexity requires that we be precise about what exactly it is whose
propagation we would like to understand. In this paper, we focus on the “bulk”
propagation of small222These wave packets must be large compared to their
dominant wavelengths but small compared to all other length scales.
electromagnetic pulses in curved spacetimes. We choose a “center” for each
pulse and ask how that center evolves in time. Pulses in geometric optics are
simple: With reasonable assumptions, they travel along null geodesics [9].
Like their constituent rays, the centers of high-frequency pulses behave, at
leading order, like massless monopolar particles. One order beyond geometric
optics, the motion depends on a pulse’s angular momentum. More subtly, it also
depends on precisely which definition is used to describe the pulse’s center.
Regardless, there is a sense in which otherwise-identical wave packets with
opposite circular polarizations can be deflected with respect to one another.
This behavior may be summarized by stating that one order beyond geometric
optics, the bulk motion is equivalent to that of a massless dipole.

In the literature on flat-spacetime optics in nontrivial materials, spin-
dependent corrections to the propagation of electromagnetic fields are
sometimes described as spin Hall effects. There are in fact a number of
different spin Hall effects which have been discussed theoretically, some of
which have also been observed experimentally [10, 11, 12]. Some spin Hall
effects are induced by, e.g., gradients in the refractive index [13, 14, 15,
16, 17, 18, 19, 20, 21, 22]. Others arise even without any material
inhomogeneities: The geometric spin Hall effect [23, 24, 25] and the related
relativistic Hall effect [26] and Wigner(-Souriau) translations [27, 28, 29]
all arise in vacuum and in flat spacetime. These three effects may be shown to
be associated with differing definitions for the “center” of a given wave
packet. More precisely, the relativistic Hall effect and the Wigner
translations are related to differences between three-dimensional centroids
which would naturally be associated with different observers. They are
essentially the same as (unnamed) effects which have long been known for
massive objects [30, 31, 32, 33]. The geometric spin Hall effect is somewhat
different, being instead concerned with differences between centroids which
are defined on different two-dimensional cross sections.

The spin-dependent propagation effects discussed in this paper arise in vacuum
but in generic spacetimes, and are sometimes referred to as gravitational spin
Hall effects [34, 35]. Various approaches have been taken before to understand
the motion of electromagnetic, and also gravitational, wave packets in this
context. Some approaches have been based on classical high-frequency
expansions in the spirit of geometric optics [36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47]. Others have taken a semiclassical approach, using the
Bargmann-Wigner equations or Weyl equations [48, 49, 50]. Still other
approaches have not made any direct contact with an underlying field theory,
but have instead claimed that the motion of a wave packet could be described
using massless versions of the Mathisson-Papapetrou (MP) equations [51, 52,
53, 54, 55, 56, 57, 18], equations which are known to describe classical
spinning objects in curved spacetimes.

This paper focuses on the gravitational spin Hall effect of light, as
described in Ref. [42]. While the derivation there was based on a high-
frequency approximation, it differs from other high-frequency approaches by
being applicable in arbitrary spacetimes and by avoiding specific $3+1$
foliations. This paper endeavors to better understand the meaning, the domain
of applicability, and the limitations of the gravitational spin Hall
equations. It also unifies those equations with others which have appeared in
different contexts: The gravitational spin Hall equations are shown to be a
special case of the MP equations, and the flat-spacetime gradient-index spin
Hall effect, the relativistic Hall effect, and the Wigner(-Souriau)
translations are all shown to be special cases of the gravitational spin Hall
equations.

But before any equations of motion can be sensibly discussed, it is necessary
to first explain what exactly those equations describe. This leads us to
consider what can be meant by the centroid of an extended wave packet. One
definition which has appeared in the literature is shown to be untenable. A
large number of others remain, however, and we show the set of all such
possible centroids is unbounded for massless—but not massive—objects. While
this might at first appear to be a failure of the definitions, it is in fact a
failure of masslessness. We show that wave packets with nonzero angular
momentum cannot be massless, and this is essential to their localizability.
While the massless approximation can be useful for many purposes, ignoring its
limitations can result in qualitatively incorrect conclusions.

The paper begins in Sec. II by reviewing the gravitational spin Hall effect of
light, as presented in Ref. [42]. For comparison with the spin Hall effect of
light in flat-spacetime optics [11], we emphasize the role of the Berry phase
and the Berry connection in describing polarization, as well as the role of
the Berry curvature in the gravitational spin Hall equations.

Section III shows that with particular initial and spin supplementary
conditions, the gravitational spin Hall equations emerge as a special case of
the MP equations. This relation allows us to use the well-developed theory
associated with the MP equations to clarify the meanings of the worldline and
the momentum which arise in the spin Hall equations. It also allows us to
write down conservation laws for those equations and to discuss their regimes
of validity.

The spin Hall equations involve an arbitrary choice of timelike vector field,
and we show in Sec. IV that this parametrizes different definitions for the
centroid of an extended wave packet. Our main result regarding these centroids
is that although massive spinning objects can be localized, massless ones
cannot.

Section V examines whether or not the initial conditions which reduce the MP
equations to the gravitational spin Hall equations are in fact realized by
reasonable wave packets. We use a high-frequency approximation to explicitly
construct a large class of electromagnetic wave packets. For many members of
this class, the appropriate conditions are indeed satisfied. However, we also
find wave packets which do not have the expected properties. This implies that
there are nontrivial assumptions on the nature of the wave packet which have
been implicitly (and unknowingly) imposed in the prior literature. We also
show that our approximate wave packets can fail to satisfy the dominant energy
condition. This is an unphysical artifact of the high-frequency approximation,
and is what leads to the apparent delocalization of spinning wave packets
discussed in Sec. IV.

In Sec. VI, we discuss an analogy between light propagation through an optical
medium and light propagation through vacuum but in an effective optical
metric. Using a standard optical metric, we recover the spin Hall effect of
light in an inhomogeneous medium from the gravitational spin Hall equations.

Finally, the Appendix demonstrates that at least in flat spacetime, the spin
of any massless object which satisfies the dominant energy condition must
vanish. It follows that, e.g., electromagnetic wave packets with nonzero spin
cannot be exactly massless.

Notation and conventions: We work on an arbitrary smooth Lorentzian manifold
$(M,g_{\alpha\beta})$, where the metric tensor $g_{\alpha\beta}$ has signature
$(-\,+\,+\,+)$. Greek letters are used for spacetime indices and run from $0$
to $3$. We use bold symbols to denote $3$-vectors, and their components are
labeled by Latin letters from the middle of the alphabet, $(i,j,k,\ldots)$,
that run from $1$ to $3$. Units are used in which $G=c=1$, the Einstein
summation convention is assumed, and we use the notation
$a_{\alpha}b^{\alpha}=a\cdot b$, $a_{\alpha}a^{\alpha}=a\cdot a=a^{2}$. The
Riemann tensor is defined such that
$2\nabla_{[\alpha}\nabla_{\beta]}\omega_{\gamma}=R_{\alpha\beta\gamma}{}^{\lambda}\omega_{\lambda}$
for any $\omega_{\gamma}$. When working with tensors $T$ defined at different
spacetime points $x^{\alpha},\tilde{x}^{\alpha}\in M$, we use the usual
notation, $T\indices{{}_{\alpha}^{\beta}}(x)$, when the tensor is defined at
$x^{\alpha}$, while we use primed indices,
$T\indices{{}_{\alpha^{\prime}}^{\beta^{\prime}}}(\tilde{x})$, for tensors
defined at $\tilde{x}^{\alpha}$.

##  II Gravitational spin Hall effect of light

The equations of motion which describe the spin Hall effect for
electromagnetic waves propagating through curved spacetimes were derived in
Ref. [42]. They were obtained by performing a covariant high-frequency
analysis of the vacuum Maxwell equations. A similar approach was used in Ref.
[45] to describe the spin Hall effect for gravitational waves propagating on
curved backgrounds. For electromagnetic waves, the derivation starts with the
WKB ansatz

$A_{\alpha}=\mathrm{Re}\left[\epsilon\left(\psi
a_{\alpha}+\epsilon\psi^{(1)}_{\alpha}+\mathcal{O}(\epsilon^{2})\right)e^{iu/\epsilon}\right]$
(2.1)

for the electromagnetic potential, where $\epsilon$ is a small parameter
related to the wavelength, $u$ is a real phase function, $\psi$ is a real
scalar amplitude, and $a_{\alpha}$ is a complex polarization vector normalized
such that $a_{\alpha}\bar{a}^{\alpha}=1$. Higher-order terms, such as
$\psi^{(1)}_{\alpha}$, do not play any role in this section. The overall
factor of $\epsilon$ is for convenience and ensures that the field strength
$F_{\alpha\beta}=2\nabla_{[\alpha}A_{\beta]}$ is nontrivial and finite in the
$\epsilon\to 0$ limit. If $u$ increases as with time, it is convenient to
define the future-directed wave vector $k_{\alpha}=-\nabla_{\alpha}u$, and in
terms of that, a timelike observer with $4$-velocity $t^{\alpha}$ will measure
the wave frequency

$\omega=-t\cdot k/\epsilon.$ (2.2)

The derivation of the spin Hall effect in Ref. [42] relies on an analysis of
the overall phase factor of the field, which consists of $u$ at the lowest
order in $\epsilon$, together with a higher-order phase factor, referred to as
the Berry phase, which comes from the polarization vector $a_{\alpha}$. Using
the WKB ansatz above, together with the Maxwell equation

$\left(\nabla^{\beta}\nabla_{\alpha}-\delta^{\beta}_{\alpha}\nabla^{\gamma}\nabla_{\gamma}\right)A_{\beta}=0,$
(2.3)

and the Lorenz gauge condition $\nabla_{\alpha}A^{\alpha}=0$, the wave vector
$k_{\alpha}$ must be be null and orthogonal to the polarization vector,

$k\cdot k=k\cdot a=0.$ (2.4)

Additionally, the scalar amplitude $\psi$ must satisfy the transport equation

$\nabla_{\alpha}\left(k^{\alpha}\psi^{2}\right)=0,$ (2.5)

and the polarization vector $a_{\alpha}$ must be parallel transported,

$k^{\beta}\nabla_{\beta}a_{\alpha}=0.$ (2.6)

These are the usual equations of geometric optics. The null geodesic rays of
geometric optics are integral curves of $k^{\alpha}$.

It is convenient to expand the polarization vector in terms of a tetrad
$\\{k_{\alpha},t_{\alpha},m_{\alpha},\bar{m}_{\alpha}\\}$, where the real
covector $t_{\alpha}$ is timelike, $k_{\alpha}$, $m_{\alpha}$, and its complex
conjugate $\bar{m}_{\alpha}$ are null, $t\cdot m=0$, and $m\cdot\bar{m}=1$.
Given Eq. (2.4), there must exist complex scalars $z_{1}$, $z_{2}$ and $z_{3}$
such that

$a_{\alpha}=z_{1}m_{\alpha}+z_{2}\bar{m}_{\alpha}+z_{3}k_{\alpha}.$ (2.7)

The complex covectors $m_{\alpha}$ and $\bar{m}_{\alpha}$ form a circular
polarization basis, and the considered electromagnetic wave is circularly
polarized when $z_{1}=0$ or $z_{2}=0$. The term proportional to $k_{\alpha}$
is pure gauge, not fixed by the Lorenz gauge condition, and it will not play
any role in what follows.

The only element of the tetrad which is interpreted as being fixed by the
field is $k_{\alpha}$. Supplementing that with $t_{\alpha}$ fixes the
$2$-plane spanned by $m_{\alpha}$ and $\bar{m}_{\alpha}$. And within that
plane, there is still an additional freedom associated with the spin rotations
$m_{\alpha}\mapsto e^{i\phi}m_{\alpha}$, where $\phi$ is any real scalar. Any
change in $t_{\alpha}$ will result in a shift with the form $m_{\alpha}\mapsto
m_{\alpha}+ck_{\alpha}$, which can only affect $z_{3}$ in (2.7). Although that
is interpreted as a gauge transformation here, changes in $t_{\alpha}$ will
act nontrivially and play an important role in the spin Hall
equations333Changes of $t_{\alpha}$ are analogous to the Wigner translations
discussed in Ref. [27], which act as gauge transformations on plane waves, but
act nontrivially on finite wave packets.. Spin rotations instead affect the
values of $z_{1}$ and $z_{2}$. Nevertheless, they do not affect the spin Hall
equations.

We can now obtain a transport equation for $z_{1}$ and $z_{2}$ along the rays.
Viewing $m_{\alpha}$ as a covector-valued field over the cotangent bundle,
depending on both position and on $k_{\alpha}$, the parallel transport
equation (2.6) implies that

$\frac{d}{d\tau}\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}=i(k^{\alpha}B_{\alpha})\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}z_{1}\\\ z_{2}\end{pmatrix},$ (2.8)

where

$B_{\alpha}=i\bar{m}^{\beta}\left(\nabla_{\alpha}+k_{\gamma}\Gamma^{\gamma}_{\alpha\lambda}\frac{\partial}{\partial
k_{\lambda}}\right)m_{\beta}$ (2.9)

is the Berry connection. The operator in brackets in the Berry connection may
be seen to be a horizontal covariant derivative on the cotangent bundle.
Regardless, the transport equation (2.8) can be integrated to yield

$z_{1}(\tau)=e^{i\gamma}z_{1}(\tau_{0}),\qquad
z_{2}(\tau)=e^{-i\gamma}z_{2}(\tau_{0}),$ (2.10)

where

$\gamma=\int_{\tau_{0}}^{\tau}d\tau^{\prime}k^{\alpha}B_{\alpha}$ (2.11)

is the Berry phase. The Berry phase represents a higher-order correction to
the overall phase of the WKB potential, and is generally responsible for the
spin Hall effect of light [16, 21]. For circularly polarized electromagnetic
waves, the leading-order field takes the form
$A_{\alpha}=\mathrm{Re}\left[\epsilon\psi
m_{\alpha}e^{i(u+\epsilon\gamma)/\epsilon}\right]$ or
$A_{\alpha}=\mathrm{Re}\left[\epsilon\psi\bar{m}_{\alpha}e^{i(u-\epsilon\gamma)/\epsilon}\right]$,
depending on the handedness of circular polarization state. The total phase at
this order is therefore proportional to $u+\epsilon s\gamma$, where $s=\pm 1$.
Note that both the Berry connection and the Berry phase depend on spin
rotations $m_{\alpha}\mapsto e^{i\phi}m_{\alpha}$, transforming as
$B_{\alpha}\mapsto B_{\alpha}-\nabla_{\alpha}\phi$ and
$\gamma(\tau)\mapsto\gamma(\tau)-\phi(\tau)+\phi(\tau_{0})$.

In geometric optics, the dispersion relation $k\cdot k=0$ may be viewed as a
Hamilton-Jacobi equation for the phase function $u$. If that is solved using
the method of characteristics, one recovers the null geodesic rays of
geometric optics. In Ref. [42], the strategy was to generalize this procedure,
deriving the spin Hall effect by looking for an effective dispersion relation
involving the gradient of the corrected phase function $u+\epsilon s\gamma$.
Letting $K_{\alpha}=-\nabla_{\alpha}(u+\epsilon s\gamma)$, we use $k\cdot k=0$
and the definition of the Berry phase $\gamma$ to arrive at the following
effective dispersion relation:

$K\cdot K-2\epsilon sK\cdot B=\mathcal{O}(\epsilon^{2}).$ (2.12)

This can be viewed as a Hamilton-Jacobi equation for the total phase function
$u+\epsilon s\gamma$. Using the method of characteristics, we can solve this
Hamilton-Jacobi equation and obtain ray equations with polarization-dependent
corrections to the geodesic equations of geometric optics. However, since
these equations depend on the Berry connection $B_{\alpha}$, they are not
invariant under spin rotations. This gauge dependence can be removed by
switching to noncanonical coordinates444A similar approach is also used for
the description of charged particles in an external electromagnetic field,
where a coordinate transformation is used to rewrite the equations of motion
in terms of the gauge-invariant Faraday tensor instead of the gauge-dependent
vector potential., as described in Ref. [42, Sec. IV.B.1] (see also Ref.
[58]).

The spin Hall equations which result from the use of these coordinates, which
describe the polarization-dependent propagation of circularly polarized light,
can be written as [42]

$\begin{split}\dot{x}^{\alpha}&=p^{\alpha}+\epsilon
sp^{\beta}\Big{[}\left(F_{px}\right)\indices{{}_{\beta}^{\alpha}}+\Gamma^{\gamma}_{\lambda\beta}p_{\gamma}\left(F_{pp}\right)^{\lambda\alpha}\Big{]},\\\
\dot{p}_{\alpha}&=\Gamma^{\gamma}_{\alpha\beta}p_{\gamma}p^{\beta}-\epsilon
sp^{\beta}\Big{[}\left(F_{xx}\right)_{\alpha\beta}+\Gamma^{\gamma}_{\beta\lambda}p_{\gamma}\left(F_{xp}\right)\indices{{}^{\lambda}_{\alpha}}\Big{]},\end{split}$
(2.13)

where $x^{\alpha}$ denotes a position, $p_{\alpha}$ a momentum, the dot an
ordinary (noncovariant) derivative $d/d\tau$, and $s=\pm 1$ depending on the
handedness of the circular polarization state. The same equations, but with
$s=\pm 2$ instead, describe the spin Hall effect for circularly polarized
gravitational waves [45]. They are understood to be valid up to terms of order
$\epsilon^{2}$. The spin Hall equations are expressed in terms of the Berry
curvature components

$\begin{split}(F_{pp})^{\beta\alpha}&=2\mathrm{Im}\left(\frac{\partial
m_{\gamma}}{\partial p_{\alpha}}\frac{\partial\bar{m}^{\gamma}}{\partial
p_{\beta}}\right),\\\
(F_{xx})_{\beta\alpha}&=2\mathrm{Im}\left(\nabla_{\alpha}m_{\gamma}\nabla_{\beta}\bar{m}^{\gamma}+m_{\gamma}\nabla_{[\alpha}\nabla_{\beta]}\bar{m}^{\gamma}\right),\\\
(F_{px})_{\alpha}{}^{\beta}&=-(F_{xp})^{\beta}{}_{\alpha}=2\mathrm{Im}\left(\frac{\partial
m_{\gamma}}{\partial
p_{\beta}}\nabla_{\alpha}\bar{m}^{\gamma}\right),\end{split}$ (2.14)

which are invariant with respect to spin rotations.

The procedure leading to the spin Hall equations has removed any dependence on
spin rotations. However, it has introduced a physical dependence on the
timelike covector $t_{\alpha}$. This can be made explicit by calculating the
components of the Berry curvature using the properties of the tetrad555Here,
$p_{\alpha}$ has replaced the $k_{\alpha}$ which appeared in the above
discussion. $m_{\alpha}$ and $\bar{m}_{\alpha}$ are now viewed as functions of
position and of $p_{\alpha}$.
$\\{p_{\alpha},t_{\alpha},m_{\alpha},\bar{m}_{\alpha}\\}$, which results in
[42, Appendix C]

$\begin{split}\left({F_{pp}}\right)^{\beta\alpha}&=\frac{\Sigma^{\alpha\beta}}{(p\cdot
t)^{2}},\\\
\left({F_{xx}}\right)_{\beta\alpha}&=\frac{\Sigma^{\gamma\lambda}}{2}\bigg{[}R_{\gamma\lambda\alpha\beta}+\frac{2}{(p\cdot
t)^{2}}p_{\rho}\Gamma^{\rho}_{\gamma[\alpha}\\\
&\qquad\qquad\leavevmode\nobreak\
\times\bigg{(}\Gamma^{\sigma}_{\beta]\lambda}p_{\sigma}-2(p\cdot
t)\nabla_{\beta]}t_{\lambda}\bigg{)}\bigg{]},\\\
\left({F_{xp}}\right)\indices{{}^{\alpha}_{\beta}}&=\frac{\Sigma^{\alpha\gamma}}{(p\cdot
t)^{2}}\left(p_{\rho}\Gamma^{\rho}_{\beta\gamma}-(p\cdot
t)\nabla_{\beta}t_{\gamma}\right).\end{split}$ (2.15)

Each term here is linear in the real bivector

$\Sigma^{\alpha\beta}=2i\bar{m}^{[\alpha}m^{\beta]}=\frac{\varepsilon^{\alpha\beta\gamma\lambda}p_{\gamma}t_{\lambda}}{p\cdot
t},$ (2.16)

which is invariant under spin rotations and is uniquely determined by
$p_{\alpha}$ and $t_{\alpha}$. We shall see in Sec. III that
$\Sigma^{\alpha\beta}$ is proportional to the angular momentum tensor of the
wave packet. Regardless, substituting Eq. (2.15) into (2.13) shows that the
spin Hall equations can be written in the more compact form

$\displaystyle\dot{x}^{\alpha}$ $\displaystyle=p^{\alpha}+\frac{1}{p\cdot
t}(\epsilon s\Sigma^{\alpha\beta})p^{\gamma}\nabla_{\gamma}t_{\beta},$ (2.17a)
$\displaystyle\frac{Dp_{\alpha}}{d\tau}$
$\displaystyle=-\frac{1}{2}R_{\alpha\beta\gamma\lambda}p^{\beta}(\epsilon
s\Sigma^{\gamma\lambda}),$ (2.17b)

where $D/d\tau=\dot{x}^{\alpha}\nabla_{\alpha}$ denotes the covariant
derivative along the worldline. It is now manifest that the only external
choice relevant to these equations is the timelike vector field $t^{\alpha}$.
We shall see below that that choice parametrizes the definition for
$x^{\alpha}$.

Unlike the integral curves of $k^{\alpha}$, which are interpreted as rays
within (say) a wave packet, the position $x^{\alpha}$ which appears in the
spin Hall equations (2.17) is interpreted as describing the position of the
wave packet as a whole: its “centroid.” Similarly, the momentum $p_{\alpha}$
is interpreted as the net momentum of the wave packet, not as a momentum
density within that wave packet. One consequence of the spin Hall equations is
that the worldline is not necessary tangent to the momentum. Systems with this
feature are sometimes referred to as having hidden momentum [59, 60, 61, 62,
33] or anomalous velocity [63, 15, 48, 64, 65]. Also note that inspection of
Eq. (2.17) shows that the affine parameter $\tau$ is dimensionless and that
$\epsilon$ has units $\mbox{(length)}^{2}$. Examination of the spin Hall
equations shows that $\tau$ has been chosen such that $\dot{x}\cdot t=p\cdot
t$. It is straightforward to see from Eq. (2.17b) that if $p_{\alpha}$ is
initially null, it remains null for all time. Equation (2.17a) implies that
when the momentum is null, so too is the worldline:
$\dot{x}\cdot\dot{x}=\mathcal{O}(\epsilon^{2})$. These equations are assumed
to be used only for initial data in which $p_{\alpha}$, and therefore
$\dot{x}^{\alpha}$, are indeed null.

There is not always a hidden momentum. As a particular case, suppose that
$t^{\alpha}$ is parallel transported in the sense that

$\dot{x}^{\beta}\nabla_{\beta}t_{\alpha}=\mathcal{O}(\epsilon).$ (2.18)

With this choice, $\dot{x}^{\alpha}=p^{\alpha}+\mathcal{O}(\epsilon^{2})$ and
Eq. (2.17) reduces to the polarization-dependent ray equations obtained by
Frolov in Ref. [44, Eq. 110-112]. In this sense, the spin optics approximation
in that paper describes a particular case of the gravitational spin Hall
equations obtained in [42]. However, since one of our goals is to understand
the role of $t^{\alpha}$ in Eq. (2.17), we do not assume any special choices
for it in the remainder of this paper.

It was not clear in the derivation of the spin Hall equations precisely what
$x^{\alpha}$, $p_{\alpha}$, or $t_{\alpha}$ are, what types of wave packets
these equations describe, or what sorts of approximations are implicit in them
(beyond the assumption of high-frequencies). These issues will be addressed
below.

##  III Mathisson-Papapetrou equations and their implications for the spin
Hall equations

The spin Hall equations (2.17) are interpreted as describing the motion of
circularly polarized electromagnetic wave packets. However, it is known from
separate arguments that the motion of any sufficiently compact spinning object
is governed by the Mathisson-Papapetrou (MP) equations666These are variously
referred to as the Papapetrou, Mathisson-Papapetrou, and Mathisson-Papapetrou-
Dixon equations. The same labels are commonly applied also to more general
equations which involve the quadrupole and higher-order moments of the
relevant object. As recounted in [66], Mathisson [67] appears to have been the
first to obtain the pole-dipole equations (3.1). He did so before Papapetrou
[68] and using a superior method. Mathisson also derived some of the
quadrupole terms which are not included here. Dixon [69] derived all
quadrupole and higher-order terms, developing a full theory of multipole
moments to all orders. This was later generalized to also allow for self-
interaction [70, 71]. Here we refer to the test body pole-dipole
equations—without quadrupole or higher-order moments—as the MP equations.

$\displaystyle\frac{Dp_{\alpha}}{d\tau}=-\frac{1}{2}R_{\alpha\beta\gamma\lambda}\dot{x}^{\beta}S^{\gamma\lambda},$
(3.1a)
$\displaystyle\frac{DS^{\alpha\beta}}{d\tau}=2p^{[\alpha}\dot{x}^{\beta]}.$
(3.1b)

These equations evolve an object’s linear momentum $p_{\alpha}$ and its
angular momentum $S^{\alpha\beta}=S^{[\alpha\beta]}$ along a specified
worldline. They are very general: As long as the quadrupole and higher-order
multipole moments of an object’s stress-energy tensor can be ignored, the MP
equations hold for all sufficiently compact objects with conserved stress-
energy tensors [69, 71]. In particular, although much of the literature on
these equations assumes that $p_{\alpha}$ is timelike, their derivation makes
no use of that condition; null momenta are also admissible. Whether or not the
momentum is null depends only on the nature of the underlying stress-energy
tensor.

The generality of the MP equations can be understood, in part, from the fact
that they are essentially kinematic. They arise as consequences of attempting
to maintain Poincaré invariance as much as possible along the given worldline
[72]. Indeed, they imply the presence of ten conserved quantities along that
worldline, which correspond locally to the four translations, three rotations,
and three boosts of a four-dimensional Minkowski spacetime (even when the
actual spacetime is not Minkowski). The nontrivial physics which enters into
this is that corrections to the MP equations—deviations due to the breakdown
of Poincaré invariance—depend only an object’s quadrupole and higher-order
moments. It is expected from the equivalence principle that “sufficiently
compact” objects should behave, at least locally, as though the spacetime is
flat, and a calculation shows that the breakdown of the flat-spacetime
conservation laws first occurs at quadrupolar order.

What is relevant here is that electromagnetic wave packets are associated with
conserved stress-energy tensors. Their bulk motion can therefore be described
not only by the spin Hall equations, but also by the MP equations. We show in
Sec. III.1 that there is a precise sense in which the spin Hall equations
arise as a special case of the MP equations. Section III.2 exploits this
connection between the spin Hall equations and the MP equations to relate
quantities in the spin Hall equations to an underlying stress-energy tensor.
Section III.3 uses known results for the MP equations to write down previously
unknown conservation laws associated with the spin Hall equations. Finally,
Sec. III.4 explores the approximations used in the spin Hall equations and
discusses when those approximations hold.

###  III.1 Spin Hall equations from MP equations

Our first task is to show that the spin Hall equations are a special case of
the MP equations. We now show that the spin Hall equations arise after
choosing appropriate initial data for the MP equations, fixing an appropriate
definition for the centroid of an extended wave packet, and imposing a
particular parameterization for the worldline of that centroid.

A priori, it may appear that the spin Hall and MP equations do not even
describe the same physical quantities. The spin Hall equations evolve
$x^{\alpha}$ and $p_{\alpha}$ while $t^{\alpha}$ is specified independently.
By contrast, the MP equations evolve $p_{\alpha}$ and $S^{\alpha\beta}$ while
$x^{\alpha}$ is specified independently777The worldline which appears in the
MP equations is to be interpreted as a choice of origin for a multipole
expansion. It does not necessarily have any interpretation as a centroid. That
interpretation arises only when additional conditions are imposed on the
worldline. However, except in maximally symmetric spacetimes, ignoring the
quadrupole and higher-order moments cannot be justified unless there is some
sense in which the worldline lies near an object’s “center.”. This discrepancy
is resolved by showing that in the present context, i) the angular momentum
equation (3.1b) can be trivially solved, and ii) the specification of
$t^{\alpha}$ is equivalent to the specification of $x^{\alpha}$.

To summarize our result, given any future-directed timelike vector field
$t^{\alpha}$ and any constant “spin parameter” $s\epsilon$, the MP equations
reduce to the spin Hall equations, at least up to terms of order
$\epsilon^{2}$, when:

  1. 1.

The worldline parameter $\tau$ is chosen such that

$\dot{x}\cdot t=p\cdot t$ (3.2)

for all time.

  2. 2.

The momentum $p_{\alpha}$ is at least initially null.

  3. 3.

The angular momentum satisfies

$S^{\alpha\beta}p_{\beta}=0$ (3.3)

at least initially, and

$S^{\alpha\beta}t_{\beta}=0$ (3.4)

for all time.

  4. 4.

The magnitude of the angular momentum is at least initially given by

$S^{\alpha\beta}S_{\alpha\beta}=2(s\epsilon)^{2}.$ (3.5)

The MP equations are reparametrization-invariant, so no generality is lost by
imposing condition 1 for all time, at least so long as the worldline is not
orthogonal to $t^{\alpha}$. Equation (3.2) serves merely to use the object’s
energy to nondimensionalize the time parameter.

The interpretation of $S^{\alpha\beta}t_{\beta}=0$, also known as the
Corinaldesi-Papapetrou spin supplementary condition [73, 33], is more
substantial. As discussed in more detail in Sec. IV.1 below, it is an implicit
definition for $x^{\alpha}$. The angular momentum of any object depends on the
choice of origin888Recall that this is true even in Newtonian physics., and
certain components of $S^{\alpha\beta}$ can always be eliminated by an
appropriate choice of origin. Here, $x^{\alpha}$ is chosen to eliminate
$S^{\alpha\beta}t_{\beta}$, which is proportional to the body’s mass dipole
moment with respect to an observer whose 4-velocity is tangent to
$t^{\alpha}$. This definition allows $x^{\alpha}$ to be interpreted as a kind
of centroid. However, that centroid clearly depends on $t^{\alpha}$. Different
choices for $t^{\alpha}$ generically result in different centroids, and each
of these is in principle observable. The different centroids represent
slightly different notions of “center” for an extended object. Relations
between them are discussed in Secs. IV.2 and IV.3 below.

A particular worldline can be fixed by choosing a particular $t^{\alpha}$.
There should therefore exist an evolution equation for the tangent vector
$\dot{x}^{\alpha}$ to that worldline. To derive that evolution equation, first
combine (3.1b), (3.2), and (3.4) to see that

$\displaystyle 0$
$\displaystyle=\frac{D}{d\tau}\left(S^{\alpha\beta}t_{\beta}\right)$
$\displaystyle=(p\cdot
t)(p^{\alpha}-\dot{x}^{\alpha})+S^{\alpha\beta}\dot{x}^{\gamma}\nabla_{\gamma}t_{\beta}.$
(3.6)

Rearranging then results in the momentum-velocity relation

$\dot{x}^{\alpha}=p^{\alpha}+\frac{1}{p\cdot
t}S^{\alpha\beta}\dot{x}^{\gamma}\nabla_{\gamma}t_{\beta}.$ (3.7)

It follows that $\dot{x}^{\alpha}$ and $p^{\alpha}$ are not necessarily
collinear. As long as the operator $\delta^{\alpha}_{\beta}-(p\cdot
t)^{-1}S^{\alpha\gamma}\nabla_{\beta}t_{\gamma}$ can be inverted, (3.7)
determines $\dot{x}^{\alpha}$ uniquely in terms of $p^{\alpha}$,
$S^{\alpha\beta}$, and $t^{\alpha}$. In the small-angular momentum context
considered here, the invertibility requirement is trivially satisfied as long
as $p\cdot t$ is not too small. In particular, $t^{\alpha}$ cannot be null and
proportional to $p^{\alpha}$. This is discussed further in Sec. IV.1 below.

With the centroid fixed by (3.4), the superficially similar spin constraint
(3.3) plays a very different role: It is interpreted as a genuine physical
restriction on the types of systems which can be described by the spin Hall
equations. Combining (3.3) and (3.4) with (3.5) shows that the angular
momentum is at least initially999The given constraints determine
$S^{\alpha\beta}$ only up to an overall sign. Here we fix the sign in order
for the MP and spin Hall equations to agree, which may also be viewed as
fixing the definition for the sign of $s$.

$\displaystyle S^{\alpha\beta}=\epsilon s\Sigma^{\alpha\beta}=\frac{\epsilon
s}{p\cdot t}\varepsilon^{\alpha\beta\gamma\lambda}p_{\gamma}t_{\lambda}.$
(3.8)

The spin constraint $S^{\alpha\beta}p_{\beta}=0$ therefore amounts to a
particular choice of initial condition for the angular momentum tensor. It
implies that the spin is purely longitudinal. As explained in Sec. V.5 below,
this is consistent with a wide class of high-frequency electromagnetic wave
packets. However, it is also shown there that there are reasonable high-
frequency wave packets which are not consistent with (3.8). It is a genuine
physical restriction on the types of wave packets which can be described by
the spin Hall equations.

Our next task is to show that if $S^{\alpha\beta}$ is initially given by
(3.8), it retains that form for all time. This can be demonstrated by showing
that the constraints (3.3) and (3.5), as well as the null character of
$p_{\alpha}$, are preserved under time evolution. First consider the null
character of $p_{\alpha}$. If we assume that
$S^{\alpha\beta}=\mathcal{O}(\epsilon)$ for all time, (3.1a) and (3.7)
immediately imply that

$\frac{d}{d\tau}(p^{\alpha}p_{\alpha})=\mathcal{O}(\epsilon^{2}).$ (3.9)

If $p_{\alpha}p^{\alpha}$ is initially zero, it can therefore grow to be at
most of order $\epsilon^{2}$. Again applying the MP equations and the
momentum-velocity relation,

$\displaystyle\frac{D}{d\tau}(S^{\alpha\beta}p_{\beta})=-\frac{1}{p\cdot
t}\left[(S^{\gamma\beta}p_{\beta})\dot{x}^{\lambda}\nabla_{\lambda}t_{\gamma}\right]p^{\alpha}+\mathcal{O}(\epsilon^{2}).$
(3.10)

If $S^{\alpha\beta}p_{\beta}$ is initially zero, as is assumed in condition
3.4 above, it therefore remains zero up to terms of order $\epsilon^{2}$.
Equation (3.3) is thus preserved under time evolution. Lastly, use of these
results together with (3.1b) shows that

$\frac{d}{d\tau}(S^{\alpha\beta}S_{\alpha\beta})=\mathcal{O}(\epsilon^{3}).$
(3.11)

This implies that the initial spin magnitude (3.5) is preserved under time
evolution. Combining these results shows that $S^{\alpha\beta}$ retains the
form (3.8) for all time, at least up to terms of order $\epsilon^{2}$.

The spin Hall equations (2.17) now follow, up to terms of order
$\epsilon^{2}$, by substituting (3.8) into (3.1a) and (3.7). They may be
viewed as the MP equations (3.1) specialized to conditions 1-4 above. This
result can also be established by directly showing that (3.8) is a solution to
(3.1b) and then using that to deduce the momentum-velocity relation (3.7) [35,
Sec. 2.4.4.]. Regardless, the spin Hall equations of motion are equivalent to
the equations of motion satisfied by a massless dipolar particle.

###  III.2 The meaning of the momentum

Now that we have established that the spin Hall equations follow from the MP
equations, results known for the latter may be applied to the former. It is
natural to ask what exactly is meant by the $p_{\alpha}$ and the $x^{\alpha}$
which appear in the spin Hall equations. The fundamental object in classical
electromagnetism is the electromagnetic field, so there must be a relation
between that field and (say) the momentum. Such a relation is not necessarily
clear from the derivation of the spin Hall equations in Ref. [42]. However,
the MP equations can be derived by first defining $p_{\alpha}$ and
$S^{\alpha\beta}$ as integrals over an object’s stress-energy tensor and then
using stress-energy conservation to deduce the evolution equations for those
quantities [74, 69, 71]. Imposition of a centroid condition then provides a
definition for $x^{\alpha}$ in terms of the underlying stress-energy tensor.
To summarize, the field can be used to construct the stress-energy tensor,
which can in turn be used to construct the momenta and the centroid.

As the spin Hall equations are special cases of the MP equations, we may
identify momenta in the former with momenta in the latter. There are however
subtleties. In particular, different definitions for the momenta may satisfy
formally identical evolution equations. This is especially clear when the
definitions differ by terms which are considered “higher order.” However, it
can also occur in other cases. For example, if the triple
$(x^{\alpha},p_{\alpha},S^{\alpha\beta})$ satisfies the MP equations together
with an appropriate centroid condition, so does
$(x^{\alpha},cp_{\alpha},cS^{\alpha\beta})$, where $c$ is any nonzero
constant. It follows that at best, the momenta in the two frameworks can be
identified only up to an overall constant.

Despite this, we choose to interpret the momenta in the spin Hall equations to
be exactly those which are typically used in derivations of the MP equations
and their generalizations: If the object of interest has stress-energy tensor
$T^{\alpha\beta}$, and if that object’s worldtube is foliated by the
1-parameter family of hypersurfaces $\Sigma_{\tau}$, the linear and angular
momenta at time $\tau$ are given by [74, Eqs. (5.1) and (5.2)]

$\displaystyle p_{\alpha}$
$\displaystyle=\int_{\Sigma_{\tau}}K_{\alpha}{}^{\alpha^{\prime}}T^{\beta^{\prime}}{}_{\alpha^{\prime}}dS_{\beta^{\prime}},$
(3.12a) $\displaystyle S^{\alpha\beta}$
$\displaystyle=2\int_{\Sigma_{\tau}}\sigma^{[\beta}H^{\alpha]\alpha^{\prime}}T^{\beta^{\prime}}{}_{\alpha^{\prime}}dS_{\beta^{\prime}}.$
(3.12b)

Unprimed indices here are associated with $x^{\alpha}(\tau)$, which is assumed
to lie in $\Sigma_{\tau}$. Primed indices are associated with the integration
point $x^{\prime}$. The bitensors
$K_{\alpha}{}^{\alpha^{\prime}}(x,x^{\prime})$ and
$\sigma^{\beta}(x,x^{\prime})H^{\alpha\alpha^{\prime}}(x,x^{\prime})$ are
Jacobi propagators; they can be used to form a basis for solutions to the
geodesic deviation (or Jacobi) equation along the geodesic segment which
connects $x$ to $x^{\prime}$. The Jacobi propagators can be computed
explicitly [74] using derivatives of Synge’s world function
$\sigma(x,x^{\prime})$, which is defined to be one half of the squared
geodesic distance between its arguments [75, 76]. In terms of this world
function, $\sigma_{\alpha}=\nabla_{\alpha}\sigma$ and

$H^{\alpha\alpha^{\prime}}=[-\nabla_{\alpha^{\prime}}\sigma_{\alpha}]^{-1},\qquad
K_{\alpha}{}^{\alpha^{\prime}}=H^{\beta\alpha^{\prime}}\nabla_{\alpha}\sigma_{\beta},$
(3.13)

where $[\ldots]^{-1}$ denotes an inverse operation.

In flat spacetime and in inertial coordinates, the above propagators reduce to

$K_{\alpha}{}^{\alpha^{\prime}}=\delta^{\alpha^{\prime}}_{\alpha},\qquad\sigma^{\beta}H^{\alpha\alpha^{\prime}}=(x^{\beta}-x^{\prime\beta})\eta^{\alpha\alpha^{\prime}},$
(3.14)

where $\eta^{\alpha\alpha^{\prime}}=\mathrm{diag}(-1,1,1,1)$ is the usual
Minkowski metric. Substituting these expressions into (3.12) recovers the
standard special-relativistic definitions [1, 32] for the linear and angular
momenta. Even in curved spacetimes, the special relativistic expressions for
the momenta remain good approximations to the exact expressions (3.12) if the
coordinates are taken to be Riemann normal coordinates with origin
$x^{\alpha}$.

###  III.3 Conservation laws

The specific choice of propagators appearing in the momenta (3.12) may appear
to be obscure. They were chosen, in part, so that if $\kappa^{\alpha}$ is
Killing,

$\int_{\Sigma_{\tau}}T^{\beta^{\prime}}{}_{\alpha^{\prime}}\kappa^{\alpha^{\prime}}dS_{\beta^{\prime}}=p_{\alpha}\kappa^{\alpha}+\frac{1}{2}S^{\alpha\beta}\nabla_{\alpha}\kappa_{\beta}.$
(3.15)

This relation is exact. It is useful because the integral on the left-hand
side—which is now identified as a linear combination of the linear and angular
momenta—is conserved.

In fact, if $\kappa^{\alpha}$ is interpreted not as an ordinary Killing field,
but as a generalized Killing field101010A generalized Killing field requires
for its construction a choice of worldline and a foliation [72, 71]. Each
generalized Killing field is exactly Killing on the worldline in the sense
that $\mathcal{L}_{\kappa}g_{ab}=\nabla_{a}\mathcal{L}_{\kappa}g_{bc}=0$
there. Away from the worldline, the generalized Killing fields are exact
symmetries for separation vectors (defined via the exponential map) rather
than for the metric itself. Although the generalized Killing fields do not
necessarily satisfy Killing’s equation there, they do satisfy certain
projections of it., the momentum definitions (3.12) ensure that (3.15) remains
valid in any spacetime; cf. Refs. [72, 71]. The space of generalized Killing
fields is always ten dimensional in four spacetime dimensions. It also
includes all ordinary Killing fields which may exist. If a generalized Killing
field $\kappa^{\alpha}$ is not an ordinary Killing field,
$p_{\alpha}\kappa^{\alpha}+\frac{1}{2}S^{\alpha\beta}\nabla_{\alpha}\kappa_{\beta}$
is not necessarily conserved, at least exactly. However, that quantity is
approximately conserved in the pole-dipole context in which the quadrupole and
higher-order moments of a body are neglected. That is, the approximation in
which the MP equations hold, and indeed, those equations are equivalent to the
statement that

$p_{\alpha}\kappa^{\alpha}+\frac{1}{2}S^{\alpha\beta}\nabla_{\alpha}\kappa_{\beta}=\mathrm{const}.$
(3.16)

for all generalized Killing fields $\kappa^{\alpha}$. There are ten
independent constants associated with the ten generalized Killing fields, and
these completely determine the four components of $p_{\alpha}$ and the six
components of $S^{\alpha\beta}$. The coupling of the linear and angular
momenta in the MP equations (3.1) is interpreted as a consequence of the fact
that, e.g., a local rotation about one point on the worldline is equivalent to
both a rotation and translation when viewed at another point on the worldline.

Regardless, (3.16) is most useful when $\kappa^{\alpha}$ is an ordinary
Killing field. In that case, the associated conservation law can be simplified
in the spin Hall context. There, the angular momentum is given by Eq. (3.8) so

$p_{\alpha}\left[\kappa^{\alpha}+\left(\frac{\epsilon s}{2p\cdot
t}\right)\varepsilon^{\alpha\beta\gamma\lambda}t_{\beta}\nabla_{\gamma}\kappa_{\lambda}\right]=\mathrm{const}.$
(3.17)

A particular component of the linear momentum is therefore conserved.
Precisely which component is conserved generically depends on both the spin
magnitude $\epsilon s$ and on the choice of $t^{\alpha}$.

There are, however, cases where the $s$-dependent terms vanish in Eq. (3.17).
First, the Killing field may be covariantly constant. This occurs for all
translations in Minkowski spacetime, and also for null translations along the
direction of gravitational wave propagation in pp-wave spacetimes. For
different reasons, there can be no spin correction to the conservation of
energy in static spacetimes, at least when $t^{\alpha}$ and $\kappa^{\alpha}$
are both identified with the static Killing field. The spin-dependent term in
(3.17) will then be proportional to $t_{[\alpha}\nabla_{\beta}t_{\gamma]}$,
which vanishes on account of the spacelike hypersurface which is orthogonal to
$t^{\alpha}$ in that context. Statements of energy conservation in the
Schwarzschild spacetime are therefore independent of spin in the spin Hall
context. This is not the case in Kerr spacetimes with nonzero angular
momentum, which are stationary but not static.

We have only discussed conservation laws associated with Killing vector
fields. Other conserved quantities, associated with the existence of Killing-
Yano tensors, are known (at least approximately) for the massive MP equations
coupled to appropriate centroid conditions [77, 78, 79, 80, 81, 82]. However,
we have not investigated whether or not these laws also hold for the massless
case of interest here.

###  III.4 Neglected terms

The spin Hall equations (2.17) are expected to be valid only through first
order in the “small” (although dimensionful) parameter $\epsilon$. Terms
nonlinear in the spin have been ignored, as have any contributions from the
quadrupole and higher-order moments of a wave packet’s stress-energy tensor.
Dixon has however found all multipolar corrections to the MP equations [69],
and using his results, the neglected terms in the spin Hall equations can be
estimated. We now discuss how to perform these estimates and under which
conditions the spin Hall equations can be justified.

Dixon’s laws of motion may be found in, e.g., [69, Eqs. (13.7) and (13.8)].
See also [66, Eqs. (283), (284), and (290)] for a version of those laws which
is truncated at quadrupolar order. Inspecting them shows that the MP evolution
equation (3.1a) for $p_{\alpha}$ is corrected by a force term proportional to

$J^{\beta\gamma\lambda\rho}\nabla_{\alpha}R_{\beta\gamma\lambda\rho},$ (3.18)

where $J^{\beta\gamma\lambda\rho}$ denotes the quadrupole moment of the wave
packet’s stress-energy tensor. The evolution equation (3.1b) for
$S^{\alpha\beta}$ is corrected as well, acquiring a torque term proportional
to

$J^{\gamma\lambda\rho[\alpha}R^{\beta]}{}_{\gamma\lambda\rho}.$ (3.19)

There are further forces and torques which couple a wave packet’s octupole and
higher-order moments to higher-order derivatives of the Riemann tensor, and
together, these corrections provide a complete description for the evolution
of the linear and angular momentum along a given worldline. If the worldline
is fixed by adopting the centroid condition (3.4), there is, in addition, an
evolution equation for $x^{\alpha}$ which generalizes the spin Hall equation
(2.17a). That generalization differs from its spin Hall counterpart due to the
presence of terms involving the quadrupole and higher-order moments, as well
as terms which are nonlinear in $S^{\alpha\beta}$.

To begin to estimate the consequences of neglecting all of these corrections
to the spin Hall equations, it is first necessary to introduce a number of
scales. To begin, the background spacetime is assumed to be characterized by a
radius of curvature $\ell_{R}$ and a scale $\ell_{\nabla R}$ over which that
radius varies,

$R_{\alpha\beta\gamma\lambda}\sim\frac{1}{\ell_{R}^{2}},\qquad\nabla_{\rho}R_{\alpha\beta\gamma\lambda}\sim\frac{1}{\ell_{R}^{2}\ell_{\nabla
R}}.$ (3.20)

These estimates, and all the similar ones below, are assumed to hold in a
locally inertial frame which is instantaneously at rest with respect to
$t^{\alpha}$, the vector field chosen to fix the centroid. Another important
scale is $\ell_{t}$, which characterizes variations in $t^{\alpha}$,

$\frac{\nabla_{\alpha}t_{\beta}}{\sqrt{-t^{2}}}\sim\frac{1}{\ell_{t}}.$ (3.21)

The length scales $\ell_{R}$, $\ell_{\nabla R}$, and $\ell_{t}$ characterize
the external environment.

We now introduce several additional scales which characterize the wave packet
itself. The first of these is the energy

$E\equiv-\frac{p\cdot t}{\sqrt{-t^{2}}}.$ (3.22)

In terms of a wave packet’s characteristic frequency $\omega$, it is suggested
by (2.2) [and by (5.27) below] that $E=\epsilon\omega$. Another relevant scale
is provided by the angular momentum. Using (3.5), this is given by

$S^{\alpha\beta}\sim\epsilon|s|=\frac{|s|}{\omega}E.$ (3.23)

While the derivation of the spin Hall equations in [42] suggested that $s=\pm
1$ for circularly polarized electromagnetic wave packets, we shall see in Sec.
V below that this is true only when the wave packet has a relatively simple
structure; it must have “spin angular momentum” but not “orbital angular
momentum.” More generally, it follows from the integral expression (3.12b) for
$S^{\alpha\beta}$ that a rough bound is given by $|s|<\omega\ell_{w}$, where
$\ell_{w}$ denotes a characteristic width for the wave packet. Below, we
assume that $|s|$ remains well below this bound in order not to violate the
high-frequency approximation. But even so, it may still be that $|s|\gg 1$.

Our final estimate involves the quadrupole moment
$J^{\alpha\beta\gamma\lambda}$. Unlike $p_{\alpha}$ and $S^{\alpha\beta}$,
this rescales under changes in the worldline parameter. Using the
dimensionless parameter $\tau$ which is associated with the normalization
condition (3.2), it may be shown that the quadrupole moment has dimension
$(\mbox{length})^{4}$. It is generically of order

$J^{\alpha\beta\gamma\lambda}\sim(E\ell_{w})^{2}.$ (3.24)

Magnitudes of the higher-order moments are essentially the same except for the
involvement of higher powers of $\ell_{w}$. Regardless, these scalings imply
that a wave packet can be characterized by $s$, $\omega$, $E$, and $\ell_{w}$.

The approximations inherent in the spin Hall equations may now be summarized
as

  1. 1.

Terms nonlinear in the spin can be neglected in the momentum-velocity
relation: $\ell_{t}\gg|s|/\omega$.

  2. 2.

The instantaneous quadrupole force can be neglected in comparison with the
spin-curvature contribution to the linear momentum evolution:
$\ell_{w}^{2}\ll(|s|/\omega)\ell_{\nabla R}$.

  3. 3.

The instantaneous quadrupole torque can be neglected in comparison with the
$p^{[\alpha}\dot{x}^{\beta]}$ contribution to the angular momentum evolution:
$\ell_{w}^{2}\ll(|s|/\omega)\ell_{R}^{2}/\ell_{t}$.

  4. 4.

The quadrupole torque negligibly affects the spin over the dimensionless
integration timescale $\Delta\tau$:
$E\Delta\tau\ll(|s|/\omega)(\ell_{R}/\ell_{w})^{2}$.

These constraints are all related to terms which are neglected when going from
Dixon’s laws of motion to the MP equations, and finally to the spin Hall
equations. Separately, it is also necessary to assume that

  1. 5.

The wave packet is large compared with its wavelength and it does not have
nontrivial structure on very small scales, $\omega\ell_{w}\gg|s|$.

This is required for the approximate validity of geometric optics, which was
used in the derivation of the spin Hall equations in [42]. Alternatively, if
the MP equations are used as a starting point, geometric optics must be used
to motivate the initial data considered here—for example the null character of
$p_{\alpha}$. This viewpoint is discussed further in Sec. V below. Regardless,
condition 5 must be imposed in order for a pulse to maintain its structure. If
it were violated, a wave packet would rapidly diffract away.

Let us now examine the consequences of assumptions 1–5. The first of these
implies that $t^{\alpha}$ must vary sufficiently slowly that
$\ell_{t}\gg|s|(\mbox{wavelengths})$. Assumptions 1, 2, 3, and 5 imply that
the wave packet must be small compared to the curvature scales,

$\ell_{w}\ll\min(\ell_{R},\ell_{\nabla R}).$ (3.25)

This guarantees that, e.g., the octupole terms in the laws of motion are
negligible compared with quadrupole terms. It is therefore unnecessary to
impose that restriction separately from the ones above.

Regardless, (3.25) does not exhaust the content of the first three
assumptions. When $|s|\sim 1$, for example, they imply much stricter bounds on
$\ell_{w}$. Writing those bounds in dimensionless form while also
incorporating assumption 5,

$1\ll\omega\ell_{w}\ll\min\left[\ell_{R}(\omega/\ell_{t})^{\frac{1}{2}},(\omega\ell_{\nabla
R})^{\frac{1}{2}}\right].$ (3.26)

In this sense, $\ell_{w}$ cannot be either too large or too small when
compared with one wavelength.

Assumptions 1, 2, and 3 arise from comparing the instantaneous magnitudes of
different terms in the equations of motion. Assumption 4 tells us how long
those equations can be reliably integrated. If $\ell_{w}$ is approximately
constant, it implies that the integrations remain valid over (dimensionful)
timescales—or equivalently distances—of order

$\Delta t\equiv
E\Delta\tau\ll\frac{|s|}{\omega}(\ell_{R}/\ell_{w})^{2}<\ell_{R}^{2}/\ell_{w}.$
(3.27)

Note that the allowable integration time here is much smaller when $|s|\sim 1$
than it is for a maximally spinning wave packet. This is because smaller spin
effects are more easily overwhelmed by quadrupole corrections.

One subtlety in this discussion is that it is not necessarily justified to
assume that $\ell_{w}$ remains constant over an integration timescale.
Electromagnetic wave packets almost111111There are nondiverging beams in flat
spacetime, but these must be specially prepared and most do not decay rapidly
enough to have well-defined momenta. In a curved spacetime, it is likely that
except in very special circumstances, beam divergence will be even more rapid
due to the defocusing of null geodesics. invariably diffract and spread out as
they propagate. This differs from the behavior of (some) solids and strongly
self-gravitating fluids, which can—at least approximately—maintain their
dimension over long timescales. We estimate the divergence of an
electromagnetic wave packet by analogy with a Laguerre–Gauss beam in flat
spacetime. If such a beam has minimum width $\ell^{*}_{w}$, its width at a
distance $L\gg\ell^{*}_{w}$ away from where that minimum occurs is of order
[83]

$\ell_{w}\sim\frac{L|s|}{\omega\ell^{*}_{w}}.$ (3.28)

We assume that this relation holds not only for Laguerre–Gauss beams, but
generically. Further assuming that the equations of motion are integrated
beginning near the point where the beam has attained its minimum width, so
$L\sim\Delta t$, substituting (3.28) into (3.26) while also using (3.27) shows
that

$\Delta
t\ll\min\left[\left(\frac{\omega\ell_{R}^{2}}{\ell_{t}}\right)^{\frac{1}{2}},\left(\frac{|s|^{\frac{1}{2}}\omega\ell_{R}^{2}}{\ell^{*}_{w}}\right)^{\frac{1}{3}},(\omega\ell_{\nabla
R})^{\frac{1}{2}}\right]\frac{\ell^{*}_{w}}{|s|^{\frac{1}{2}}}.$ (3.29)

Increasing $\omega$ at fixed $\ell^{*}_{w}$ therefore increases the upper
bound on the integration time. This is consistent with what might have been
expected from improving the high-frequency approximation. However, it is not
possible to increase $\Delta t$ indefinitely: As shown by, e.g., (3.26) in the
$|s|\sim 1$ case, increasing $\omega$ results in a decreasing upper bound on
$\ell^{*}_{w}$. Also note that the maximum allowable integration time
decreases when $|s|\gg 1$.

One example which may be considered is that of a wave packet propagating at a
distance $r$ from a static gravitating object of mass $M$. In this case, the
curvature scales are $\ell_{R}\sim r(r/M)^{1/2}$ and $\ell_{\nabla R}\sim r$.
Furthermore, if $t^{\alpha}$ is chosen to be parallel to the static Killing
field, $\ell_{t}\sim r(r/M)$. This implies that
$\ell_{t}\gg\ell_{R}\gg\ell_{\nabla R}$ in an approximately Newtonian regime.
In that regime, it follows from assumptions 2, 3, and 5 that the minimum beam
width is bounded by

$1\ll\omega\ell^{*}_{w}\ll(|s|\omega r)^{\frac{1}{2}}.$ (3.30)

It also follows from (3.29) that if $r$ does not change too much over the
integration time,

$\Delta t\ll\min\left[\left(\frac{\omega
r}{|s|}\right)^{\frac{1}{2}},\left(\frac{\omega}{|s|M\bar{\ell}_{w}}\right)^{\frac{1}{3}}r\right]\ell^{*}_{w}.$
(3.31)

Both bounds together imply that $\Delta t\sim L\ll r$, which significantly
limits the applicability of the spin Hall equations in astrophysical systems.

Except for assumption 5 above, our discussion has focused only on neglected
terms in the spin Hall equations of motion. However, there are separate errors
incurred by using inaccurate initial data in those equations. As discussed in
Sec. III.1, it is assumed in the spin Hall context that $p_{\alpha}$ is null,
$S^{\alpha\beta}S_{\alpha\beta}>0$, and $S^{\alpha\beta}p_{\beta}=0$. However,
we show in the Appendix that in fact, there does not exist any exact wave
packet with these properties. An electromagnetic field with nonzero angular
momentum must have a timelike momentum, not a null one. Nevertheless, there
are large classes of electromagnetic fields for which the spin Hall initial
data is approximately valid, and it is in that context that the spin Hall
equations should be understood. We do not, however, attempt to estimate the
errors incurred by this aspect of the approximation.

##  IV The many centroids of extended objects

Whether an extended object is composed of “ordinary” matter, electromagnetic
fields, or anything else, it is not possible to fully describe its location
using only a single worldline. There are nevertheless situations in which it
is useful to use a single worldline to describe the “averaged” location of an
extended object. This is the role of a centroid. However, unlike in Newtonian
mechanics, there are many centroids which might reasonably be associated with
relativistic systems. One of these centroids might be more useful in one
context, while another might be more useful in another context. The various
centroids may be interpreted as a particular class of observables.

The centroids considered here are associated with timelike vector fields. As
stated in Sec. III.1 above, any such vector field may be associated with a
centroid by requiring that $S^{\alpha\beta}t_{\beta}=0$. This interpretation
is verified in Sec. IV.1 below. Distinctions between the various centroids and
the implications of those distinctions are discussed in Secs. IV.2–IV.4.

Many of the ideas described in this section were introduced long ago by, e.g.,
Pryce [30] and Møller [31]. Some of those ideas have been rediscovered more
recently by different communities, who have introduced different terminologies
and interpretations. For example, displacements between different centroids
have, in certain contexts, been described as relativistic Hall effects [26]
and also as Wigner [27] or Wigner-Souriau [28, 29] translations. Regardless,
properties of different centroids are well-understood for massive objects,
where $p_{\alpha}$ is timelike.

What has not been so carefully explored in the literature is the massless
case, where $p_{\alpha}$ is null. This section discusses both the massive and
massless cases together. Our main new finding is concerned with the maximum
possible separation between different centroids associated with the same
physical object. For massive bodies, we recover the classical result [31, 32,
84, 33] that all possible centroids are confined to a disk with finite radius.
The set of all centroids therefore localizes a massive object to a finite
region, providing some reassurance that the centroid definition is a
reasonable one. The massless case is different, however. We find that massless
spinning objects cannot be localized in this way; they possess centroids
separated by arbitrarily large distances. This is potentially problematic, and
resolving it involves examining certain subtleties of the approximations used
to describe, e.g., electromagnetic wave packets. Our conclusion is that the
delocalization of massless objects is not physically relevant because a wave
packet cannot truly be massless.

The strategy taken in this section is to first discuss all issues in flat
spacetime and in inertial coordinates. All arguments are then straightforward
and all results are exact. There are no subtleties involving neglected higher-
order terms in the laws of motion. Later, in Sec. IV.4, we discuss how—with
appropriate caveats—the same results carry over for sufficiently small objects
in curved spacetime.

###  IV.1 Defining a centroid

Our first task is to show that, as claimed above, the choice of $t^{\alpha}$
is equivalent to a choice of worldline. For simplicity, we work in flat
spacetime and use inertial coordinates.

To begin, recall that the definitions (3.12) for an object’s linear and
angular momentum supposed that a particular worldline had been fixed and that
$p_{\alpha}$ and $S^{\alpha\beta}$ depended only upon a parameter $\tau$ which
had been associated with that worldline. Those definitions are easily
generalized to avoid the introduction of any particular worldline. Instead, if
the hypersurfaces $\Sigma_{\tau}$ are replaced by $\Sigma_{x}$, where
$x^{\alpha}$ is now an arbitrary point (not yet associated with any particular
worldline), $p_{\alpha}$ and $S^{\alpha\beta}$ may be viewed as functions of
that point. With this redefinition in mind, as long as the $\Sigma_{x}$
foliate the support of the stress-energy tensor, stress-energy conservation
implies that the left-hand side of (3.15) must be independent of $x^{\alpha}$
for each Killing field $\kappa^{\alpha}$. The quantities
$p_{\alpha}\kappa^{\alpha}+\frac{1}{2}S^{\alpha\beta}\nabla_{\alpha}\kappa_{\beta}$
are therefore conserved in the sense that they are independent of
$x^{\alpha}$. Using this together with the fact that the flat spacetime
Killing fields can be written as
$\kappa^{\alpha}=\mathcal{T}^{\alpha}+\mathcal{B}^{\alpha\beta}x_{\beta}$,
where the translation $\mathcal{T}^{\alpha}$ and the rotation or boost
$\mathcal{B}^{\alpha\beta}=\mathcal{B}^{[\alpha\beta]}$ are arbitrary
constants, the linear and angular momenta associated with two different
points, $x$ and $\tilde{x}$, must be related via

$\displaystyle p_{\alpha}(\tilde{x})$ $\displaystyle=p_{\alpha}(x),$ (4.1a)
$\displaystyle S^{\alpha\beta}(\tilde{x})$
$\displaystyle=S^{\alpha\beta}(x)+2(x-\tilde{x})^{[\alpha}p^{\beta]}(x).$
(4.1b)

This describes how the linear and angular momenta transform under a shift of
origin. In flat spacetime, these relations are exact. They have the same form
as the Wigner-Souriau translations which arose in the study of chiral fermions
in Refs. [28, Eq. 3.7] and [29, Eq. 2.7]. They could also have been derived
straightforwardly from the momentum definitions (3.12) as well as (3.14).

Defining $\tilde{S}^{\alpha\beta}\equiv S^{\alpha\beta}(\tilde{x})$ together
with the deviation vector $\xi^{\alpha}\equiv\tilde{x}^{\alpha}-x^{\alpha}$,
it follows from (4.1b) that

$S^{\alpha\beta}t_{\beta}=\tilde{S}^{\alpha\beta}t_{\beta}-(-p\cdot
t)\xi^{\alpha}-(\xi\cdot t)p^{\alpha}.$ (4.2)

The claimed centroid condition (3.4) now amounts to the vanishing of the left-
hand side of this equation. And no matter how $\tilde{x}^{\alpha}$ has been
chosen or what form $\tilde{S}^{\alpha\beta}$ may have, that can be arranged
by choosing $x^{\alpha}$ such that

$x^{\alpha}=\tilde{x}^{\alpha}+Tp^{\alpha}-\frac{1}{(-p\cdot
t)}\tilde{S}^{\alpha\beta}t_{\beta},$ (4.3)

where $T$ is an arbitrary parameter. Varying over all possible values of this
parameter recovers a worldline: what we call the centroid associated with
$t^{\alpha}$. This is true regardless of whether $p_{\alpha}$ is timelike or
null. It may also be seen that if $\nabla_{\beta}t^{\alpha}=0$, the centroid
is tangent to $p^{\alpha}$ and $T$ may be identified with the worldline
parameter $\tau$ which is associated with the normalization condition (3.2).
Both of these statements can fail when $t^{\alpha}$ is not constant.

Next, we verify that the centroid is deserving of its name. First, recall that
stress-energy conservation implies that $S^{\alpha\beta}(x)$ does not depend
on the hypersurface $\Sigma_{x}$, as long as all fields fall off sufficiently
rapidly and all relevant hypersurfaces completely cut through the support of
the stress-energy tensor. We may therefore choose $\Sigma_{x}$ to be the
hyperplane which is orthogonal to $t^{\alpha}$ at $x^{\alpha}$. Doing so,
while temporarily adopting inertial coordinates which are comoving with
$t^{\alpha}$, use of (3.12) and (3.14) shows that $S^{\alpha\beta}t_{\beta}=0$
holds only when

$x^{i}=\frac{1}{E}\int x^{\prime i}T^{00}(x^{\prime})d^{3}x^{\prime},$ (4.4)

where

$E=\int T^{00}(x^{\prime})d^{3}x^{\prime}$ (4.5)

is the energy (3.22). This is the standard nonrelativistic center of mass
definition, but with the nonrelativistic mass density replaced by the
relativistic energy density $T_{\alpha\beta}t^{\alpha}t^{\beta}$. It follows
that as long as $T_{\alpha\beta}t^{\alpha}t^{\beta}\geq 0$, the centroid must
lie inside the convex hull of the spatial support of the stress-energy tensor.
That the energy density should not be negative could be viewed as a
consequence of, e.g., the dominant energy condition. That condition is
satisfied by essentially all standard classical fields, including
electromagnetic ones [85].

One subtlety which does not appear to have been recognized before is that
although the dominant energy condition is satisfied by exact electromagnetic
field configurations, it is not necessarily satisfied by the approximate
fields which might be used to describe high-frequency wave packets. As
discussed further in Sec. V.5 below, there can exist timelike $t^{\alpha}$ for
which the approximate energy density is positive in some regions and negative
in others. This has a dramatic consequence: The centroid of an approximate
wave packet can appear to lie arbitrarily far from the wave packet itself.
Those centroids are of course spurious. They are a consequence of neglecting
higher-order terms in the stress-energy tensor. See further discussion in
Secs. IV.3 and V.5 below.

Another comment which can be made is concerned with the fact that it is common
in the literature [31, 69, 86, 87, 71] to use $S^{\alpha\beta}p_{\beta}=0$ as
a centroid condition instead of $S^{\alpha\beta}t_{\beta}=0$, particularly—but
not exclusively [53, 88, 89, 90]—for objects with timelike momenta. This has
the apparent advantage that the results do not depend on extraneous choices
such as that of $t^{\alpha}$. And in the massive case, there is nothing wrong
with this; Eq. (3.7) remains valid with $t^{\alpha}=p^{\alpha}$. But this
fails for massless objects. In that case, (4.2) remains valid so replacing
$t^{\alpha}$ there by $p^{\alpha}$ shows that $x$ must be a solution to

$\tilde{S}^{\alpha\beta}p_{\beta}=[(\tilde{x}-x)\cdot p]p^{\alpha}.$ (4.6)

If the left-hand side here is nonzero and not proportional to $p^{\alpha}$, no
such solution exists. If the left-hand side is instead proportional to
$p^{\alpha}$, any $x^{\alpha}$ which satisfies $(x-\tilde{x})\cdot
p=\mbox{const}$ will do. That restricts the centroid only to a three-
dimensional null hypersurface, not a worldline. In either case,
$S^{\alpha\beta}p_{\beta}=0$ cannot be interpreted as a centroid condition for
massless objects. While this has been noted before, details were scant [64,
39].

In some of the literature which does attempt to use
$S^{\alpha\beta}p_{\beta}=0$ as a centroid condition for massless objects [52,
88, 89, 90], there is a relation derived between the momentum and the velocity
which suggests that a centroid does indeed exist. However, that relation
involves a ratio whose denominator (in a curved spacetime) is
$R_{\alpha\beta\gamma\lambda}S^{\alpha\beta}S^{\gamma\lambda}$. The momentum-
velocity relation therefore fails in the flat spacetime context of our present
discussion. It also fails at least somewhere on many worldlines which might be
considered in more general spacetimes. It does not appear to us to be viable
to attempt to impose a condition which fails to be robust or to have
reasonable limits. In particular, the lack of a viable flat-spacetime limit
implies that even when $S^{\alpha\beta}p_{\beta}=0$ does result in a unique
worldline, it will not describe a centroid in the sense of (4.4).

###  IV.2 Displacements between different centroids

As there are many different centroids which may be used to describe an
extended object, it is natural to ask how these are related to one another.
The answer has long been known for massive objects, as described in, e.g.,
Refs. [91, 32, 84, 33, 92]. There, a canonical centroid was defined via
$S^{\alpha\beta}p_{\beta}=0$ and separations were derived between this
centroid and others. As noted above, a canonical centroid cannot be defined in
this way when considering massless objects. Nevertheless, only minor changes
are needed to consider the differences between arbitrary reference centroids.
We discuss both the massless and massive cases below. For simplicity, we also
continue to work in flat spacetime and to use inertial coordinates.

Consider two future-directed timelike vector fields $t^{\alpha}$ and
$\tilde{t}^{\alpha}$ and the corresponding centroid conditions

$S^{\alpha\beta}t_{\beta}=\tilde{S}^{\alpha\beta}\tilde{t}_{\beta}=0.$ (4.7)

These define two worldlines, the points on which may be denoted by
$x^{\alpha}$ and $\tilde{x}^{\alpha}$. Finding a unique displacement
$\xi^{\alpha}=\tilde{x}^{\alpha}-x^{\alpha}$ between them requires that points
on each worldline be identified in a particular way. It is convenient to do so
by supposing that

$\xi\cdot t=0,$ (4.8)

in which case (4.2) and (4.7) imply that $\xi\cdot\tilde{t}=0$ and

$\xi^{\alpha}=\frac{S^{\alpha\beta}\tilde{t}_{\beta}}{p\cdot\tilde{t}}=-\frac{\tilde{S}^{\alpha\beta}t_{\beta}}{p\cdot
t}.$ (4.9)

This displacement vector is exact in flat spacetime, is valid for both massive
and massless objects, and there is no constraint on the nature of the angular
momentum. One immediate consequence is that all centroids coincide for
nonspinning objects.

Given Eq. (4.7), it is always possible to introduce a spin vector $S^{\alpha}$
such that

$S^{\alpha\beta}=\frac{\varepsilon^{\alpha\beta\gamma\lambda}S_{\gamma}t_{\lambda}}{\sqrt{-t^{2}}},$
(4.10)

which is unique only up to arbitrary multiples of $t^{\alpha}$. In terms of
any such spin vector, the displacement (4.9) can be written as

$\xi^{\alpha}=\frac{\varepsilon^{\alpha\beta\gamma\lambda}S_{\beta}t_{\gamma}\tilde{t}_{\lambda}}{(p\cdot\tilde{t})\sqrt{-t^{2}}}.$
(4.11)

This is a spacelike vector orthogonal to $t^{\alpha}$, $\tilde{t}^{\alpha}$,
and $S^{\alpha}$. It may be used to relate any two centroids to one another.
It still does not make any assumptions regarding the nature of the spin or the
object’s mass.

If we now specialize to the spin Hall case where $S^{\alpha\beta}$ is given by
Eq. (3.8) and $p_{\alpha}$ is null, the spin vector may be be identified with

$S_{\alpha}=\epsilon s\left(\frac{\sqrt{-t^{2}}}{p\cdot t}\right)p_{\alpha}.$
(4.12)

As this is proportional to $p_{\alpha}$, it may be described as a
“longitudinal spin.” The displacements in this case are given by

$\xi^{\alpha}=\epsilon
s\left[\frac{\varepsilon^{\alpha\beta\gamma\lambda}p_{\beta}t_{\gamma}\tilde{t}_{\lambda}}{(p\cdot
t)(p\cdot\tilde{t})}\right],$ (4.13)

which are transverse to the momentum.

###  IV.3 Localization of extended objects

As the choice of $t^{\alpha}$ is essentially arbitrary, one might hope that
the centroids associated with different vector fields are not too different.
In particular, it is natural to ask if they are all confined to a finite
region—perhaps within the convex hull of the spacelike support of the object’s
stress-energy tensor. As noted above, this does indeed follow from (4.3) when
$T_{\alpha\beta}t^{\alpha}t^{\beta}\geq 0$. It is also possible to show,
without using $T_{\alpha\beta}$, that the set of all possible centroids is
localized whenever $p_{\alpha}$ is timelike; cf. [91, Sec. 6.3] or [32, Sec.
3.1b]. We now discuss both the massive and the massless cases and show that in
the latter context, some “centroids” can be arbitrarily distant from one
another.

Assume that some future-directed timelike $t^{\alpha}$ has been fixed and
measure all deviations as being with respect to the centroid for which
$S^{\alpha\beta}t_{\beta}=0$. If points on the centroids associated with
$\tilde{t}^{\alpha}$ and $t^{\alpha}$ are identified using (4.8), it follows
from (4.11) that the square of the proper distance between those points is

$\xi^{2}=\left[\frac{(t\cdot\tilde{t})^{2}-t^{2}\tilde{t}^{2}}{(-t^{2})(p\cdot\tilde{t})^{2}}\right]h_{\alpha\beta}S^{\alpha}S^{\beta},$
(4.14)

where

$h_{\alpha\beta}\equiv
g_{\alpha\beta}+\frac{t^{2}\tilde{t}_{\alpha}\tilde{t}_{\beta}+\tilde{t}^{2}t_{\alpha}t_{\beta}-2(t\cdot\tilde{t})t_{(\alpha}\tilde{t}_{\beta)}}{(t\cdot\tilde{t})^{2}-t^{2}\tilde{t}^{2}}$
(4.15)

projects vectors into the space orthogonal to both $t^{\alpha}$ and
$\tilde{t}^{\alpha}$ (when those vectors are not parallel).

To discuss the implications of this in the massive case, it is convenient to
now choose $t^{\alpha}=p^{\alpha}$ so all deviations are measured with respect
to the centroid defined by $S^{\alpha\beta}p_{\beta}=0$. Then (4.14) implies
that

$\xi=\frac{\mathcal{S}}{m}(V\sin\theta),$ (4.16)

where $\mathcal{S}\equiv(\frac{1}{2}S^{\alpha\beta}S_{\alpha\beta})^{1/2}$
characterizes the magnitude of the spin, $m\equiv(-p^{2})^{1/2}$ is the mass,

$V\equiv[1-t^{2}\tilde{t}^{2}/(t\cdot\tilde{t})^{2}]^{1/2}<1$ (4.17)

is the relative speed between $t^{\alpha}$ ($=p^{\alpha}$) and
$\tilde{t}^{\alpha}$, and $\theta\in[0,\pi]$ is the angle between
$\tilde{t}^{\alpha}$ and $S^{\alpha}$ which would be measured by an observer
whose 4-velocity is tangent to $t^{\alpha}$. It is evident from (4.16) that
the magnitude of the centroid displacement can be no larger than the Møller
radius $\mathcal{S}/m$. All centroids are therefore confined to a disk with
that radius. Unless energy conditions are violated, there is a sense in which
the disk of centroids must be smaller than the object itself.

The massless case is more subtle. For simplicity, we do not discuss the most
general massless case, but only the spin Hall case in which the angular
momentum is restricted via $S^{\alpha\beta}p_{\beta}=0$. As it is not possible
to choose $t^{\alpha}$ to be proportional to $p^{\alpha}$ in this context, we
assume that $t^{\alpha}$ and its associated centroid have been fixed in some
other way and that all other centroids are measured with respect to it. The
distance between the centroid determined by $t^{\alpha}$ and the one
determined by $\tilde{t}^{\alpha}$ is then found by substituting (4.12) into
(4.14). This yields

$\xi=\frac{\mathcal{S}}{E}\left(\frac{V\sin\theta}{1-V\cos\theta}\right)$
(4.18)

for a massless object, where $E$ is the energy (3.22), and $\mathcal{S}$, $V$,
and $\theta$ have the same meanings as in (4.16). In this case, it also
follows from (3.5) that $\mathcal{S}=\epsilon|s|$.

The prefactors are essentially the same in the massless displacement (4.18)
and its massive counterpart (4.16); the energy $E$ which appears in the
massless case is simply replaced by the mass $m$, which is of course the
energy in the zero-momentum frame. Up to this replacement, both displacements
coincide when $V\ll 1$. Indeed, all centroids determined by “nearly comoving”
observers satisfy $\xi\leq(\mathcal{S}/E)V$. In both the massless and the
massive cases, they are contained within the “generalized Møller radius”
$\mathcal{S}/E$.

If the magnitude of $V$ is not restricted, the massless and massive
displacements still coincide when $\tilde{t}^{\alpha}$ is aligned,
antialigned, or orthogonal to $S^{\alpha}$ in a frame comoving with
$t^{\alpha}$. In the aligned and antialigned cases, there is no effect at all:
$\xi=0$. In the orthogonal case where $\theta=\pi/2$, we have instead that the
proper distance between two centroids is $\xi=(\mathcal{S}/E)V$. Since $V<1$,
this is again bounded by generalized Møller radius. In the massless case and
for a high-frequency wave packet, (5.27) below shows that this bound can be
written as

$\xi<\frac{\mathcal{S}}{E}=\frac{|s|}{\omega},$ (4.19)

where $\omega$ denotes the angular frequency of the field in the frame
comoving with $t^{\alpha}$. The displacement is therefore less than
approximately $|s|$ wavelengths. It is in agreement with discussions of the
relativistic Hall effect [26] and the Wigner translations [27], where the
energy centroid of a beam with nonzero angular momentum was shown to
experience a similar shift after applying a boost orthogonal to the direction
of propagation.

What does not appear to have been noticed before is that the maximum
displacement in the massless case does not occur at $\theta=\pi/2$ (except in
the $V\ll 1$ limit). For fixed $V$, the angle which maximizes $\xi$ in (4.18)
is instead

$\theta=\cos^{-1}V.$ (4.20)

Using that, the maximum displacement between massless centroids is found to be

$\xi=\frac{\mathcal{S}}{E}\left(\frac{V}{\sqrt{1-V^{2}}}\right).$ (4.21)

This diverges as $V\to 1$. Unlike in the massive case, the set of all possible
centroids is not bounded for a massless spinning body. Arbitrarily large
displacements can occur between the centroids associated with $t^{\alpha}$ and
$\tilde{t}^{\alpha}$ when those vectors differ by ultrarelativistic boosts
which are almost—but not quite—parallel to the momentum.

This presents an apparent problem for the formalism. One interpretation is
simply that massless spinning objects, whatever those may be, cannot be
localized. However, this is unacceptable if we interpret certain
electromagnetic wave packets as examples of massless spinning objects.
Physically realizable wave packets clearly can be localized, and any
worldlines which fail to lie near the support of their stress-energy tensors
are hardly deserving to be called “centroids”. Therefore, either our centroid
definition is inappropriate or there is something wrong with our
interpretation of electromagnetic wave packets as spinning objects with null
momenta. The first possibility can be discounted by recalling the discussion
following (4.3).

The resolution is that the momentum of a spinning electromagnetic wave packet
is not actually null. It must be timelike. We have been assuming above that
the momentum is null, and given reasonable assumptions, this is approximately
true for a high-frequency wave packet. Indeed, it is true through leading and
subleading orders in a high-frequency approximation, and that is all that the
spin Hall equations can describe (as they omit terms of order $\epsilon^{2}$).
However, it is demonstrated in Sec. V.4 below, using an explicit family of
wave packets, that the momentum is always timelike when going to one higher
order. The mass is found to be order $\epsilon/\ell_{w}$, where $\ell_{w}$ is
again a characteristic width for the wave packet. Using Eq. (4.16), this
implies that the maximum deviation between centroids is of order $\ell_{w}$.
Although that is the intuitively expected result, establishing it requires
that calculations be performed to a relatively high order. Truncating the
approximation too early results in a conclusion which is not even
qualitatively correct.

At lower orders in the high-frequency approximation, one can say only that the
momentum is approximately null. Mathematically, we are considering 1-parameter
families of wave packets in which, e.g., $\lim_{\epsilon\to
0}p^{\alpha}p_{\alpha}=0$. However, what is physically interesting is an
example of such a family at a particular (“small”) value of $\epsilon$. In
that context, a vector can be “approximately” null only with respect to some
restricted class of observers. If a vector is actually timelike, for example,
there clearly exist some observers for whom it appears to be stationary and
some observers for whom it appears to be “nearly null.” There is therefore a
sense in which the high-frequency approximation implicitly selects a kind of
rest frame. It is reliable only in frames which are not too highly boosted
with respect to that rest frame.

###  IV.4 Centroids in curved spacetimes

The main results obtained thus far in this section are that i) the
displacements between different centroids are given by Eq. (4.11), ii) the
maximum magnitude of the displacement is given by Eq. (4.16) when $p_{\alpha}$
is timelike, and iii) the displacement is unbounded when $p_{\alpha}$ is null.
These results were derived in flat spacetime, and in that context, they are
exact. There are no corrections due to higher-order spin effects, quadrupole
moments, or anything else. We now discuss the sense in which our results
remain at least approximately valid for sufficiently small objects in generic
spacetimes. One would expect from the equivalence principle that everything
remains at least approximately valid even in curved spacetimes, and indeed it
does.

We begin by obtaining a curved spacetime form for the transformation law
(4.1b) between the angular momentum evaluated about different points
$x^{\alpha}$ and $\tilde{x}^{\alpha}$. It is useful to first use the
exponential map to define a deviation vector $\xi^{\alpha}$ between those
points, so

$\tilde{x}^{\alpha}=\exp_{x}\xi^{\beta}.$ (4.22)

In a Riemann normal coordinate system with origin $x^{\alpha}$, this takes the
standard form $\xi^{\alpha}=\tilde{x}^{\alpha}-x^{\alpha}$. Regardless of the
coordinate system, the displacement vector is the negative gradient of Synge’s
world function: $\xi^{\alpha}=-\sigma^{\alpha}(x,\tilde{x})$. We can use this
to find covariant Taylor expansions in the style of, e.g., [76, Sec. 6].
Letting primed indices be associated with $\tilde{x}^{\alpha}$ and unprimed
ones with $x^{\alpha}$, the relevant expansion for the angular momentum tensor
is

$\tilde{S}^{\alpha^{\prime}\beta^{\prime}}=g^{\alpha^{\prime}}{}_{\alpha}g^{\beta^{\prime}}{}_{\beta}[S^{\alpha\beta}+\xi^{\gamma}\nabla_{\gamma}S^{\alpha\beta}+\mathcal{O}(\xi^{2})],$
(4.23)

where $g^{\alpha^{\prime}}{}_{\alpha}$ denotes the bitensor which parallel
propagates vectors from $x^{\alpha}$ to $\tilde{x}^{\alpha}$ along the
geodesic segment which connects those points. A similar expansion may also be
used to relate the linear momentum at $x^{\alpha}$ to the linear momentum at
$\tilde{x}^{\alpha}$.

Regardless, continuing requires that we compute the gradients of $p_{\alpha}$
and $S^{\alpha\beta}$. While the argument can be generalized, consider for
simplicity displacements $x^{\alpha}\mapsto\tilde{x}^{\alpha}$ which lie
entirely within the same “constant-time” hypersurface, so
$\Sigma_{x}=\Sigma_{\tilde{x}}$. Then, associating double-primed indices with
an integration point $x^{\prime\prime}$, it follows from (3.12) that

$\displaystyle\xi^{\gamma}\nabla_{\gamma}p_{\alpha}$
$\displaystyle=\xi^{\gamma}\int_{\Sigma_{x}}dS_{\beta^{\prime\prime}}T^{\beta^{\prime\prime}}{}_{\alpha^{\prime\prime}}\nabla_{\gamma}K^{\alpha^{\prime\prime}}{}_{\alpha},$
(4.24a) $\displaystyle\xi^{\gamma}\nabla_{\gamma}S_{\alpha\beta}$
$\displaystyle=2\xi^{\gamma}\int_{\Sigma_{x}}dS_{\beta^{\prime\prime}}T^{\beta^{\prime\prime}}{}_{\alpha^{\prime\prime}}(H^{\alpha^{\prime\prime}}{}_{[\alpha}\sigma_{\beta]\gamma}$
$\displaystyle\qquad\qquad\qquad\leavevmode\nobreak\
+\nabla_{\gamma}H^{\alpha^{\prime\prime}}{}_{[\alpha}\sigma_{\beta]}).$
(4.24b)

All bitensors here are evaluated at $(x,x^{\prime\prime})$. If the maximum
distance, within $\Sigma_{x}$, between $x$ and any integration point where
$T^{\beta^{\prime\prime}}{}_{\alpha^{\prime\prime}}\neq 0$ is of order
$\ell_{w}$, standard coincidence limits for the world function [76] imply that
$\sigma_{\alpha\beta}=g_{\alpha\beta}+\mathcal{O}(\ell_{w}^{2}/\ell_{R}^{2})$,
$K^{\alpha^{\prime}}{}_{\alpha}=H^{\alpha^{\prime}}{}_{\alpha}+\mathcal{O}(\ell^{2}_{w}/\ell_{R}^{2})$,
and $\nabla_{\gamma}H^{\alpha^{\prime}}{}_{\alpha}$ and
$\nabla_{\gamma}K^{\alpha^{\prime}}{}_{\alpha}$ are both of order
$\ell_{w}/\ell_{R}^{2}$, where $\ell_{R}$ denotes the curvature length scale
introduced in Sec. III.4. Moreover, using the energy (3.22) to estimate the
error terms, it follows that through the first order in $\xi^{\alpha}$,

$\displaystyle\tilde{p}_{\alpha^{\prime}}$
$\displaystyle=g^{\alpha}{}_{\alpha^{\prime}}p_{\alpha}+\mathcal{O}(E\xi\ell_{w}/\ell_{R}^{2}),$
(4.25a) $\displaystyle\tilde{S}^{\alpha^{\prime}\beta^{\prime}}$
$\displaystyle=g^{\alpha^{\prime}}{}_{\alpha}g\indices{{}^{\beta^{\prime}}_{\beta}}(S^{\alpha\beta}+2p^{[\alpha}\xi^{\beta]})+\mathcal{O}(E\xi\ell_{w}^{2}/\ell_{R}^{2}).$
(4.25b)

In flat spacetime, the error terms here are exactly zero; cf. (4.1b).

We would now like to use (4.7) to associate one centroid $\tilde{x}^{\alpha}$
with the timelike vector field $\tilde{t}^{\alpha^{\prime}}$ and another
centroid $x^{\alpha}$ with the timelike vector field $t^{\alpha}$. Repeating
the same steps as in Secs. IV.1 and IV.2, it is again convenient to identify
points on both worldlines using $\xi\cdot t=0$, which we now assume to be
compatible with the assumption that $\Sigma_{x}=\Sigma_{\tilde{x}}$. Equation
(4.25) can then be shown to imply that
$g_{\alpha\alpha^{\prime}}\xi^{\alpha}\tilde{t}^{\alpha^{\prime}}=\mathcal{O}(\xi\ell_{w}^{2}/\ell_{R}^{2})$.
It also follows that (4.9) generalizes to

$\xi^{\alpha}=\frac{g_{\beta\beta^{\prime}}S^{\alpha\beta}\tilde{t}^{\beta^{\prime}}}{g^{\gamma}{}_{\gamma^{\prime}}p_{\gamma}\tilde{t}^{\gamma^{\prime}}}+\mathcal{O}(\xi\ell_{w}^{2}/\ell_{R}^{2}).$
(4.26)

This assumes that $\xi$ is not so large that terms of order $\xi^{2}$ become
important in (4.25). More generally, the basic special relativistic form for
this expression remains valid when $\ell_{w}$ and $\xi$ are both much smaller
than $\ell_{R}$. Most of the above special-relativistic results remain valid
in this context. Technically, however, one can no longer conclude that there
are massless centroids which are arbitrarily distant from one another, as then
$\xi$ must be large. It is nevertheless clear from the flat spacetime limit
that massless objects must still be problematic in curved spacetime.

Figure 1: A massless spinning object, as described by two different families
of timelike observers, $t^{\alpha}$ and $\tilde{t}^{\alpha}$. The displacement
between two points on the worldlines $x^{\alpha}$ and $\tilde{x}^{\alpha}$ is
described by the shift vector $\xi^{\alpha}$. Since
$t^{\alpha}=\tilde{t}^{\alpha}$ near the emitter and the receiver, we have
$\xi^{\alpha}=0$ in these regions, and the two worldlines coincide, up to
relative error terms of order $(\ell_{w}/\ell_{R})^{2}$.

###  IV.5 The irrelevance of centroid conditions

One important consequence of Eq. (4.26) is that the centroid depends only
(quasi)locally on the timelike vector field used to define it. If $t^{\alpha}$
and $\tilde{t}^{\alpha}$ coincide in, say, neighborhoods of emission and
observation points, whatever they do in between the emitter and the observer
is irrelevant: The associated centroids will coincide at both the beginnings
and ends of their journeys. This is illustrated schematically in Fig. 1.
Physical meaning can be attributed to $t^{\alpha}$ only in the neighborhoods
of the emission and the observation events. What it does elsewhere is
essentially irrelevant.

As a consequence, the effects of different spin supplementary conditions can
be understood without repeatedly solving the equations of motion with
different conditions. Solving the Mathisson-Papapetrou equations—whether
massive or massless—with the spin supplementary condition
$S^{\alpha\beta}t_{\beta}=0$ requires an apparently extraneous specification
of the vector field $t^{\alpha}$. Indeed, that vector field must be specified
not only at a point, but in a neighborhood of the entire trajectory. The
observation of the previous paragraph shows that all that matters is the
specification of the vector field near the emission and the observation
points, at least if the displacement never gets too large. It may also be
noted that it is only near the emitter and source that there necessarily
exists a natural choice for $t^{\alpha}$: It may be identified near those
points with the 4-velocities of the emitter and the observer.

The transformations (4.25) and (4.26) may also be interpreted as a way to
generate new solutions to the equations of motion. Given a triple
$(x^{\alpha},p_{\alpha},S^{\alpha\beta})$ which satisfies the MP equations
with the centroid condition $S^{ab}t_{b}=0$, the triple
$(\exp_{x}\xi,\tilde{p}_{\tilde{\alpha}},\tilde{S}^{\tilde{\alpha}\tilde{\beta}})$
satisfies those same equations but with the centroid condition
$\tilde{S}^{\tilde{\alpha}\tilde{\beta}}\tilde{t}_{\tilde{\beta}}=0$. This
statement is exact in flat spacetime and approximate more generally.

##  V Momenta of circularly polarized wave packets

As discussed in Sec. III, there are essentially only two conditions required
for the validity of the spin Hall equations in the form considered here.
First, the effect of the quadrupole and higher-order moments must be
negligible. When this occurs can be estimated using the arguments in Sec.
III.4. However, the spin Hall equations also require for their validity that
$p_{\alpha}$ be null and that $S^{\alpha\beta}$ have the form (3.8). These may
be viewed as restrictions on the initial data for the MP equations. The claim
has been that such conditions model an electromagnetic wave packet. However,
this connection appears in the literature as an unsubstantiated (and usually
unstated) hypothesis. It is implied by results in the Appendix that the spin
Hall initial data cannot hold exactly, at least when $s\neq 0$. The purpose of
this section is to understand if there is an appropriate approximate sense in
which the spin Hall initial data is actually associated with electromagnetic
wave packets.

We show that to the expected orders, there are indeed generic wave packets
which are compatible with the spin Hall initial data. Nevertheless, we show
that those data inevitably break down at one higher order; the momentum
becomes timelike, for example. We also show that there are reasonable wave
packets which are not even approximately described by the spin Hall initial
data. For them, $S^{\alpha\beta}p_{\beta}$ is nonzero even at the lowest
nontrivial order. Said differently, the spin is not purely longitudinal. The
existence of these exceptions emphasizes that in applications, the connection
between the “microscopic” (the electromagnetic field structure) and the
“macroscopic” (the spin Hall equations or generalizations) is nontrivial and
must be considered on a case-by-case basis.

All calculations in this section are performed in flat spacetime and in
inertial coordinates. However, as we are concerned only with finding initial
linear and angular momenta for the MP equations, all calculations for
sufficiently small wave packets are confined to small regions in spacetime.
Flat spacetime calculations therefore remain excellent approximations for
sufficiently compact wave packets even in curved spacetimes, as long as the
inertial Minkowski coordinates are reinterpreted as an appropriate system of
Riemann normal coordinates.

###  V.1 A family of wave packets

Our first task is to construct a sufficiently general class of approximate
electromagnetic wave packets. We work in a high-frequency approximation and
consider a family of vector potentials $A_{\alpha}$. These vector potentials
are assumed to be given by the asymptotic series

$A_{\alpha}=\mathrm{Re}\left[\sum_{n=0}^{\infty}\tilde{\epsilon}^{n+1}\psi_{\alpha}^{(n)}e^{iu/\tilde{\epsilon}}\right],$
(5.1)

where the amplitudes $\psi^{(n)}_{\alpha}$ and the eikonal $u$ are independent
of the small parameter $\tilde{\epsilon}>0$. Note that the $\tilde{\epsilon}$
which appears here is related to, but generically distinct from, the
$\epsilon$ which appears in the spin Hall equations [and in (2.1)]. If $c$ is
any constant, $A_{\alpha}/\tilde{\epsilon}$ is invariant, at leading order,
under all transformations where $u\mapsto cu$ and $\tilde{\epsilon}\mapsto
c\tilde{\epsilon}$. It is convenient for now to avail of this ambiguity by
allowing $\tilde{\epsilon}$ to differ from $\epsilon$ by a convenient
constant. Then $u$ is not restricted to have units $(\mbox{length})^{2}$.

Again defining the leading-order wave vector
$k_{\alpha}\equiv-\nabla_{\alpha}u$, Maxwell’s equations and the Lorenz gauge
condition imply that $u$ must be a solution to the eikonal equation

$\nabla u\cdot\nabla u=k\cdot k=0.$ (5.2)

Maxwell’s equations and the gauge condition also imply that, for all $n\geq
0$, the amplitudes must satisfy the transport equations

$[2(k\cdot\nabla)+(\nabla\cdot
k)]\psi^{(n)}_{\alpha}=-i\Box\psi^{(n-1)}_{\alpha},$ (5.3)

and the constraint equations

$k^{\alpha}\psi_{\alpha}^{(n)}=-i\nabla^{\alpha}\psi_{\alpha}^{(n-1)},$ (5.4)

where $\psi^{(-1)}_{\alpha}\equiv 0$ [39, 42]. These equations are
hierarchical. A solution to the $n=0$ equation is required to solve the $n=1$
equation, an $n=1$ solution is required to solve the $n=2$ equation, etc.

We now specialize to fields which are, at least at the leading order, plane
fronted121212A plane fronted wave is not necessarily a plane wave. It is not
necessarily uniform on each wavefront. and traveling in the $+z$ direction in
the inertial coordinate system $(t,x,y,z)$. This can be represented
mathematically by choosing the eikonal

$u=t-z.$ (5.5)

It is then convenient to solve the transport and constraint equations by
interpreting $u$ as a null coordinate and by defining

$v\equiv\frac{1}{2}(t+z),\qquad\zeta\equiv\frac{1}{\sqrt{2}}(x+iy).$ (5.6)

The four scalars $(u,v,\zeta,\bar{\zeta})$ form a null coordinate system in
which the Minkowski line element reduces to

$ds^{2}=2(d\zeta d\bar{\zeta}-dudv).$ (5.7)

These coordinates can be associated with a complex null tetrad
$(k_{\alpha},n_{\alpha},m_{\alpha},\bar{m}_{\alpha})$ via

$k_{\alpha}\equiv-\nabla_{\alpha}u,\quad
n_{\alpha}\equiv-\nabla_{\alpha}v,\quad m_{\alpha}\equiv\nabla_{\alpha}\zeta.$
(5.8)

The only nonvanishing inner products among this tetrad are $m\cdot\bar{m}=1$
and $k\cdot n=-1$. In terms of it, the metric is
$g_{\alpha\beta}=2[m_{(\alpha}\bar{m}_{\beta)}-k_{(\alpha}n_{\beta)}]$. Note
that unlike in Sec. II, the $m_{\alpha}$ and $\bar{m}_{\alpha}$ here are
ordinary fields on spacetime. They are not defined over the cotangent bundle.

We now restrict to waves which are not only plane fronted but also circularly
polarized at leading order. Mathematically, this is taken to mean that
$\psi^{(0)}_{\alpha}$ is assumed to be null (and nonzero). The constraint
equation (5.4) then implies that the leading-order amplitude must be
proportional either to $m_{\alpha}+\chi k_{\alpha}$ or to
$\bar{m}_{\alpha}+\chi k_{\alpha}$, where $\chi$ is any scalar [see (2.7)].
Terms proportional to $k_{\alpha}$ are pure gauge at leading order, although
not necessarily at higher orders131313Terms in $\psi^{(0)}_{\alpha}$ which are
proportional to $k_{\alpha}$ are physically equivalent to an ordinary (gauge-
invariant) geometric optics field added to $\psi^{(1)}_{\alpha}$ [39, Appendix
B].. Regardless, we set $\chi=0$ for simplicity. Circularly polarized fields
are then described by

$\psi^{(0)}_{\alpha}=\psi m_{\alpha}$ (5.9)

or $\psi^{(0)}_{\alpha}=\psi\bar{m}_{\alpha}$, depending on the handedness of
the field. Unlike in Sec. II, we do not assume that $\psi$ which appears here
is necessarily real. Use of Eq. (5.3) shows that it is independent of $v$ but
otherwise arbitrary: $\psi=\psi(u,\zeta,\bar{\zeta})$.

We may now substitute the leading-order amplitude (5.9) into the transport
equation (5.3) and the constraint equation (5.4) in order to derive the
subleading amplitude $\psi^{(1)}_{\alpha}$. One solution is

$\psi^{(1)}_{\alpha}=i\partial_{\bar{\zeta}}\left(\psi
n_{\alpha}-vm_{\alpha}\partial_{\zeta}\psi\right).$ (5.10)

Two orders beyond geometric optics, we find that

$\displaystyle\psi^{(2)}_{\alpha}=\tfrac{1}{4}\partial_{\bar{\zeta}}\Box\left[v\left(2\psi
n_{\alpha}-v\partial_{\zeta}\psi m_{\alpha}\right)\right].$ (5.11)

Together, (5.1), (5.9), (5.10), and (5.11) describe an approximate family of
circularly polarized electromagnetic fields in flat spacetime. That family is
parametrized by the geometric-optics scalar amplitude $\psi$ and by the
constant $\tilde{\epsilon}$, which is related to the inverse frequency of the
field. Its properties are explored in the remainder of this section. If
desired, fields with the opposite helicity can be considered by swapping
$m_{\alpha}$ and $\bar{m}_{\alpha}$ in the amplitudes (5.9), (5.10), and
(5.11).

Vector potentials are not directly measurable. More interesting are the field
strengths $F_{\alpha\beta}$, and for any vector potential with the form (5.1),
a direct calculation shows that these are given by

$\displaystyle
F_{\alpha\beta}=2\mathrm{Im}\sum_{n=0}^{\infty}\tilde{\epsilon}^{n}\big{(}k_{[\alpha}\psi_{\beta]}^{(n)}+i\nabla_{[\alpha}\psi_{\beta]}^{(n-1)}\big{)}e^{iu/\tilde{\epsilon}}.$
(5.12)

For the amplitudes constructed above, this evaluates to

$\displaystyle
F_{\alpha\beta}=2\mathrm{Im}\big{\\{}\big{[}\left(\psi-\tfrac{1}{2}i\tilde{\epsilon}v^{2}\Box(v^{-1}\psi)-\tfrac{1}{8}\tilde{\epsilon}^{2}v^{2}\Box^{2}\psi\right)$
$\displaystyle{}\leavevmode\nobreak\ \times
k_{[\alpha}m_{\beta]}+\tilde{\epsilon}^{2}\partial_{\bar{\zeta}}^{2}\psi
n_{[\alpha}\bar{m}_{\beta]}+i\tilde{\epsilon}\partial_{\bar{\zeta}}\left(\psi-\tfrac{1}{2}i\tilde{\epsilon}v\Box\psi\right)$
$\displaystyle\leavevmode\nobreak\
\times\left(k_{[\alpha}n_{\beta]}-m_{[\alpha}\bar{m}_{\beta]}\right)\big{]}e^{iu/\tilde{\epsilon}}\big{\\}}+\mathcal{O}(\tilde{\epsilon}^{3}).$
(5.13)

At leading (geometric optics) order, $F_{\alpha\beta}$ is a linear combination
of $k_{[\alpha}m_{\beta]}$ and $k_{[\alpha}\bar{m}_{\beta]}$. At subleading
order, its tensorial structure changes; it acquires terms proportional to
$k_{[\alpha}n_{\beta]}$ and $m_{[\alpha}\bar{m}_{\beta]}$. These corrections
may be interpreted as modifying the apparent “polarization state” at higher
orders. More broadly, the tensorial structure of the electromagnetic field may
alternatively be understood by noting that the Newman-Penrose scalars
associated with the tetrad
$(k_{\alpha},n_{\alpha},m_{\alpha},\bar{m}_{\alpha})$ satisfy
$\Phi_{i}=\mathcal{O}(\tilde{\epsilon}^{2-i})$ for all $i=0,1,2$. At leading
order, there is only $\Phi_{2}$. At subleading order, there is also
$\Phi_{1}$. At two orders beyond geometric optics, there is also $\Phi_{0}$.
This is a special case of the peeling result for high-frequency fields which
was obtained in [39].

###  V.2 Stress-energy tensors

Our goal is to compute linear and angular momenta, which are determined by
integrals of stress-energy tensors. The electromagnetic stress-energy tensor
is

$T_{\alpha\beta}=\frac{1}{4\pi}\left(F_{\alpha\gamma}F_{\beta}{}^{\gamma}-\tfrac{1}{4}g_{\alpha\beta}F_{\gamma\lambda}F^{\gamma\lambda}\right),$
(5.14)

and through leading and subleading orders, substitution of Eq. (5.13) into
this expression results in

$\displaystyle
T_{\alpha\beta}=\frac{1}{8\pi}\big{\\{}\\!\left[|\psi|^{2}+\tilde{\epsilon}v^{2}\mathrm{Im}\left(\bar{\psi}\Box(\psi/v)\right)\right]k_{\alpha}k_{\beta}$
$\displaystyle\leavevmode\nobreak\
{}-4\tilde{\epsilon}\mathrm{Im}\left(\bar{\psi}\partial_{\bar{\zeta}}\psi
k_{(\alpha}\bar{m}_{\beta)}\right)\big{\\}}+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.15)

In regions where $|\psi|\neq 0$, this can be written more suggestively as

$\displaystyle
T_{\alpha\beta}=\frac{1}{8\pi}\left[|\psi|^{2}+\tilde{\epsilon}v^{2}\mathrm{Im}\left(\bar{\psi}\Box(\psi/v)\right)\right]$
$\displaystyle{}\leavevmode\nobreak\
\times\tilde{k}_{\alpha}\tilde{k}_{\beta}+\mathcal{O}(\tilde{\epsilon}^{2}),$
(5.16)

where the covector

$\tilde{k}_{\alpha}\equiv
k_{\alpha}-2\tilde{\epsilon}\mathrm{Im}\left[(\partial_{\bar{\zeta}}\ln\psi)\bar{m}_{\alpha}\right]$
(5.17)

which appears here can be interpreted as a modified wave vector. Like the
leading-order wave vector, this is null:
$\tilde{k}_{\alpha}\tilde{k}^{\alpha}=\mathcal{O}(\tilde{\epsilon}^{2})$. At
leading order, $T_{\alpha\beta}\propto
k_{\alpha}k_{\beta}+\mathcal{O}(\tilde{\epsilon})$, and it is somewhat
remarkable that at one order higher, this is modified only to
$T_{\alpha\beta}\propto\tilde{k}_{\alpha}\tilde{k}_{\beta}+\mathcal{O}(\tilde{\epsilon}^{2})$.
All observers therefore agree on the direction of momentum density, even at
the subleading order.

Similar factorizations of the stress-energy tensor141414Ref. [39] discusses
these factorizations for the averaged stress-energy tensor. However, averaging
has no effect in the circularly polarized case considered here. have been
discussed in more general contexts in Ref. [39]. A result like the one found
here was shown to arise whenever $k_{\alpha}$ is shear-free. If there is shear
in the leading-order rays, different observers generically disagree on the
direction of the subleading momentum density. Moreover, even in the shear-free
case, observers typically disagree on the direction of the momentum density
once terms of order $\tilde{\epsilon}^{2}$ are included; stress-energy tensors
at that order are generically more complicated.

###  V.3 Linear and angular momenta

The stress-energy tensor in Eq. (5.16) may be used to compute the net linear
momentum on a $t=\mbox{const.}$ hypersurface. Using Eq. (3.12), and putting
primes on integration variables and on the objects which depend on the
integration variables,

$\displaystyle p_{\alpha}=\frac{1}{8\pi}\int
d^{3}x^{\prime}\left[|\psi^{\prime}|^{2}+\tilde{\epsilon}v^{\prime
2}\mathrm{Im}\left(\bar{\psi}^{\prime}\Box^{\prime}(\psi^{\prime}/v^{\prime})\right)\right]$
$\displaystyle\leavevmode\nobreak\
{}\times\tilde{k}_{\alpha}+\mathcal{O}(\tilde{\epsilon}^{2}).$ (5.18)

It follows from Eq. (5.17) that
$p_{\alpha}p^{\alpha}=\mathcal{O}(\tilde{\epsilon}^{2})$; the momentum is null
through leading and subleading orders. However, it is not necessarily true
that $p_{\alpha}$ is proportional to $k_{\alpha}$ beyond leading order.

More interesting is the angular momentum $S^{\alpha\beta}$, which we compute
about a point $x^{\alpha}=(t,x^{i})$ which lies within the hypersurface of
integration. There are two interesting sets of components. First, using Eqs.
(3.12) and (5.16),

$\displaystyle S^{i0}=\frac{1}{8\pi}\int
d^{3}x^{\prime}\big{[}|\psi^{\prime}|^{2}+\tilde{\epsilon}v^{\prime
2}\mathrm{Im}\left(\bar{\psi}^{\prime}\Box^{\prime}(\psi^{\prime}/v^{\prime})\right)\big{]}$
$\displaystyle{}\leavevmode\nobreak\
\times(x^{\prime}-x)^{i}+\mathcal{O}(\tilde{\epsilon}^{2}).$ (5.19)

This can be interpreted as the dipole moment of the energy density with
respect to a static observer at $x^{\alpha}$. The other relevant components of
the angular momentum tensor are

$\displaystyle S^{ij}=\frac{\tilde{\epsilon}}{2\pi}\mathrm{Im}\int
d^{3}x^{\prime}\left(\bar{\psi}^{\prime}\partial_{\bar{\zeta}^{\prime}}\psi^{\prime}\right)\bar{m}^{[i}(x^{\prime}-x)^{j]}$
$\displaystyle\leavevmode\nobreak\
-2k^{[i}S^{j]0}+\mathcal{O}(\tilde{\epsilon}^{2}).$ (5.20)

###  V.4 Vanishing phase gradients

The linear and angular momenta simplify considerably when the complex phase of
$\psi$ is constant. Looking first at the linear momentum, if
$\nabla_{\alpha}\arg\psi=0$ and if $|\psi|$ decays to zero sufficiently
rapidly at large transverse distances, Eq. (5.18) reduces to

$p_{\alpha}=\frac{1}{8\pi}\left(\int
d^{3}x|\psi|^{2}\right)k_{\alpha}+\mathcal{O}(\tilde{\epsilon}^{2}).$ (5.21)

The subleading contribution to the linear momentum vanishes and $p_{\alpha}$
is seen to be proportional to the constant leading-order wave vector
$k_{\alpha}$. It is important to note, however, that the net momentum is in
general distinct from the momentum density. The former is proportional to
$k_{\alpha}$ while the latter is proportional to $\tilde{k}_{\alpha}$. An
observer with a high-resolution detector might therefore ascribe an apparent
“direction of propagation” which differs, at $\mathcal{O}(\tilde{\epsilon})$,
from the direction of $p^{\alpha}$. More than this, the direction of the
momentum density varies slightly across the wave packet.

When the phase gradient vanishes, Eq. (5.19) simplifies to

$\displaystyle S^{i0}=\frac{1}{8\pi}\int
d^{3}x^{\prime}(x^{\prime}-x)^{i}|\psi^{\prime}|^{2}+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.22)

Here too the subleading contribution vanishes. Furthermore, Eq. (5.20) reduces
to

$S^{ij}=2i\tilde{\epsilon}p^{0}\bar{m}^{[i}m^{j]}-2k^{[i}S^{j]0}+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.23)

One particularly simple choice for $x^{i}$ arises by enforcing the centroid
condition (3.4). Temporarily assume that $t^{\alpha}=(1,0,0,0)$ so that
condition reduces to $S^{i0}=0$. It then follows from Eq. (5.22) that this
centroid condition implies that

$x^{i}=\frac{1}{8\pi E}\int d^{3}x^{\prime}|\psi^{\prime}|^{2}x^{\prime
i}+\mathcal{O}(\tilde{\epsilon}^{2}),$ (5.24)

where $E$ is again given by Eq. (3.22). In terms of the bivector
$\Sigma^{\alpha\beta}$ which is defined by Eq. (2.16),

$S^{\alpha\beta}=\tilde{\epsilon}E\Sigma^{\alpha\beta}+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.25)

The angular momentum therefore satisfies
$S^{\alpha\beta}p_{\beta}=\mathcal{O}(\tilde{\epsilon}^{2})$. An angular
momentum which differs from this only by a sign can be obtained by considering
an otherwise-identical field with opposite helicity, which is accomplished by
replacing the $m_{\alpha}$ which appears in Eq. (5.9) with $\bar{m}_{\alpha}$.

Equation (5.25) matches the form (3.8) for $S^{\alpha\beta}$, assuming that
$s=1$ and

$\epsilon=\tilde{\epsilon}E.$ (5.26)

The two small parameters we have introduced are therefore proportional to one
another. Physically, either one can be interpreted as related to the leading-
order angular frequency which would be seen by an observer with 4-velocity
$\tilde{t}^{\alpha}$:

$\omega=-(k\cdot\tilde{t})/\tilde{\epsilon}=-(p\cdot\tilde{t})/\epsilon.$
(5.27)

To summarize, we have found sufficient conditions for the physical picture
suggested in Sec. III, namely that $p_{\alpha}$ is null, $s=\pm 1$, and
$S^{\alpha\beta}p_{\beta}=0$. This is valid, up to terms of order
$\tilde{\epsilon}^{2}$, at least for all decaying, circularly polarized
electromagnetic wave packets with planar wavefronts and vanishing phase
gradients. It is shown in Sec. V.5 below that this picture can change when
$\psi$ has a nontrivial phase gradient.

####  V.4.1 Violation of energy conditions

It was shown in Sec. IV above that if $p_{\alpha}$ is null and
$S^{\alpha\beta}$ has the form (3.8), the set of all possible centroids
determined by $S^{\alpha\beta}t_{\beta}=0$ is not bounded in spacelike
directions (when varying over all timelike $t^{\alpha}$). This suggests that
the worldlines we refer to as centroids are perhaps poorly named; it may be
that some of them are nowhere near the wave packet of interest. It is well
known that this type of situation can occur for the Newtonian center of mass
if the mass density switches sign151515As an example, suppose that two masses,
$m_{+}>0$ and $m_{-}=2\delta-m_{+}$, are placed on a line at coordinates
$\pm\ell_{w}$. Then the center of mass lies at $(m_{+}/\delta-1)\ell_{w}$,
which can be anywhere at all for appropriate choices of $\delta$..
Relativistically, avoiding this kind of pathology involves requiring that the
stress-energy tensor satisfy appropriate energy conditions. And in an exact
context, electromagnetic stress-energy tensors with the form (5.14) are known
to satisfy all standard energy conditions [85]. However, it is not necessarily
true that the approximate electromagnetic stress-energy tensors of interest
here also satisfy those energy conditions. We now show that they do not. It is
this violation of energy conditions which is behind the peculiarly distant
centroids associated with massless spinning objects.

Suppose that $\tilde{t}^{\alpha}$ has a form which maximizes the centroid
displacements in Sec. IV.3. Using our null tetrad (5.8), one possibility which
is compatible with (4.20) is

$\tilde{t}^{\alpha}=\frac{\frac{1}{2}k^{\alpha}(1+V^{2})+n^{\alpha}(1-V^{2})}{\sqrt{1-V^{2}}}+\frac{V}{\sqrt{2}}(m^{\alpha}+\bar{m}^{\alpha}),$
(5.28)

where $V\in[0,1)$ denotes the relative speed between $\tilde{t}^{\alpha}$ and
$(1,0,0,0)$. In the case of interest here, where there is no phase gradient,
(5.16) and (5.17) imply that if $\mathcal{O}(\tilde{\epsilon}^{2})$ terms are
ignored,

$\displaystyle T_{\alpha\beta}\tilde{t}^{\alpha}\tilde{t}^{\beta}$
$\displaystyle=\frac{|\psi|^{2}}{8\pi}(\tilde{k}\cdot\tilde{t})^{2},$
$\displaystyle=\frac{1-V^{2}}{8\pi}\left(|\psi|^{2}+\frac{\tilde{\epsilon}V\partial_{y}|\psi|^{2}}{\sqrt{1-V^{2}}}\right).$
(5.29)

For any nontrivial bounded wave packet, there will be some regions in which
$\partial_{y}|\psi|^{2}$ is negative and other regions in which it is
positive. Furthermore, the first term here can always be made negligible
compared to the second by choosing $V$ sufficiently close to 1. It follows
that if $\mathcal{O}(\tilde{\epsilon}^{2})$ terms are ignored, there are
timelike vectors $\tilde{t}^{\alpha}$ for which
$T_{\alpha\beta}\tilde{t}^{\alpha}\tilde{t}^{\beta}$ is negative in some parts
of the wave packet and positive in others; the energy density switches sign.
This amounts to a violation of the weak, strong, and dominant energy
conditions.

As noted above, this violation is an artifact of our approximation. All exact
electromagnetic stress-energy tensors satisfy the weak, strong, and dominant
energy conditions. That there is a problem with our approximation is not
difficult to see in this context, as we are finding a “subleading” term which
dominates over the “leading” term. It is not particularly surprising that in
such a scenario, terms of even higher order might not be negligible. It is
less clear, however, that simply assuming that a wave packet is both massless
and spinning is enough for it to be associated with spurious, arbitrarily
distant centroids. That is, however, a consequence of the fact the high-
frequency approximation breaks down for certain highly boosted observers.

####  V.4.2 Momentum is timelike, not null

It is shown in the Appendix that it is impossible for a truly massless wave
packet to have nonzero spin. It is however clear that an electromagnetic wave
packet can have nonzero spin. The conclusion is that spinning electromagnetic
wave packets cannot truly be null. We now show that our wave packets are
timelike once we include terms two orders beyond those in geometric optics.

To establish this, first note that the electromagnetic field (5.13) can be
used to compute the stress-energy tensor to one higher order than shown in Eq.
(5.16). That may in turn be used to compute $p_{\alpha}$. The full stress-
energy tensor is complicated, however. The calculation can be considerably
simplified by noting that all we need to determine the causal character of
$p_{\alpha}$ is the $\mathcal{O}(\tilde{\epsilon}^{2})$ contribution to
$k\cdot p$. That can in turn be computed by showing only that

$T_{\alpha\beta}k^{\beta}=-\frac{\tilde{\epsilon}^{2}}{16\pi}|\nabla\psi|^{2}k_{\alpha}+\mathcal{O}(\tilde{\epsilon}^{3}).$
(5.30)

This result and (5.21) imply that

$p\cdot p=-\frac{\tilde{\epsilon}^{2}E}{8\pi}\int
d^{3}x^{\prime}|\nabla^{\prime}\psi^{\prime}|^{2}+\mathcal{O}(\tilde{\epsilon}^{3}),$
(5.31)

where $E$ again denotes the energy seen by a stationary observer. Note that
because $\partial_{v}\psi=0$, this is always negative; every localized
electromagnetic wave packet of the given form has the small nonzero rest mass

$m=\tilde{\epsilon}\left(\frac{E}{8\pi}\int
d^{3}x^{\prime}|\nabla^{\prime}\psi^{\prime}|^{2}\right)^{1/2}+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.32)

If $\nabla\psi\sim\psi/\ell_{w}$ for some length scale $\ell_{w}$, this
suggest that $m\sim\tilde{\epsilon}E/\ell_{w}=\epsilon/\ell_{w}$. Results on
the localization of massive objects which were reviewed in Sec. IV.3 therefore
imply that all centroids associated with these wave packets are in fact
confined to a disk whose radius is of order $\mathcal{S}/m\sim\ell_{w}$. This
is the expected result. We emphasize, however, that it cannot be
established—even qualitatively—in the massless approximation.

###  V.5 Nonvanishing phase gradients and the limitations of the spin Hall
framework

If the phase gradient of $\psi$ does not vanish, the physical picture
associated with the spin Hall equations might not hold. First, it is not
necessarily true that $s=\pm 1$ for a circularly polarized electromagnetic
wave packet. Additional contributions to this parameter can arise. This is
referred to as orbital (as opposed to spin) angular momentum in the optics
literature [93, 94]. While the derivation of the spin Hall equations in [42]
did not allow for the possibility of orbital angular momentum, their
derivation as a special case of the MP equations makes it clear that orbital
angular momentum requires no essential changes; $s$ merely takes on different
integer values in the spin Hall equations161616An analogous discussion for
electromagnetic beams propagating in flat spacetime but in nontrivial
materials may be found in [95].. A more dramatic consequence of allowing
nontrivial phase gradients is that it becomes possible to construct wave
packets with spin vectors which are not longitudinal:
$S^{\alpha\beta}p_{\beta}\neq 0$ even at the leading nontrivial order. In
these cases, the form (3.8) for the angular momentum is incomplete and the
spin Hall equations are no longer valid. Even then, however, the MP equations
can still be applied.

We first consider the possibility of nonlongitudinal spin. If the centroid
condition is imposed with $t^{\alpha}=(1,0,0,0)$, it follows from (5.20) that

$\displaystyle S^{\alpha\beta}p_{\beta}$
$\displaystyle=ES^{\alpha\beta}k_{\beta}+\mathcal{O}(\tilde{\epsilon}^{2})$
$\displaystyle=\frac{\tilde{\epsilon}E}{4\pi}\mathrm{Im}\left[\bar{m}^{\alpha}\int
d^{3}x^{\prime}(z^{\prime}-z)\bar{\psi}^{\prime}\partial_{\bar{\zeta}^{\prime}}\psi^{\prime}\right]+\mathcal{O}(\tilde{\epsilon}^{2}).$
(5.33)

A nontrivial transverse angular momentum therefore requires that
$\bar{\psi}\partial_{\bar{\zeta}}\psi$ have a nontrivial moment along the
optical axis. That this is possible can be illustrated by examples. Suppose
that

$\psi=|\psi|e^{ixu/\ell_{\perp}^{2}},$ (5.34)

where $\ell_{\perp}>0$ is a parameter. Also assume that

$\int du^{\prime}u^{\prime}|\psi^{\prime}|^{2}=0$ (5.35)

so the $z$ component of the centroid lies at
$z=t+\mathcal{O}(\tilde{\epsilon})$. Substitution into (5.33) then shows that

$S^{\alpha\beta}p_{\beta}=-\frac{\tilde{\epsilon}E(m+\bar{m})^{\alpha}}{8\sqrt{2}\pi\ell_{\perp}^{2}}\int
d^{3}x^{\prime}|u^{\prime}\psi^{\prime}|^{2}+\mathcal{O}(\tilde{\epsilon}^{2}),$
(5.36)

This is nonzero for any nontrivial $\psi$. It follows that for this class of
wave packets, $S^{\alpha\beta}$ is not in the spin Hall form (3.8).
Equivalently, $S^{\alpha}$ cannot be parallel to $p^{\alpha}$; it must have a
nonzero $y$ component. It is not possible to use the spin Hall equations to
understand the motion of such a wave packet. However, there is no obstacle to
using the MP equations in their more general form (3.1). The conclusion here
is that $S^{\alpha\beta}p_{\beta}=0$ is a physical restriction; it is not
inevitable. The derivation of the spin Hall equations appears to have
implicitly assumed that the wave packets have, e.g., vanishing phase
gradients.

Now consider a wave packet with no transverse angular momentum, but with
potentially large amounts of longitudinal angular momentum. This can be
produced by introducing polar coordinates $(r,\theta)$ in the $xy$ plane and
then supposing that

$\psi=|\psi|e^{in\theta},$ (5.37)

where $n$ is an integer. Additionally, suppose for simplicity that $|\psi|$
depends only on $r$ and $u$ and that it satisfies (5.35). It then follows from
(5.20) that when evaluated at the centroid,

$S^{\alpha\beta}=(n+1)\tilde{\epsilon}E\Sigma^{\alpha\beta}+\mathcal{O}(\tilde{\epsilon}^{2}),$
(5.38)

where $\Sigma^{\alpha\beta}$ is again given by (2.16). If $\epsilon$ is again
related to $\tilde{\epsilon}$ via (5.26) comparison with (3.8) shows that for
these wave packets,

$s=n+1.$ (5.39)

If a wave packet with the opposite helicity had been considered, we would have
found instead that $s=n-1$. Regardless, it is clear that $s$ is not
necessarily equal to $\pm 1$. Much larger amounts of angular momentum are
possible than had been supposed in, e.g., [42]. Formally, $n$ can be
arbitrarily large here. However, the high-frequency analysis breaks down when
$\psi$ varies on the same scale as $e^{iu/\tilde{\epsilon}}$. If the spatial
extent of the wave packet is of order $\ell_{w}$, this implies that our
equations can be trusted only when
$n\ll\ell_{w}/\tilde{\epsilon}=\omega\ell_{w}$.

##  VI Spin Hall effect of light in an inhomogeneous medium

As a final application, we now show how the ray equations describing the spin
Hall effect of light in an inhomogeneous medium [13, 14, 15, 16, 17, 18, 96,
21, 97] can be recovered from the gravitational spin Hall equations (2.17).
The main tool used here is the well-known analogy between electromagnetic
waves propagating inside a dielectric medium and electromagnetic waves
propagating through vacuum but in an effective metric [98, 99, 100, 101, 102,
103, 104]. More precisely, consider a background metric
$\tilde{g}_{\alpha\beta}$ and a dielectric medium with a varying refractive
index $n$ and a 4-velocity $u^{\alpha}$. It has then been shown in Ref. [99,
75] (see also Ref. [104]) that the combined effect of the background spacetime
and the dielectric medium on light rays can be studied by considering vacuum
propagation in the optical metric

$g_{\alpha\beta}=\tilde{g}_{\alpha\beta}+(1-n^{-2})u_{\alpha}u_{\beta},$ (6.1)

where the indices on the 4-velocities here have been lowered using
$\tilde{g}_{\alpha\beta}$.

To describe the spin Hall effect of light in an inhomogeneous medium, we take
the background metric to be the Minkowski one in inertial coordinates
$(t,x,y,z)$ so $\tilde{g}_{\alpha\beta}=\eta_{\alpha\beta}$. We also suppose
that the medium is stationary in these coordinates so $\partial_{t}n=0$ and
$u^{\alpha}=(1,0,0,0)$. The spin Hall equations (2.17) additionally require
the choice of a timelike vector field $t^{\alpha}$ to fix the centroid
definition, and this may be identified here with $u^{\alpha}$. A calculation
then shows that with the effective metric $g_{\alpha\beta}$,
$\Sigma^{\alpha\beta}R_{\alpha\beta\gamma}{}^{\lambda}=0$. The spin Hall
equations in this metric therefore reduce to

$\displaystyle\dot{x}^{\alpha}$
$\displaystyle=g^{\alpha\beta}\left(p_{\beta}+\frac{1}{p\cdot
t}S_{\beta\rho}g^{\gamma\lambda}p_{\gamma}\nabla_{\lambda}t^{\rho}\right),$
(6.2a) $\displaystyle\dot{p}_{\alpha}$
$\displaystyle=\Gamma^{\beta}_{\alpha\gamma}g^{\gamma\lambda}p_{\beta}p_{\lambda}.$
(6.2b)

As $u^{\alpha}$ is Killing and the effective metric is static, the spin-
dependent terms in the conservation law (3.17) vanish so
$E=-p_{\alpha}u^{\alpha}=\mbox{const}$. Introducing a $3$-vector notation, the
momentum must be null with respect to $g_{\alpha\beta}$, meaning that

$p_{i}=\bm{p},\qquad p_{0}=-\frac{\sqrt{\bm{p}\cdot\bm{p}}}{n}=-\frac{p}{n}.$
(6.3)

Energy conservation in the effective metric therefore implies that $E=p/n$ is
constant. It is only the direction of $\bm{p}$ which must be determined from
the equations of motion.

A calculation shows that the nontrivial components of (6.2) reduce to

$\displaystyle\frac{dt}{d\tau}$ $\displaystyle=np=n^{2}E,$ (6.4a)
$\displaystyle\frac{d\bm{x}}{d\tau}$ $\displaystyle=\bm{p}+\frac{\epsilon
s}{p^{3}}\left(\frac{d\bm{p}}{d\tau}\times\bm{p}\right),$ (6.4b)
$\displaystyle\frac{d\bm{p}}{d\tau}$
$\displaystyle=\frac{p^{2}}{n}\boldsymbol{\nabla}n=\frac{1}{2}\bm{\nabla}(nE)^{2},$
(6.4c)

To obtain the same form of the ray equations as in the optics literature, we
can reparametrize everything in terms of $t$ instead of $\tau$. Doing so,

$\displaystyle\frac{d\bm{x}}{dt}$
$\displaystyle=\frac{\bm{p}}{np}+\frac{\epsilon
s}{p^{3}}\left(\frac{d\bm{p}}{dt}\times\bm{p}\right),$ (6.5a)
$\displaystyle\frac{d\bm{p}}{dt}$
$\displaystyle=\frac{p}{n^{2}}\boldsymbol{\nabla}n,$ (6.5b)

These are the ray equations describing the spin Hall effect of light in an
inhomogeneous medium, as obtained in Refs. [15, 105, 97]. They can be
rewritten in the form presented in Refs. [16, 17, 21, 22] by rescaling the
momentum and time, as mentioned in Ref. [97] (see also Ref. [18]).

Deriving the spin Hall effect of light in an inhomogeneous medium from (2.17)
is important for several reasons. First, it establishes that the gravitational
spin Hall equations really are related to the spin Hall effects described in
flat-spacetime optics; the gravitational spin Hall equations thus have been
given an appropriate name. Second, the ray equations usually used in the
optical literature implicitly fix $t^{\alpha}$ at the outset and do not allow
it to vary. Beginning instead with the gravitational spin Hall equations,
where $t^{\alpha}$ is arbitrary, instead allows a unified description of the
spin Hall effect of light, determined by the gradient of $n$, and the
relativistic Hall effect [26, 27], determined by changes in $t^{\alpha}$.
Lastly, the spin Hall effect of light, as described by Eqs. (6.5), has been
confirmed experimentally in Refs. [19, 21]. The present connection between
Eqs. (2.17) and Eqs. (6.5) gives some level of confidence in the theoretical
predictions of Eqs. (2.17) and in the existence of a genuinely gravitational
spin Hall effect of light.

As another application of the type of analysis presented in this section, one
might consider the propagation of light in a plasma which is in a curved
spacetime, perhaps near a black hole. In some regimes, the plasma can be
expected to have an effective refractive index [106, 107, 108, 109, 110]. The
spin Hall equations (2.17) together with the optical metric (6.1) could then
be used to derive polarization-dependent corrections to the propagation of
electromagnetic pulses in the presence of an astrophysical plasma.

##  VII Conclusions

This paper has investigated the implications, properties, and limitations of
the gravitational spin Hall equations derived in Refs. [42, 45]. In the
electromagnetic case of interest here, these equations describe the motion of
high-frequency circularly polarized electromagnetic pulses which propagate in
vacuum but in arbitrary background spacetimes. In this context, the spin Hall
effect refers to the transverse deflection of a pulse due to its spin.

Our first class of results concern the meanings of the position and the
momentum which appear in the gravitational spin Hall equations. In general, a
spinning wave packet must be extended and the adoption of any equation of
motion must be associated with a particular choice of centroid. We have found
that the position appearing in the gravitational spin Hall equations is a
centroid whose definition is parametrized by the timelike vector field
$t^{\alpha}$ which appears in those equations. That position may be
interpreted as a spin supplementary condition, chosen to ensure that the
angular momentum satisfies $S^{\alpha\beta}t_{\beta}=0$. More physically, the
centroid is the center of energy of the wave packet in a frame which is
instantaneously at rest with respect to $t^{\alpha}$.

Different choices for $t^{\alpha}$ are in general associated with different
centroids, and we have computed the shifts between those centroids. For
massive objects with timelike momentum, we recover the known result that these