# Rotational Doppler Effect and Spin-Rotation Coupling

Bahram Mashhoon1,2<EMAIL_ADDRESS>1Department of Physics and
Astronomy, University of Missouri, Columbia, Missouri 65211, USA
2School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5531, Iran

###### Abstract

The Rotational Doppler Effect (RDE) involves both the orbital angular momentum
of electromagnetic radiation as well as its helicity. The RDE phenomena
associated with photon helicity go beyond the standard theory of relativity.
The purpose of this paper is to elucidate the theoretical basis of the
helicity-dependent RDE in terms of the general phenomenon of spin-rotation
coupling. The physical implications of this coupling are briefly pointed out
and the nonlocal theory of rotating observers is described. For an observer
rotating uniformly about the direction of incidence of a plane wave of
definite helicity, the nonlocally measured amplitude of the wave is larger
(smaller) if the observer rotates in the same (opposite) sense as the
electromagnetic field. The implications of the nonlocal helicity dependence of
the measured amplitude of the radiation are briefly discussed.

Special relativity, Nonlocality of accelerated observers, Inertia of intrinsic
spin

###### pacs:

03.30.+p, 11.10.Lm

##  I Introduction

The rotational Doppler effect (RDE) is a variant of the standard Doppler
effect when the receiver rotates and is thus noninertial. This aspect of the
Doppler effect has now gained prominence in experimental physics; see, for
instance, LCT ; LZZ ; ZFD ; FPW ; DLLL ; CWL ; E+E  and the references cited
therein. The theoretical basis of RDE has been widely discussed in the context
of the quantum theory of the interaction of the electromagnetic radiation with
matter; see, for instance, B+B ; BSV ; LBS ; SLP ; ABM  and references
therein. The main purpose of the present work is to discuss RDE in connection
with the relativistic physics of rotating observers. Accelerated systems play
a fundamental role in the interface connecting the physics of the special and
general theories of relativity. It is therefore necessary to discuss carefully
the extension of Lorentz invariance to accelerated observers. In the rest of
this introductory section, the derivation of the relativistic Doppler effect
based on Lorentz invariance is presented. The standard extension of Lorentz
invariance to accelerated systems in Minkowski spacetime is discussed in
Section II. The limitations of the standard theory are pointed out and
attempts to mitigate them are discussed in Sections III-V. Section VI contains
a brief discussion of our results.

In a global inertial frame in Minkowski spacetime with Cartesian coordinates
$x^{\mu}=(ct,\mathbf{x})$, we consider the propagation of an electromagnetic
wave packet. Inertial observers that are spatially at rest in this background
inertial frame can determine via Fourier analysis that the packet consists of
a spectrum of plane electromagnetic waves each with null propagation vector
$k^{\mu}=(\omega/c,\mathbf{k})$, where $\omega=c|\mathbf{k}|$. We are
interested in the reception of the radiation by the inertial observers that
are all at rest in a global inertial frame moving with unit 4-velocity vector
$u^{\mu}=\gamma(1,\mathbf{v}/c)$. Here, $u^{\mu}=dx^{\mu}/d\tau$, where
$d\tau$ is given by

$d\tau=c\,\left(1-\frac{v^{2}}{c^{2}}\right)^{1/2}dt\,,$ (1)

$v=|\mathbf{v}|$ and $\gamma=(1-v^{2}/c^{2})^{-1/2}$ is the Lorentz factor.
The Lorentz transformation from the background frame to the moving frame with
Cartesian coordinates $x^{\prime\mu}=(ct^{\prime},\mathbf{x}^{\prime})$ that
is a pure boost with no rotation is given by

$t=\gamma\left(t^{\prime}+\frac{1}{c^{2}}\mathbf{v}\cdot\mathbf{x}^{\prime}\right)\,,$
(2)
$\mathbf{x}=\mathbf{x}^{\prime}+\frac{\gamma-1}{v^{2}}(\mathbf{v}\cdot\mathbf{x}^{\prime})\mathbf{v}-\frac{1}{c^{2}}\gamma\mathbf{v}t^{\prime}\,.$
(3)

Here, we have assumed that at $t=t^{\prime}=0$, the spatial origins of the two
systems coincide, namely, $\mathbf{x}=\mathbf{x}^{\prime}=0$. We define the
invariant proper time $\tau$ of an inertial observer to be the time it
measures in its own rest frame multiplied by the speed of light $c$. For
instance, it follows from Eq. (2) that $cdt^{\prime}=d\tau=cdt/\gamma$ for the
moving observer fixed at $\mathbf{x}^{\prime}$, in agreement with Eq. (1).

The inertial observers at rest in the moving frame measure a similar wave
spectrum with components
$k^{\prime\mu}=(\omega^{\prime}/c,\mathbf{k}^{\prime})$. The Lorentz
transformation connecting the measurements of the two classes of inertial
observers, each class at rest in a global inertial frame, leaves the phase
$\phi$,

$\phi(t,\mathbf{x})=-\omega t+\mathbf{k}\cdot\mathbf{x}\,,$ (4)

of each plane-wave component of the packet invariant; that is,
$\phi(t,\mathbf{x})=\phi^{\prime}(t^{\prime},\mathbf{x}^{\prime})$ or

$-\omega
t+\mathbf{k}\cdot\mathbf{x}=-\omega^{\prime}t^{\prime}+\mathbf{k}^{\prime}\cdot\mathbf{x}^{\prime}\,.$
(5)

From this phase invariance and Eqs. (2) and (3), one can simply derive the
standard relativistic formulas for the Doppler effect and aberration of
electromagnetic radiation, namely,

$\omega^{\prime}=\gamma(\omega-\mathbf{v}\cdot\mathbf{k})\,,$ (6)
$\mathbf{k}^{\prime}=\mathbf{k}+\frac{\gamma-1}{v^{2}}(\mathbf{v}\cdot\mathbf{k})\mathbf{v}-\frac{1}{c^{2}}\gamma\omega\mathbf{v}\,.$
(7)

In particular, we note that the Doppler effect has to do with the change in
the frequency of the wave due to the relative motion of the source and the
receiver, while aberration involves the change in the measured direction of
propagation of the wave.

Let us note that we can write the Doppler formula as

$\omega^{\prime}=-k_{\mu}u^{\mu}\,,$ (8)

which illustrates the fact that what is measured is an invariant under Lorentz
transformations.

We now turn to the standard theory of noninertial systems.

##  II Measurements of Accelerated Observers

To simplify matters, we generally assume in this paper that the gravitational
interaction is turned off; therefore, the spacetime under consideration is
Minkowskian. In general, an observer in Minkowski spacetime has access to an
orthonormal tetrad $e^{\mu}{}_{\hat{\alpha}}$, such that
$e^{\mu}{}_{\hat{0}}=dx^{\mu}/d\tau$ is the 4-velocity of the observer and
$\tau$ is its proper time; moreover, $e^{\mu}{}_{\hat{i}}$, $i=1,2,3$,
constitute the observer’s local spatial frame. Hatted indices enumerate the
tetrad axes in the local tangent space. The orthonormality condition can be
expressed as

$\eta_{\mu\nu}\,e^{\mu}{}_{\hat{\alpha}}\,e^{\mu}{}_{\hat{\beta}}=\eta_{\hat{\alpha}\hat{\beta}}\,,$
(9)

where $\eta_{\mu\nu}$ is the Minkowski metric tensor and is given by
diag$(-1,1,1,1)$ in our convention.

For the hypothetical ideal inertial observers, we have

$\frac{de^{\mu}{}_{\hat{\alpha}}}{d\tau}=0\,.$ (10)

The temporal variation of a component of an observer’s tetrad frame would
indicate that the observer is noninertial and is therefore subject to
translational or rotational acceleration. Actual observers in Minkowski
spacetime are all accelerated. At each instant of proper time $\tau$, the
tetrad frame of an accelerated observer $e^{\mu}{}_{\hat{\alpha}}(\tau)$ also
belongs to an otherwise identical momentarily comoving _inertial_ observer and
it is natural to assume that their measurements of physical quantities give
the same results at that instant. In this way, Lorentz invariance has been
extended to accelerated systems in the standard theory of special relativity
Ein . We discuss this approach in some detail in the rest of this section.

###  II.1 Locality

The standard theory of relativity is based on two basic assumptions: Lorentz
invariance and the hypothesis of locality. Lorentz invariance is a basic
symmetry of nature and connects the measurements of hypothetical ideal
inertial observers in Minkowski spacetime. Inertial observers remain forever
at rest in global inertial frames of reference and have been of fundamental
significance for the formulation of the basic laws of physics since their
introduction by Isaac Newton. However, all real observers are more or less
accelerated. Therefore, we need to discuss the measurements of accelerated
observers in Minkowski spacetime. Let us recall that an accelerated Newtonian
point particle is always momentarily inertial; that is, the classical particle
follows the law of inertia if its acceleration is instantaneously turned off.
Hence, an accelerated path is the envelope of the osculating straight lines
tangent to the trajectory. The Newtonian theory of accelerated observers can
be straightforwardly extended to the relativistic regime by replacing Galilean
invariance with Lorentz invariance. The accelerated observer is thus
momentarily equivalent to an otherwise identical instantaneously comoving
inertial observer. Therefore, Lorentz transformation can be applied point by
point along the world line of the accelerated observer in order to determine
what the accelerated observer measures. In other words, the accelerated
observer has a velocity $\mathbf{v}(t)$ at each instant of time $t$ and the
instantaneous Lorentz transformation associated with $\mathbf{v}(t)$ can be
employed to connect the accelerated observer to inertial observers; in this
way, relativity can be extended to accelerated systems Ein .

In this “small-scale” approach ROB , the acceleration of the observer is
ignored locally, but is taken into consideration via the variation in the
velocity of motion embodied in the pointwise Lorentz transformations. This
hypothesis of locality determines what accelerated observers measure according
to the standard special theory of relativity Ein . Measuring devices that work
in accordance with this basic assumption will be called “standard”. Measuring
devices carried by accelerated observers are subject to various inertial
forces and effects that could possibly integrate over proper time to render
such devices non-standard. Nevertheless, we limit our considerations
throughout to measuring devices that are robust enough to function as standard
instruments to a sufficient degree of accuracy. In connection with the notion
of standard rods and clocks, it is interesting to note that Einstein made a
similar assumption in the footnote on page 60 of his book Einstein .

The locality postulate implies that the accelerated observer along its world
line passes continuously through an infinite sequence of inertial observers
and their associated inertial frames. The Lorentz transformation connecting
one such inertial _rest_ frame with Cartesian coordinates
$x^{\prime\mu}=(ct^{\prime},x^{\prime},y^{\prime},z^{\prime})$ to the
background inertial frame with Cartesian coordinates $x^{\mu}=(ct,x,y,z)$
leaves the Minkowski spacetime interval,

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\,,$ (11)

invariant. Therefore,

$c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=c^{2}dt^{\prime 2}\,,$ (12)

where $d\mathbf{x}^{\prime}=0$ and $dt^{\prime}$ is the infinitesimal time
interval experienced by the accelerated observer in the comoving inertial
frame at time $t^{\prime}$ in accordance with the locality postulate. From
$d\tau=c\,dt^{\prime}$, we find the proper time measured by the _standard
clock_ carried by the accelerated observer, namely,

$\tau=c\int_{0}^{t}\left[1-\frac{v^{2}(\xi)}{c^{2}}\right]^{1/2}d\xi\,,$ (13)

where $\mathbf{v}(t)$ is the velocity of the observer with respect to the
background global inertial frame and we have assumed that $\tau=0$ when $t=0$.
That is, the accelerated observer’s proper time $\tau$ is the accumulation of
local infinitesimal comoving time intervals. If the accelerated observer
travels from an initial event $x_{I}^{\mu}=(ct_{I},\mathbf{x})$ in the
background inertial frame and finally returns at a later time to the same
place, i.e. $x_{F}^{\mu}=(ct_{F},\mathbf{x})$, we find

$\frac{1}{c}(\tau_{F}-\tau_{I})=\int_{t_{I}}^{t_{F}}\left[1-\frac{v^{2}(t)}{c^{2}}\right]^{1/2}dt<t_{F}-t_{I}\,,$
(14)

where $t_{F}-t_{I}$ is the elapsed temporal interval according to an inertial
observer at rest in the background global inertial frame in Minkowski
spacetime. In this way, Einstein explained away the twin paradox Ein .

Next, imagine the reception of electromagnetic radiation by the accelerated
observer. The instantaneous Lorentz transformation connecting the background
global frame with the local inertial rest frame of the accelerated observer
affects the field amplitudes, but the phase of the wave remains invariant,
i.e. $d\phi=\eta_{\mu\nu}k^{\mu}dx^{\nu}$, where
$k^{\mu}=(\omega/c,\mathbf{k})$ is the propagation 4-vector of the wave in the
background frame. That is, $d\phi=\eta_{\mu\nu}k^{\prime\mu}dx^{\prime\nu}$,
where

$\omega=-\left(\frac{\partial\phi}{\partial
t}\right)_{\mathbf{x}}\,,\qquad\mathbf{k}=\left(\frac{\partial\phi}{\partial\mathbf{x}}\right)_{t}\,$
(15)

and

$\omega^{\prime}=-\left(\frac{\partial\phi}{\partial
t^{\prime}}\right)_{\mathbf{x}^{\prime}}\,,\qquad\mathbf{k}^{\prime}=\left(\frac{\partial\phi}{\partial\mathbf{x}^{\prime}}\right)_{t^{\prime}}\,.$
(16)

The relativistic Doppler effect and aberration for accelerated observers
follow from phase invariance. In particular, the corresponding Doppler
frequency is given by

$\omega^{\prime}_{\rm D}(\tau)=\gamma[\omega-\mathbf{v}(t)\cdot\mathbf{k}]\,,$
(17)

where $\mathbf{v}(t)$ is the instantaneous velocity of the accelerated
observer. The locally measured frequency (17) can also be expressed in the
form of an invariant as in Eq. (8); that is, $\omega^{\prime}_{\rm D}(\tau)$
is proportional to the projection of the incident propagation 4-vector of the
wave upon the 4-velocity vector of the accelerated observer.

The basic limitations of the locality postulate have been studied in detail
BM1 ; BM2 ; Mashhoon:2003st ; indeed, this postulate is physically reasonable
if the motion is sufficiently slow such that the velocity of the accelerated
system is essentially constant during an elementary act of measurement. In
connection with the approximation involved in the use of standard rods and
clocks in the measurement process, there is a brief discussion in Weyl’s book
Weyl . After pointing out that under acceleration the classic rigid bodies no
longer exist in relativity theory due to the finite speed of interactions,
Weyl discussed accelerated observers on pages 176 and 177 of his book Weyl
and concluded that the approximation basically involved slow steady change in
the observer’s velocity.

A curved path is the envelope of the straight lines tangent to the path;
similarly, a pointlike accelerated observer passes through an infinite
sequence of otherwise identical momentarily comoving inertial observers. Is
this geometric fact also true for a physical observer that carries along
measuring instruments? Standard ideal rods and clocks are assumed to work in
accordance with the hypothesis of locality. More generally, we assume
throughout that accelerated observers have access to _ideal standard measuring
devices_. This approach is consistent with the notion that measuring devices
generally follow the laws of classical physics Mashhoon:2008vr .

The temporal dependence of the measured frequency $\omega^{\prime}_{\rm
D}(\tau)$ in Eq. (17) indicates that the domain of applicability of the
Doppler formula for accelerated observers is rather limited Mashhoon:2008vr ;
BMB . Rotating observers will be considered in the rest of this paper.

###  II.2 Doppler Effect for Rotating Observers

Let us consider an observer that rotates about the $z$ axis in a global
inertial frame in Minkowski spacetime. The observer is located at position
$\mathbf{r}$ and rotates with angular velocity
$\bm{\Omega}=\Omega\,\hat{\mathbf{z}}$ and velocity
$\mathbf{v}=\bm{\Omega}\times\mathbf{r}$, where $\hat{\mathbf{z}}$ is a unit
vector along the $z$ axis.

The application of the relativistic Doppler formula (17) to the motion of the
rotating observer results in

$\omega_{\rm
D}^{\prime}=\gamma[\omega-(\bm{\Omega}\times\mathbf{r})\cdot\mathbf{k}]=\gamma(\omega-\bm{\Omega}\cdot\bm{\ell})\,,$
(18)

where $\bm{\ell}=\mathbf{r}\times\mathbf{k}$. Multiplying Eq. (18) by $\hbar$,
we get

$E^{\prime}=\gamma(E-\bm{\Omega}\cdot\mathbf{L})\,,$ (19)

where

$E=\hbar\,\omega\,,\qquad\mathbf{L}=\hbar\,\bm{\ell}\,.$ (20)

Here, $E$ is the energy and $\mathbf{L}$ is the orbital angular momentum of
the light ray with wave vector $\mathbf{k}$. Equation (19) is the relativistic
generalization of the well-known classical mechanical result for the
Hamiltonian in a rotating frame of reference L+L ; Mashhoon:2021qtc . It
follows that the orbital angular momentum aspects of the RDE are in accord
with the special theory of relativity.

On the other hand, the generator of rotations is the total angular momentum
$\mathbf{J}=\mathbf{L}+\mathbf{S}$, where
$\mathbf{S}=\pm\hbar\,\hat{\mathbf{k}}$ represents the photon spin. Therefore,
we expect

$E^{\prime}=\gamma(E-\bm{\Omega}\cdot\mathbf{L}-\bm{\Omega}\cdot\mathbf{S})\,,$
(21)

which is the more general result giving rise to the helicity-rotation coupling
that is part of the observational basis of the rotational Doppler effect
(RDE), but which goes beyond the standard relativistic Doppler effect. The
next section is devoted to a detailed discussion of the photon helicity-
rotation coupling.

##  III Beyond Locality: Helicity-Rotation Coupling

The main purpose of this paper is to point out that the helicity-rotation
aspect of RDE goes beyond the locality postulate and phase invariance. The
standard special theory of relativity is based on the hypothesis of locality;
therefore, it is necessary to provide a proper theoretical basis for the
observed helicity-rotation coupling. We show that the validity of the locality
hypothesis is limited to the WKB approximation. Locality is clearly valid for
pointlike coincidences of classical point particles and rays of radiation.
According to the Huygens principle, wave phenomena are intrinsically nonlocal;
indeed, this nonlocal wave aspect of RDE is illustrated in this section.

###  III.1 Frequency Measurements of a Noninertial Observer

Imagine a global inertial frame with Cartesian coordinates
$x^{\mu}=(ct,x,y,z)$ in Minkowski spacetime and an observer that is at rest at
the origin of spatial coordinates. The observer is noninertial, since it
refers its measurements to axes that rotate with constant angular speed
$\Omega>0$ about the $z$ axis. The rotating axes therefore have unit
directions

$\hat{\mathbf{x}}^{\prime}=\hat{\mathbf{x}}\,\cos\Omega
t+\hat{\mathbf{y}}\,\sin\Omega
t\,,\quad\hat{\mathbf{y}}^{\prime}=-\hat{\mathbf{x}}\,\sin\Omega
t+\hat{\mathbf{y}}\,\cos\Omega
t\,,\quad\hat{\mathbf{z}}^{\prime}=\hat{\mathbf{z}}\,$ (22)

and the observer employs coordinates
$x^{\prime\mu}=(\tau,x^{\prime},y^{\prime},z^{\prime})$, where $\tau=ct$ is
the proper time of the noninertial observer and

$x^{\prime}=x\,\cos\Omega t+y\,\sin\Omega t\,,\quad y^{\prime}=-x\,\sin\Omega
t+y\,\cos\Omega t\,,\quad z^{\prime}=z\,.$ (23)

Consider now a plane electromagnetic wave of frequency $\omega$ and wave
vector $\mathbf{k}=(\omega/c)\hat{\mathbf{z}}$ incident along the $z$
direction. The electromagnetic 4-potential $A_{\mu}$ of the wave is given by

$A_{0}=0\,,\quad\mathbf{A}=a_{\pm}(\hat{\mathbf{x}}\pm
i\hat{\mathbf{y}})e^{-i\,\omega(t-z/c)}\,,$ (24)

where $a_{\pm}$ is the complex constant wave amplitude and the upper (lower)
sign indicates incident positive (negative) helicity radiation. Only linear
fields are considered throughout; therefore, we use complex fields and adhere
to the convention that their real parts are physically significant. Throughout
this paper, we employ for the sake of simplicity the vector potential
$A_{\mu}$ instead of the electromagnetic field tensor $F_{\mu\nu}$, since it
turns out that they give the same results regarding the frequency content of
the field. Moreover, we work in the radiation gauge, where
$A^{\mu}=(0,\mathbf{A})$ and $\nabla\cdot\mathbf{A}=0$.

Equation (24) represents the vector potential according to the fundamental
ideal inertial observers at rest in the background global inertial frame; that
is, what is measured is the projection of the vector potential upon the tetrad
frame of the inertial observer, $e^{\mu}{}_{\hat{\alpha}}$,

$A_{\hat{\alpha}}=A_{\mu}\,e^{\mu}{}_{\hat{\alpha}}\,,\qquad
e^{\mu}{}_{\hat{\alpha}}=\delta^{\mu}_{\alpha}\,.$ (25)

However, for the measurements of the noninertial observer,

$A^{\prime}_{\hat{\alpha}}=A_{\mu}\,e^{\prime\mu}{}_{\hat{\alpha}}\,;$ (26)

that is, the vector potential $A_{\mu}$ is projected upon the orthonormal
tetrad frame $e^{\prime\mu}{}_{\hat{\alpha}}$ adapted to the noninertial
observer. The corresponding tetrad axes are given explicitly in
$x^{\mu}=(ct,x,y,z)$ coordinates by

$e^{\prime\mu}{}_{\hat{0}}=(1,0,0,0)\,,$ (27)
$e^{\prime\mu}{}_{\hat{1}}=(0,\cos\Omega t,\sin\Omega t,0)\,,$ (28)
$e^{\prime\mu}{}_{\hat{2}}=(0,-\sin\Omega t,\cos\Omega t,0)\,,$ (29)
$e^{\prime\mu}{}_{\hat{3}}=(0,0,0,1)\,.$ (30)

The adapted tetrad frame is thus the spacetime generalization of the uniformly
rotating unit triad (22) employed by the noninertial observer. Using Eqs.
(26)–(30), we find $A^{\prime}_{\hat{0}}=0$, $A^{\prime}_{\hat{3}}=A_{3}$ and

$A^{\prime}_{\hat{1}}=A_{1}\cos\Omega t+A_{2}\sin\Omega t\,,$ (31)
$A^{\prime}_{\hat{2}}=-A_{1}\sin\Omega t+A_{2}\cos\Omega t\,.$ (32)

Next, from Eq. (24) we find

$A^{\prime}_{\hat{1}}=a_{\pm}\,e^{-i(\omega\mp\Omega)t}\,,$ (33)
$A^{\prime}_{\hat{2}}=\pm i\,a_{\pm}\,e^{-i(\omega\mp\Omega)t}\,,$ (34)
$A^{\prime}_{\hat{3}}=0\,.$ (35)

The noninertial observer refers its observations to axes rotating with
_constant_ angular velocity $\Omega$ about the $z$ axis. Consequently, the
local spacetime of the noninertial observer is _stationary_ ; therefore,
Fourier analysis of the measured vector potential reveals that

$\omega^{\prime}=\omega\mp\Omega\,,$ (36)

where the upper (lower) sign refers to positive (negative) helicity radiation.
In the positive (negative) helicity radiation that propagates along the $z$
direction, the electric and magnetic fields rotate in the positive (negative)
sense about the direction of propagation with angular speed $\omega$. These
fields are then naturally perceived by the noninertial observer to be rotating
with angular speed $\omega-\Omega$ ($\omega+\Omega$) in the positive
(negative) sense about the $z$ axis. This composition of angular speeds is
reminiscent of the composition of linear speeds in the Doppler effect.

Taking the intuitive explanation of Eq. (36) one step further, it becomes
clear that for an incident linearly polarized wave, the plane of linear
polarization would appear to rotate in the negative sense about the $z$ axis
with angular speed $\Omega$ as perceived by the noninertial observer. This
circumstance is the analogue of the magneto-optic Faraday effect via the
Larmor theorem Larmor .

For the noninertial observer under consideration here, its world line is the
temporal axis and the hypothesis of locality implies that the measured Doppler
frequency is

$\omega^{\prime}_{\rm D}=\omega\,,$ (37)

since the noninertial observer is fixed at $(x,y,z)=0$ and $\mathbf{v}=0$ in
this case. Writing

$\omega^{\prime}=\omega^{\prime}_{\rm
D}\left(1\mp\frac{\Omega}{\omega}\right)\,,\qquad\frac{\Omega}{\omega}=\frac{\lambdabar}{c/\Omega}\,,$
(38)

where $\lambdabar$ is the reduced wavelength of the incident radiation, we
recognize that the helicity-dependent part of the RDE is a wave effect and
hence goes beyond the ray (WKB) picture of the standard relativistic RDE.
Moreover, multiplication of Eq. (36) by $\hbar$ results in

$E^{\prime}=E\mp\hbar\,\Omega=E-\mathbf{S}\cdot\bm{\Omega}\,,$ (39)

which illustrates the helicity-rotation coupling that is responsible for the
deviation from the locality postulate.

###  III.2 Oblique Incidence

Let us now consider the case of oblique incidence. With no loss in generality,
we can assume an incident plane electromagnetic wave of frequency $\omega$ and
unit wave vector

$\hat{\mathbf{k}}=(-\sin\theta,0,\cos\theta)\,$ (40)

in Cartesian $(x,y,z)$ coordinates, where $\theta=0$ corresponds to the
previous case of incidence along the $z$ axis. The wave has definite helicity
and electromagnetic 4-potential $A_{\mu}$ given by

$A_{0}=0\,,\quad\mathbf{A}=a_{\pm}(\hat{\mathbf{n}}\pm
i\hat{\mathbf{y}})e^{-i\,(\omega/c)(ct+\sin\theta\,x-\cos\theta\,z)}\,,$ (41)

where the unit vector

$\hat{\mathbf{n}}=(\cos\theta,0,\sin\theta)\,$ (42)

is an axis in the orthonormal triad
$(\hat{\mathbf{n}},\hat{\mathbf{y}},\hat{\mathbf{k}})$.

For the measurements of the noninertial observer fixed at the origin of
spatial coordinates, we find $A^{\prime}_{\hat{0}}=0$ and, instead of Eqs.
(33)–(35),

$A^{\prime}_{\hat{1}}=a_{\pm}\,e^{-i\,\omega t}(\cos\theta\cos\Omega t\pm
i\,\sin\Omega t)\,,$ (43) $A^{\prime}_{\hat{2}}=a_{\pm}\,e^{-i\,\omega
t}(-\cos\theta\sin\Omega t\pm i\,\cos\Omega t)\,,$ (44)
$A^{\prime}_{\hat{3}}=a_{\pm}\,e^{-i\,\omega t}\,\sin\theta\,.$ (45)

Fourier analysis of these fields reveals the presence of three frequencies

$\omega^{\prime}_{m^{\prime}}=\omega-m^{\prime}\,\Omega\,,\qquad
m^{\prime}=+1,-1,0\,;$ (46)

moreover, Eqs. (43)–(45) can be expressed in terms of amplitudes
$\psi_{(\hat{i},m^{\prime})}$ as

$A^{\prime}_{\hat{1}}=\psi_{(\hat{1},+1)}\,e^{-i\,\omega^{\prime}_{+1}t}+\psi_{(\hat{1},-1)}\,e^{-i\,\omega^{\prime}_{-1}t}\,,$
(47)
$A^{\prime}_{\hat{2}}=\psi_{(\hat{2},+1)}\,e^{-i\,\omega^{\prime}_{+1}t}+\psi_{(\hat{2},-1)}\,e^{-i\,\omega^{\prime}_{-1}t}\,$
(48)

and

$A^{\prime}_{\hat{3}}=\psi_{(\hat{3},0)}\,e^{-i\,\omega^{\prime}_{0}t}\,.$
(49)

That is, the component of the 4-potential along $e^{\prime\mu}{}_{\hat{1}}$
contains frequencies $\omega^{\prime}_{+1}=\omega-\Omega$ and
$\omega^{\prime}_{-1}=\omega+\Omega$ with amplitudes

$\psi_{(\hat{1},+1)}=\pm
a_{\pm}\frac{1\pm\cos\theta}{2}\,,\qquad\psi_{(\hat{1},-1)}=\mp
a_{\pm}\frac{1\mp\cos\theta}{2}\,,$ (50)

respectively, while along $e^{\prime\mu}{}_{\hat{2}}$ we have the same
frequencies but with amplitudes

$\psi_{(\hat{2},+1)}=\pm
i\,a_{\pm}\frac{1\pm\cos\theta}{2}\,,\qquad\psi_{(\hat{2},-1)}=\pm
i\,a_{\pm}\frac{1\mp\cos\theta}{2}\,,$ (51)

respectively. Finally, along $e^{\prime\mu}{}_{\hat{3}}$, the observer finds
frequency $\omega^{\prime}_{0}=\omega$ with amplitude

$\psi_{(\hat{3},0)}=a_{\pm}\sin\theta\,.$ (52)

The average frequency measured by the noninertial observer may be calculated
in the manner of wave mechanics as

$<\omega^{\prime}>\,=\frac{\sum\omega^{\prime}_{m^{\prime}}|\psi_{(\hat{i},m^{\prime})}|^{2}}{\sum|\psi_{(\hat{i},m^{\prime})}|^{2}}\,,$
(53)

where the summation is over $i=1,2,3$ and $m^{\prime}=-1,0,1$. A simple
calculation reveals that the numerator in Eq. (53) is given by
$2|a_{\pm}|^{2}(\omega\mp\cos\theta\,\Omega)$ and the corresponding
denominator is given by $2|a_{\pm}|^{2}$. Hence, we find

$<\omega^{\prime}>\,=\omega\mp\cos\theta\,\Omega\,.$ (54)

Multiplication by $\hbar$ again reveals the coupling of helicity with rotation

$<E^{\prime}>\,=E-\mathbf{S}\cdot\bm{\Omega}\,,$ (55)

where $\mathbf{S}=\pm\hbar\,\hat{\mathbf{k}}$ for the photon. The same result
can be obtained from a detailed WKB treatment Mashhoon:2002fq ; Hauck:2003gy .

Let us note that we can write Eq. (53) in the form

$<\omega^{\prime}>\,=(\omega-\Omega)P_{+}+\omega
P_{0}+(\omega+\Omega)P_{-}\,,$ (56)

where the probability that the observer measures
$\omega^{\prime}_{m^{\prime}}$ for $m^{\prime}=+1,0,-1$, is given by $P_{+}$,
$P_{0}$ and $P_{-}$, respectively. Here,

$P_{+}=\frac{1}{4}(1\pm\cos\theta)^{2}\,,\qquad
P_{-}=\frac{1}{4}(1\mp\cos\theta)^{2}\,,\qquad
P_{0}=\frac{1}{2}\sin^{2}\theta\,,$ (57)

where $P_{+}+P_{-}+P_{0}=1$. In these relations, the upper (lower) sign
corresponds to positive (negative) helicity radiation that is incident along
the $\hat{\mathbf{k}}$ direction given by Eq. (40). Let us recall that the
eigenstates of the incident particle of spin $\hbar$ with respect to the
$(\hat{\mathbf{n}},\hat{\mathbf{y}},\hat{\mathbf{k}})$ system can be
transformed to the $(\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}})$
system of the observer via the well-known matrix
$(\mathcal{D}^{j}_{mm^{\prime}})$ for $j=1$ that is given by

$\left[\begin{array}[]{ccc}\frac{1}{2}(1+\cos\theta)&-\frac{1}{\sqrt{2}}\sin\theta&\frac{1}{2}(1-\cos\theta)\cr\frac{1}{\sqrt{2}}\sin\theta&\cos\theta&-\frac{1}{\sqrt{2}}\sin\theta\cr\frac{1}{2}(1-\cos\theta)&\frac{1}{\sqrt{2}}\sin\theta&\frac{1}{2}(1+\cos\theta)\cr\end{array}\right]\,;$
(58)

see, e.g., page 57 of Ref. Edmonds . The application of this matrix on the
helicity states of the incident photon implies that the amplitude for the spin
states along the $z$ axis for an incident photon of definite helicity is
proportional to

$\left[\begin{array}[]{c}\frac{1}{2}(1\pm\cos\theta)\cr\pm\,\frac{1}{\sqrt{2}}\sin\theta\cr\frac{1}{2}(1\mp\cos\theta)\cr\end{array}\right]\,$
(59)

based on the first and third columns of matrix (58). We emphasize that the
upper (lower) sign here indicates _incident_ positive (negative) helicity
waves along the $\hat{\mathbf{k}}$ axis. The squares of the first and third
elements of (59), namely, $P_{+}$ and $P_{-}$, represent the probability that
the photon spin is $+\hbar$ and $-\hbar$ along the $z$ axis, i.e. the rotation
axis of the observer, respectively, while $P_{0}$, which is equal to the
square of the second element of (59), is the probability that the photon spin
along the $z$ axis vanishes. In this way, Eq. (56) can provide an intuitive
understanding of why the average frequency measured by the noninertial
observer depends upon the helicity of the incident radiation.

###  III.3 Rotating Observers

Consider an observer that rotates uniformly about the $z$ axis on a circular
orbit of constant radius $r$. The speed of the circular motion is
$v=r\,\Omega$ and the observer’s proper time is $\tau=t/\gamma$, where
$\gamma$ is the corresponding Lorentz factor. As before, we assume that
$\tau=0$ when $t=0$. The frequency content of incident radiation as measured
by the rotating observer has been treated in detail in Ref. BMB . If the wave
is incident along the $z$ axis, we find via the Fourier analysis of the
measured field that the frequency of the incident radiation in terms of the
proper time of the observer is given by

$\omega^{\prime}=\gamma(\omega\mp\,\Omega)\,,$ (60)

where the Lorentz factor takes due account of time dilation. Equation (60)
should be compared and contrasted in this case with the transverse Doppler
effect given by $\omega^{\prime}_{\rm D}=\gamma\omega$.

For the case of oblique incidence,

$\omega^{\prime}=\gamma(\omega-M\Omega)\,,$ (61)

where $\hbar M$, $M=0,\pm 1,\pm 2,...,$ is the component of the total angular
momentum of the electromagnetic radiation field along the rotation axis of the
observer BMB . Let us multiply Eq. (61) by $\hbar$ and write the resulting
equation in the WKB approximation as

$E^{\prime}=\gamma(E-\bm{\Omega}\cdot\mathbf{J})\,,$ (62)

where

$\mathbf{J}=\mathbf{r}\times\mathbf{p}+\mathbf{S}\,$ (63)

is the total angular momentum of the photon with momentum
$\mathbf{p}=\hbar\,\mathbf{k}$ and spin $\mathbf{S}$. Thus,

$E^{\prime}=\gamma(E-\mathbf{v}\cdot\mathbf{p})-\gamma\mathbf{S}\cdot\bm{\Omega}\,,$
(64)

where the energy due to spin-rotation coupling takes due account of time
dilation.

An important consequence of the helicity-rotation coupling is that an energy
shift occurs when electromagnetic waves of definite helicity pass through a
uniformly rotating half-wave plate. It appears that helicity-rotation coupling
first came to light through the corresponding frequency shift of
electromagnetic radiation in microwave experiments Allen . It was later
studied in the optical regime by a number of investigators G+A ; Garetz ; SKS
; MNHS ; 1Courtial ; 2Courtial . For further references and background
material, see, for instance, LCT ; LZZ ; ZFD ; FPW ; DLLL ; CWL ; E+E  and the
references therein. The topic of energy shift as radiation passes through a
spin flipper is briefly discussed in Appendix A.

##  IV Nonlocal Relativity

The standard theory of relativity is based on the pointwise application of the
Lorentz transformation for accelerated systems. This approach results, for
wave phenomena, in the invariance of the phase of the wave under Lorentz
transformations from which the standard Doppler formula is derived. On the
other hand, if the pointwise measured field is subjected to the temporally
nonlocal operation of Fourier analysis, one finds the more general results
involving helicity-rotation coupling described above. In this case, phase
invariance is violated. Furthermore, the photon helicity-rotation coupling is
an example of the general coupling of intrinsic spin with rotation that is due
to the _inertia of intrinsic spin_ BM3 . In quantum mechanics, mass and spin
determine the inertial properties of a particle, since they characterize the
irreducible unitary representations of the Poincaré group Wigner . The
inertial aspects of mass are well known from Newtonian physics. The inertial
aspects of intrinsic spin are independent of the inertial mass and depend
solely on intrinsic spin. The inertial aspects of intrinsic spin have been the
subject of recent investigations; see Mashhoon:2015nea  and the references
cited therein. There is much observational support for the general phenomenon
of spin-rotation coupling; for instance, it is known as the “phase wrap-up” in
the Global Positioning System (GPS) Ashby . For recent developments in this
general subject, see DSH ; DDKWLSH ; Yu:2022vjn  and the references therein.

There is, however, a fundamental difficulty here that must now be addressed.
It follows from Eqs. (36) and (60) that an observer that rotates uniformly in
the positive sense with angular speed $\omega$ about the direction of
propagation of a plane monochromatic electromagnetic wave of frequency
$\omega$ and positive helicity measures a frequency $\omega^{\prime}=0$; that
is, the wave stands completely still for the class of all uniformly rotating
observers with $r\geq 0$. The measured electromagnetic field is oscillatory in
space but independent of time. What happens in this case of rotational Doppler
effect (RDE) is reminiscent of pre-relativistic Doppler effect where an
observer moving at the speed of light along the direction of propagation could
stay completely at rest with the wave. This circumstance was mentioned by
Einstein in his autobiographical notes; see page 53 of EinAN . This problem
does not occur in the relativistic Doppler formula given by Eq. (6) for
inertial observers or Eq. (17) for accelerated observers; that is, the
measured frequency will vanish only if the frequency of the incident wave
vanishes.

To overcome the problem in the present case, we must reconsider our assumption
regarding the _pointwise_ measurement of the electromagnetic field. This is
impossible according to Bohr and Rosenfeld, who have argued that only
spacetime averages of electromagnetic fields can be measured.

###  IV.1 The Bohr-Rosenfeld Principle

Consider the presence of an electromagnetic field in a global inertial frame
in Minkowski spacetime. In Cartesian coordinates $x^{\mu}=(ct,\mathbf{x})$,
the electric and magnetic fields can be expressed as
$\mathbf{E}(t,\mathbf{x})$ and $\mathbf{B}(t,\mathbf{x})$, which are formally
defined via the projection of the Faraday tensor $F_{\mu\nu}$ on the tetrad
frame $e^{\mu}{}_{\hat{\alpha}}=\delta^{\mu}_{\hat{\alpha}}$ of the inertial
observers at rest in the background frame.

To simplify matters, let us imagine the actual measurement of the electric
field by the inertial observers, while ignoring the presence of the magnetic
field. A macroscopic body of linear size $\sim L$ and volume $V\sim L^{3}$
with constant uniform charge density $\rho$ is placed in the electric field.
The body experiences a force given by the Lorentz force law

$\frac{d\mathbf{P}}{dt}=\rho\int_{V}\mathbf{E}(t,\mathbf{x})\,d^{3}x\,,$ (65)

where $\mathbf{P}$ is the momentum of the body. The inertial observers measure
this momentum at the initial ($t^{\prime}$) and final ($t^{\prime\prime}$)
stages of the experiment that lasts for a total time
$T=t^{\prime\prime}-t^{\prime}$. Hence, integrating Eq. (65) from $t^{\prime}$
to $t^{\prime\prime}$, we get

$\mathbf{P}^{\prime\prime}-\mathbf{P}^{\prime}=\rho\int_{t^{\prime}}^{t^{\prime\prime}}\int_{V}\mathbf{E}(t,\mathbf{x})\,d^{3}x=\rho<\mathbf{E}>TV\,.$
(66)

What is measured is the spacetime average of the electric field given by

$<\mathbf{E}>\,=\frac{1}{\Delta}\int_{\Delta}\mathbf{E}\,d^{4}x\,,$ (67)

where $\Delta=c\,TV$. This simple argument due to Bohr and Rosenfeld B+R
assumes that momentum measurements by the inertial observers last for a time
period $\ll T$ and cause displacements that are $\ll L$.

###  IV.2 Nonlocality of Accelerated Observers

The considerations of Bohr and Rosenfeld may appear somewhat innocuous for the
measurements of inertial observers in Minkowski spacetime. Repeating the same
arguments for accelerated observers, on the other hand, would present special
difficulties. We therefore adopt a different indirect approach. The Bohr-
Rosenfeld argument can be taken to mean that the measurement of the
electromagnetic field by an accelerated observer requires the consideration of
the observer’s past world line. That is, we must replace the pointwise
measurement of the electromagnetic field by an accelerated observer with a
nonlocal ansatz that involves a certain average of the field over the past
world line of the accelerated observer. To this end, it is first necessary to
discuss briefly the relativistic theory of accelerated systems. There are
intrinsic spatial and temporal scales associated with accelerated observers;
for instance, Earth-bound observers have translational and rotational
acceleration lengths of $c^{2}/g_{\oplus}\approx 1$ light year and
$c/\Omega_{\oplus}\approx 28$ astronomical units, respectively.

###  IV.3 Acceleration Scales

In Minkowski spacetime, we consider an arbitrary accelerated observer with
proper time $\eta$ following a path $x^{\mu}(\eta)$ given by inertial
Cartesian coordinates. The observer carries along its world line an adapted
orthonormal tetrad frame $\xi^{\mu}{}_{\hat{\alpha}}(\eta)$ for measurement
purposes, where $\xi^{\mu}{}_{\hat{0}}=\mathfrak{u}^{\mu}=dx^{\mu}/d\eta$ is
the timelike unit 4-velocity vector of the accelerated observer. The manner in
which the observer carries its tetrad frame along its world line is specified
via

$\frac{d\xi^{\mu}{}_{\hat{\alpha}}}{d\eta}=\Phi_{\hat{\alpha}}{}^{\hat{\beta}}\,\xi^{\mu}{}_{\hat{\beta}}\,.$
(68)

The orthonormality of the moving tetrad frame implies that the acceleration
tensor $\Phi_{\hat{\alpha}\hat{\beta}}(\eta)$ is antisymmetric, namely,
$\Phi_{\hat{\alpha}\hat{\beta}}=-\Phi_{\hat{\beta}\hat{\alpha}}$. Therefore,
$\Phi_{\hat{\alpha}\hat{\beta}}$ is analogous to the Faraday tensor and can be
represented in terms of its “electric” and “magnetic” components. The
“electric” part is given by

$\Phi_{\hat{0}\hat{i}}=\mathfrak{A}_{\hat{i}}=\mathfrak{A}_{\mu}\,\xi^{\mu}{}_{\hat{i}}\,,$
(69)

where $\mathfrak{A}^{\mu}=d\mathfrak{u}^{\mu}/d\eta$,
$\mathfrak{A}_{\mu}\mathfrak{u}^{\mu}=0$, is the spacelike translational
acceleration of the observer. Similarly, the magnetic part can be expressed as

$\Phi_{\hat{i}\hat{j}}=\epsilon_{\hat{i}\hat{j}\hat{k}}\,\Omega^{\hat{k}}\,,$
(70)

where $\Omega^{\hat{k}}$ is the locally measured angular velocity of the
rotation of the observer’s local spatial frame with respect to a nonrotating
(i.e. Fermi-Walker transported) frame. The measured components of the
acceleration tensor $\Phi_{\hat{\alpha}\hat{\beta}}$ have dimensions of
1/length and can be employed to define the translational and rotational
acceleration scales of the observer BM2 . Using

$\Phi^{\mu\nu}=\Phi^{\hat{\alpha}\hat{\beta}}\,\xi^{\mu}{}_{\hat{\alpha}}\,\xi^{\mu}{}_{\hat{\beta}}\,,\qquad\Omega^{\mu}=\Omega^{\hat{k}}\,\xi^{\mu}{}_{\hat{k}}\,,$
(71)

where $\Omega^{\mu}$ is a spacelike 4-vector such that
$\mathfrak{u}_{\mu}\,\Omega^{\mu}=0$, we find

$\Phi^{\mu\nu}=\mathfrak{A}^{\mu}\mathfrak{u}^{\nu}-\mathfrak{A}^{\nu}\mathfrak{u}^{\mu}+\epsilon^{\mu\nu\rho\sigma}\mathfrak{u}_{\rho}\Omega_{\sigma}\,.$
(72)

In our convention, the alternating tensor is in general such that
$\epsilon_{\hat{0}\hat{1}\hat{2}\hat{3}}=1$.

###  IV.4 Nonlocal Ansatz

Regarding the arbitrary accelerated observer with proper time $\eta$ and
orthonormal tetrad frame $\xi^{\mu}{}_{\hat{\alpha}}(\eta)$ under
consideration in the previous subsection, let us assume that for
$\eta\leq\eta_{0}$ its motion is uniform and acceleration begins at
$\eta=\eta_{0}$. Thus, for $\eta>\eta_{0}$, the observer along its world line
passes through a continuous infinity of locally comoving inertial observers.
The incident external field under consideration is assumed to be
$\mathbb{A}_{\mu}$ and what an instantaneously comoving inertial observer
measures along the world line is $\mathbb{A}_{\hat{\alpha}}(\eta)$ given by

$\mathbb{A}_{\hat{\alpha}}(\eta)=\mathbb{A}_{\mu}(\eta)\,\xi^{\mu}{}_{\hat{\alpha}}(\eta)\,.$
(73)

On the other hand, what is actually measured by the accelerated observer is
$\mathbb{B}_{\hat{\alpha}}(\eta)$. The most general relationship between
$\mathbb{B}_{\hat{\alpha}}(\eta)$ and $\mathbb{A}_{\hat{\alpha}}(\eta)$ that
is consistent with linearity and causality is

$\mathbb{B}_{\hat{\alpha}}(\eta)=\mathbb{A}_{\hat{\alpha}}(\eta)+\Theta(\eta-\eta_{0})\int_{\eta_{0}}^{\eta}\mathbb{K}_{\hat{\alpha}}{}^{\hat{\beta}}(\eta^{\prime})\,\mathbb{A}_{\hat{\beta}}(\eta^{\prime})\,d\eta^{\prime}\,,$
(74)

where $\Theta$ is the unit step function such that $\Theta(t)=0$ for $t\leq 0$
and $\Theta(t)=1$ for $t>0$. Ansatz (74) goes beyond the pointwise field
measurement by including a certain average over the accelerated world line
with a weight function $\mathbb{K}(\eta)$ that presumably depends upon the
observer’s acceleration. That is, when the acceleration is turned on at
$\eta=\eta_{0}$,
$\mathbb{B}_{\hat{\alpha}}(\eta_{0})=\mathbb{A}_{\hat{\alpha}}(\eta_{0})$;
however, for $\eta>\eta_{0}$, there is in addition a certain cumulative memory
of past acceleration involving a kernel $\mathbb{K}(\eta)$. If the
acceleration is turned off at some future time $\eta_{f}>\eta_{0}$, there will
remain a constant memory of past acceleration for $\eta>\eta_{f}$ that is
entirely benign.

Along these lines, a nonlocal relativistic theory of accelerated observers was
first proposed in 1993 Mashhoon:1993zz . The main nonlocal ansatz is a
Volterra integral equation and the unique correspondence between
$\mathbb{B}_{\hat{\alpha}}(\eta)$ and $\mathbb{A}_{\mu}(\eta)$ in the space of
functions of physical interest is guaranteed by the Volterra-Tricomi theorem
Mashhoon:1993zz ; Mashhoon:2008vr .

To determine kernel $\mathbb{K}$, we assume that for an incident radiation
field, the accelerated observer will always detect a time-dependent field;
otherwise, if $\mathbb{B}_{\hat{\alpha}}$ turns out to be constant in time
$\eta$, the incident field $\mathbb{A}_{\mu}$ must have been a constant field
in the first place, just as in the case of the relativistic Doppler effect.
That is, in this nonlocal theory no observer can ever stay at rest with
respect to a pure radiation field. A constant $\mathbb{B}_{\hat{\alpha}}$ thus
uniquely corresponds to a constant $\mathbb{A}_{\mu}$ in accordance with the
Volterra-Tricomi theorem. To implement this idea, let us therefore assume that
$\mathbb{B}_{\hat{\alpha}}$ and $\mathbb{A}_{\mu}$ are constants for
$\eta\geq\eta_{0}$. Then,

$\mathbb{B}_{\hat{\alpha}}(\eta)=\mathbb{B}_{\hat{\alpha}}(\eta_{0})=\mathbb{A}_{\hat{\alpha}}(\eta_{0})=\mathbb{A}_{\mu}(\eta_{0})\,\xi^{\mu}{}_{\hat{\alpha}}(\eta_{0})\,,$
(75)

by our assumption and Eq. (74). In this case, Eq. (74) can be simplified using
Eq. (73) and for $\eta>\eta_{0}$ can be expressed for arbitrary constant
$\mathbb{A}_{\mu}$ as

$\xi^{\mu}{}_{\hat{\alpha}}(\eta_{0})=\xi^{\mu}{}_{\hat{\alpha}}(\eta)+\int_{\eta_{0}}^{\eta}\mathbb{K}_{\hat{\alpha}}{}^{\hat{\beta}}(\eta^{\prime})\,\xi^{\mu}{}_{\hat{\beta}}(\eta^{\prime})\,d\eta^{\prime}\,.$
(76)

Taking the derivative of this relation with respect to $\eta$, we find

$-\frac{d\xi^{\mu}{}_{\hat{\alpha}}(\eta)}{d\eta}=\mathbb{K}_{\hat{\alpha}}{}^{\hat{\beta}}(\eta)\,\xi^{\mu}{}_{\hat{\beta}}(\eta)\,.$
(77)

Comparing this relation with Eq. (68), we conclude

$\mathbb{K}_{\hat{\alpha}}{}^{\hat{\beta}}(\eta)=-\Phi_{\hat{\alpha}}{}^{\hat{\beta}}(\eta)\,;$
(78)

that is, the kernel of our nonlocal ansatz is minus the measured acceleration
tensor of the observer. Further developments of these ideas are contained in
Mashhoon:2008vr ; BMB ; Mashhoon:2020mgw  and the references cited therein.

In the rest of this paper, we work out the implications of this nonlocal
theory for the observation of helicity-rotation coupling by the noninertial
observer, introduced in Section III, that remains at rest at the origin of
spatial coordinates in Minkowski spacetime.

##  V Noninertial Observer: Nonlocal Ansatz

We would like to find out how the nonlocal theory of accelerated systems
developed in the previous section ameliorates the basic difficulty of the
local theory. Therefore, we return to the problem of frequency measurements of
the noninertial observer of Section III and assume that the observer is
permanently at rest at the origin of spatial coordinates and for $t\leq 0$
employs standard inertial Cartesian axes for measurement purposes. However,
for $t>0$, the observer refers its measurements to the system that rotates
uniformly about the $z$ axis and is given by Eqs. (22) and (23). A plane
electromagnetic wave of frequency $\omega$, definite helicity and 4-potential
$A_{\mu}$ given by Eq. (24) is incident along the $z$ axis. Moreover, the
noninertial observer’s tetrad frame $e^{\prime\mu}{}_{\hat{\alpha}}$ is given
by Eqs. (27)–(30) for $t>0$ and the electromagnetic field locally measured by
the comoving inertial observers is
$A^{\prime}_{\hat{\alpha}}=A_{\mu}\,e^{\prime\mu}{}_{\hat{\alpha}}$ as in Eq.
(26). The field actually measured by the noninertial observer is in this case
$\mathcal{A}_{\hat{\alpha}}(t)$ given by our general nonlocal ansatz (74),
namely,

$\mathcal{A}_{\hat{\alpha}}(t)=A^{\prime}_{\hat{\alpha}}(t)-\Theta(t)\int_{0}^{t}\Phi_{\hat{\alpha}}{}^{\hat{\beta}}(t^{\prime})\,A^{\prime}_{\hat{\beta}}(t^{\prime})\,dt^{\prime}\,,$
(79)

in accordance with the considerations of the previous section.

The measured components of the acceleration tensor can be simply worked out in
this case and we find that its nonzero components are given by

$\Phi_{\hat{1}\hat{2}}=-\Phi_{\hat{2}\hat{1}}=\Omega\,.$ (80)

For the incident field (24) that propagates along the $z$ axis,
$A^{\prime}_{\hat{\alpha}}$ has been worked out in Section III and is given by
$A^{\prime}_{\hat{0}}=A^{\prime}_{\hat{3}}=0$ and

$A^{\prime}_{\hat{1}}(t)=a_{\pm}\,e^{-i(\omega\mp\Omega)t}\,,\qquad
A^{\prime}_{\hat{2}}=\pm i\,A^{\prime}_{\hat{1}}\,.$ (81)

It is then straightforward to employ Eq. (79) to show that
$\mathcal{A}_{\hat{0}}=\mathcal{A}_{\hat{3}}=0$ and

$\mathcal{A}_{\hat{1}}(t)=\frac{a_{\pm}}{1\mp\tfrac{\Omega}{\omega}}\,\left[e^{-i(\omega\mp\Omega)t}\mp\,\tfrac{\Omega}{\omega}\right]\,,\qquad\mathcal{A}_{\hat{2}}=\pm
i\,\mathcal{A}_{\hat{1}}\,.$ (82)

The measured frequency is still $\omega^{\prime}=\omega\mp\Omega$ as in Eq.
(36); however, the measured amplitude in the nonlocal theory picks up a
helicity-dependent factor

$\frac{\omega}{\omega^{\prime}_{\pm
1}}=\frac{1}{1\mp\tfrac{\Omega}{\omega}}\,.$ (83)

In practice, $\Omega\ll\omega$, and the measured amplitude slightly increases
(decreases) if the observer rotates in the positive sense about the direction
of propagation of positive (negative) helicity radiation. For incident radio
waves of frequency $1$ GHz, we have $\Omega/\omega=10^{-7}$ for an observer
that rotates one hundred times per second. It is hoped that future
observations can verify this consequence of helicity-rotation coupling. In
particular, it would be interesting if the nonlocal helicity-dependent
amplitude effect could be detected within the context of RDE.

The nonlocal theory was introduced to ensure that electromagnetic radiation
could never stand completely still with respect to an observer. We recall that
$\Omega=\omega$ caused the problem in the case of incident positive-helicity
radiation; on the other hand, in the nonlocal theory the wave amplitude is no
longer constant as $\Omega\to\omega$, since

$\lim_{\Omega\to\omega}\frac{e^{-i(\omega-\Omega)t}-\tfrac{\Omega}{\omega}}{1-\tfrac{\Omega}{\omega}}=1-i\,\omega
t\,.$ (84)

Hence, in this resonance case the nonzero components of
$\mathcal{A}_{\hat{\alpha}}(t)$ are given by

$\mathcal{A}_{\hat{2}}(t)=i\,\mathcal{A}_{\hat{1}}(t)=a_{+}(1-i\,\omega t)\,.$
(85)

In this way, the nonlocal approach avoids the fundamental difficulty of the
local theory.

In the limit of large quantum numbers, there is a correspondence between the
behavior of electrons in atoms and accelerated observers of classical physics.
Indeed, using Bohr’s correspondence principle, it is possible to show that the
consequences of nonlocal theory given by Eqs. (83) and (85) are consistent
with the quantum theory BM4 .

###  V.1 Oblique Incidence: Nonlocal Ansatz

In the case of oblique incidence, we use Eq. (79) for $t>0$ with
$A^{\prime}_{\hat{\alpha}}$ given by Eqs. (47)–(49) to get

$\mathcal{A}_{\hat{1}}=\frac{\omega}{\omega^{\prime}_{+1}}\,\psi_{(\hat{1},+1)}\,\left[e^{-i\,\omega^{\prime}_{+1}t}-\tfrac{\Omega}{\omega}\right]+\frac{\omega}{\omega^{\prime}_{-1}}\,\psi_{(\hat{1},-1)}\,\left[e^{-i\,\omega^{\prime}_{-1}t}-\tfrac{\Omega}{\omega}\right]\,,$
(86)
$\mathcal{A}_{\hat{2}}=\frac{\omega}{\omega^{\prime}_{+1}}\,\psi_{(\hat{2},+1)}\,\left[e^{-i\,\omega^{\prime}_{+1}t}-\tfrac{\Omega}{\omega}\right]+\frac{\omega}{\omega^{\prime}_{-1}}\,\psi_{(\hat{2},-1)}\,\left[e^{-i\,\omega^{\prime}_{-1}t}-\tfrac{\Omega}{\omega}\right]\,,$
(87)

and

$\mathcal{A}_{\hat{3}}=\psi_{(\hat{3},0)}\,e^{-i\,\omega t}\,,$ (88)

where $\omega^{\prime}_{\pm 1}=\omega\mp\Omega$. The observer measures the
same frequency spectrum as before; however, the amplitudes can change in
accordance with

$\psi_{(\hat{i},m^{\prime})}\to\frac{\omega}{\omega^{\prime}_{m^{\prime}}}\,\psi_{(\hat{i},m^{\prime})}\,.$
(89)

Therefore, the only amplitude change that occurs involves
$\omega^{\prime}_{\pm 1}$, in agreement with Eq. (83).

In the nonlocal theory, the average frequency measured by the noninertial
observer may be calculated as in Eq. (53) taking due account of the variation
in amplitude given by Eq. (89); that is,

$<\omega^{\prime}>\,=\frac{\sum|\psi_{(\hat{i},m^{\prime})}|^{2}/\omega^{\prime}_{m^{\prime}}}{\sum|\psi_{(\hat{i},m^{\prime})}|^{2}/\omega^{\prime
2}_{m^{\prime}}}\,.$ (90)

It follows that

$\frac{<\omega^{\prime}>}{\omega}\,=\frac{\mathbb{N}}{\mathbb{D}}\,,$ (91)

where

$\mathbb{N}:=1\pm\epsilon\cos\theta-\epsilon^{2}(1+\tfrac{1}{2}\sin^{2}\theta)\mp\epsilon^{3}\cos\theta+\tfrac{1}{2}\epsilon^{4}\sin^{2}\theta\,,$
(92)
$\mathbb{D}:=1\pm\,2\epsilon\cos\theta+\epsilon^{2}(1-\tfrac{3}{2}\sin^{2}\theta)+\tfrac{1}{2}\epsilon^{4}\sin^{2}\theta\,,$
(93) $\epsilon:=\frac{\Omega}{\omega}\,.$ (94)

In powers of $\epsilon\ll 1$, we find

$\frac{<\omega^{\prime}>}{\omega}\,=1\mp\epsilon\cos\theta-\epsilon^{2}\sin^{2}\theta\pm\frac{1}{2}\epsilon^{3}\sin^{2}\theta\cos\theta+O(\epsilon^{4})\,,$
(95)

so that we recover the simple helicity-rotation coupling to first order in
$\epsilon$.

We have chosen the observer to be fixed at the origin of spatial coordinates,
i.e. $r=0$. In this way, the photon orbital angular momentum vanishes and we
only deal with the photon helicity. For a nonlocal treatment involving
rotating observers with $r>0$, see BM5 .

##  VI Helicity-Dependent Amplitude Effect

It turns out that the result of the nonlocal ansatz expressed by Eq. (83) is
quite general and may be amenable to experimental research. If the observer
rotates with angular velocity $\Omega$ in the positive sense about the
direction of propagation of incident positive (negative) helicity plane
electromagnetic wave of frequency $\omega\gg\Omega$, the amplitude of the wave
as measured by the rotating observer is enhanced (diminished) by
$(1\mp\Omega/\omega)^{-1}\approx 1\pm\Omega/\omega$. That is, the amplitude is
slightly larger (smaller) if the electromagnetic field rotates in the same
(opposite) sense as the observer. We explore this _nonlocal_ effect for an
incident linearly polarized wave in this section.

To see explicitly the significance of this nonlocal helicity-dependent
amplitude effect, we consider a linearly polarized plane electromagnetic wave
incident along the $z$ axis with vector potential $A_{\mu}=(0,\mathbf{A})$,
where

$\mathbf{A}=\tilde{a}\,\hat{\mathbf{x}}\,e^{-i\,\omega(t-z/c)}\,.$ (96)

Here $\tilde{a}$ is a constant complex amplitude. Assuming the locality
postulate, the noninertial observer fixed at the origin of spatial coordinates
with adapted tetrad frame (26)–(30) would measure a vector potential given by
$A^{\prime}_{\hat{0}}=0$, $A^{\prime}_{\hat{3}}=0$ and

$A^{\prime}_{\hat{1}}=\tilde{a}\,e^{-i\,\omega t}\,\cos\Omega t\,,\qquad
A^{\prime}_{\hat{2}}=-\tilde{a}\,e^{-i\,\omega t}\,\sin\Omega t\,.$ (97)

Let us note that

$A^{\prime}_{\hat{1}}\,\hat{\mathbf{x}}^{\prime}+A^{\prime}_{\hat{2}}\,\hat{\mathbf{y}}^{\prime}=\tilde{a}\,\hat{\mathbf{x}}\,e^{-i\,\omega
t}\,,$ (98)

where we have used Eq. (22). Assuming that the electromagnetic field is
pointwise measured by the noninertial observer, the plane of linear
polarization of the incident wave would appear to rotate in the negative sense
with angular speed $\Omega$ about the $z$ axis.

On the other hand, employing the nonlocal ansatz (79), we find instead of Eq.
(97),

$\mathcal{A}_{\hat{1}}=\frac{\tilde{a}}{1-\epsilon^{2}}\left[(\cos\Omega
t+i\,\epsilon\,\sin\Omega t)e^{-i\,\omega t}-\epsilon^{2}\right]\,,$ (99)
$\mathcal{A}_{\hat{2}}=\frac{\tilde{a}}{1-\epsilon^{2}}\left[(-\sin\Omega
t+i\,\epsilon\,\cos\Omega t)e^{-i\,\omega t}-i\,\epsilon\right]\,,$ (100)

where $\epsilon=\Omega/\omega$, as before. In this case,

$\mathcal{A}_{\hat{1}}\,\hat{\mathbf{x}}^{\prime}+\mathcal{A}_{\hat{2}}\,\hat{\mathbf{y}}^{\prime}=\frac{\tilde{a}}{1-\epsilon^{2}}\left[(\hat{\mathbf{x}}+i\,\epsilon\,\hat{\mathbf{y}})e^{-i\,\omega
t}-\epsilon^{2}\,\hat{\mathbf{x}}^{\prime}-i\,\epsilon\,\hat{\mathbf{y}}^{\prime}\right]\,,$
(101)

which should be compared and contrasted with Eq. (98). In practice,
$\epsilon\ll 1$ and to first order in $\epsilon$,

$\tilde{a}\,(\hat{\mathbf{x}}+i\,\epsilon\,\hat{\mathbf{y}})\,e^{-i\,\omega
t}\,$ (102)

in Eq. (101) represents an elliptically polarized wave with ellipticity
$\epsilon\ll 1$. Therefore, an incident linearly polarized electromagnetic
wave appears to the noninertial observer to be slightly elliptically polarized
while precessing in the negative sense about the direction of incidence $z$
with angular speed $\Omega$. To verify such effects observationally, it is
clearly necessary to measure the ellipticity of electromagnetic radiation to
rather high accuracy. In this connection, see, for instance, Ref. KDL  and
references therein.

##  VII Discussion

The photon helicity-rotation coupling discussed in this paper is an aspect of
the general phenomenon of the coupling of intrinsic spin with rotation as well
as the gravitomagnetic fields of rotating systems Mashhoon:2023idh .

The nonlocal approach we employ here has a marked resemblance to the nonlocal
electrodynamics of media Jackson ; LD+EM ; HeOb . To introduce history
dependence into the theories of accelerated systems and gravitational fields,
it seems prudent to follow the method that has been successful in the physics
of continuous media. The state of a physical system can be history dependent;
for instance, magnetic materials exhibit the phenomenon of hysteresis. This
general subject has a long history; see, for instance, Poi ; Hop  and the
references therein. For the nonlocal extension of Einstein’s theory of
gravitation along these lines, see Hehl:2008eu ; Hehl:2009es ; BMB ;
VanDenHoogen:2017nyy .

It seems challenging to find a physical circumstance that could allow the
measurement of nonlocal aspects of accelerated systems BaMa ; Bremm:2015wmv .
In this connection, the observational implications of the nonlocal helicity-
dependent amplitude effect discussed in the previous section could be
interesting.

##  Appendix A Energy Shifts via Spin-Rotation Coupling

Imagine a half-wave plate in the $(x,y)$ plane that rotates slowly and
uniformly with angular speed $\Omega$ about the $z$ axis. An incident plane
electromagnetic wave of frequency $\omega_{\rm in}$ and definite helicity
propagates along the $z$ axis. The rotation of the half-wave plate is uniform;
therefore, the spacetime within the rotating half-wave plate is stationary. If
we neglect time dilation, all observers at rest with respect to the half-wave
plate measure a frequency $\omega^{\prime}\approx\omega_{\rm in}\mp\,\Omega$
regardless of their distance from the $z$ axis in accordance with Eq. (60);
here, as before, the upper (lower) sign refers to incident positive (negative)
helicity radiation. This remarkable fact implies that as the wave passes
through the half-wave plate, its frequency remains constant and equal to
$\omega^{\prime}$ as measured by observers at rest in the half-wave plate but
its helicity changes as it leaves the plate. Thus, observers at rest at the
bottom of the half-wave plate measure a frequency

$\omega^{\prime}\approx\omega_{\rm in}\mp\Omega\,$ (103)

for the _incoming_ wave, while observers at rest at the top of the half-wave
plate measure the same frequency

$\omega^{\prime}\approx\omega_{\rm out}\pm\Omega\,$ (104)

for the _outgoing_ wave. It follows that

$\omega_{\rm out}-\omega_{\rm in}\approx\mp\,2\,\Omega\,.$ (105)

Multiplying this relation by $\hbar$, we get the general result MNHS

$E_{\rm out}-E_{\rm in}\approx-2\,\mathbf{S}\cdot\,\bm{\Omega}\,$ (106)

for radiation passing through a spin-flipper.

In the background inertial frame, the kinetic energy of the rotating half-wave
plate in the nonrelativistic approximation is given by

$\mathcal{E}=\frac{1}{2}\mathcal{I}\,\Omega^{2}=\frac{1}{2}\frac{\mathcal{J}^{2}}{\mathcal{I}}\,,$
(107)

where $\mathcal{I}$ is the moment of inertia of the plate and
$\mathcal{J}=\mathcal{I}\,\Omega$ is its angular momentum. Thus, assuming that
the moment of inertia of the plate remains constant as the wave passes
through,

$\frac{\partial\mathcal{E}}{\partial\mathcal{J}}=\Omega\,.$ (108)

For each photon of definite helicity that passes through the half-wave plate,
the angular momentum of the plate changes by $\delta\mathcal{J}=\pm\,2\hbar$,
where the upper (lower) sign indicates incident positive (negative) helicity
radiation; therefore, the corresponding change in the kinetic energy of the
plate is given by $\delta\mathcal{E}=\pm\,2\hbar\Omega$. Conservation of
energy then requires that the change in the frequency of the photon be given
by $\delta\omega=\omega_{\rm out}-\omega_{\rm in}=\mp\,2\,\Omega$. It follows
that the phenomenon of energy shift through helicity-rotation coupling is
consistent with the laws of conservation of energy and angular momentum.

For a corresponding treatment of the nonlocal helicity-dependent amplitude
effect in the process of energy shift due to helicity-rotation coupling, see
BM6 .

## References

  * (1) S. Lendinez, E. M. Chudnovsky and J. Tejada, “Rotational Doppler Effect in Magnetic Resonance”, Phys. Rev. B 82, 174418 (2010).
  * (2) G. Li, T. Zentgraf and S. Zhang, “Rotational Doppler effect in nonlinear optics”, Nature Phys. 12, 736-740 (2016).
  * (3) H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect”, Opt. Express 24, 10050-10056 (2016).
  * (4) L. Fang, M. J. Padgett and J. Wang “Sharing a common origin between the rotational and linear Doppler effects”, Laser & Photonics Rev. 11, 1700183 (2017).
  * (5) J. Deng, K. F. Li, W. Liu and G. Li, “Cascaded Rotational Doppler Effect”, Opt. Lett. 44, 2346-2349 (2019).
  * (6) T.-Y. Cheng, W.-Y. Wang, J.-S. Li, J.-X. Guo, S. Liu and J.-Q. Lü, “Rotational Doppler Effect in Vortex Light and Its Applications for Detection of the Rotational Motion”, Photonics 9, 441 (2022).
  * (7) O. Emile and J. Emile, “Rotational Doppler Effect: A Review”, Ann. Phys. (Berlin) 535, 2300250 (2023).
  * (8) I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational Frequency Shift”, Phys. Rev. Lett. 78, 2539 (1997).
  * (9) A. Ya. Bekshaev, M. S. Soskin and M. V. Vasnetsov, “Angular momentum of a rotating light beam”, Opt. Commun. 249, 367-378 (2005).
  * (10) M. P. J. Lavery, S. M. Barnett, F. C. Speirits and M. J. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body”, Optica 1, 1-4 (2014).
  * (11) F. C. Speirits, M. P. J. Lavery, M. J. Padgett and S. M. Barnett, “Optical angular momentum in a rotating frame”, Opt. Lett. 39, 2944-2946 (2014).
  * (12) O. V. Angelsky, A. Ya. Bekshaev, I. I. Mokhun, M. V. Vasnetsov, C. Yu. Zenkova, S. G. Hanson and J. Zheng, “Review on the structured light properties: rotational features and singularities”, Opto-Electronics Rev. 30, e140860 (2022).
  * (13) A. Einstein, “Zur Elektrodynamik bewegter Körper [On the Electrodynamics of Moving Bodies]”, Ann. Phys. (Leipzig) 17, 891-921 (1905). http://dx.doi.org/10.1002/andp.19053221004
  * (14) H. P. Robertson, “Postulate versus Observation in the Special Theory of Relativity”, Rev. Mod. Phys. 21, 378-382 (1949).
  * (15) A. Einstein, _The Meaning of Relativity_ (Princeton University Press, Princeton, NJ, USA, 1950).
  * (16) B. Mashhoon, “Limitations of Spacetime Measurements”, Phys. Lett. A 143, 176-182 (1990).
  * (17) B. Mashhoon, “Hypothesis of Locality in Relativistic Physics”, Phys. Lett. A 145, 147-153 (1990).
  * (18) B. Mashhoon, “The Hypothesis of Locality and its Limitations”, in _Relativity in Rotating Frames_ , edited by G. Rizzi and M. L. Ruggiero (Kluwer Academic Publishers, Dordrecht, 2004), pp. 43-55. [arXiv:gr-qc/0303029]
  * (19) H. Weyl, _Space-Time-Matter_ (Dover, New York, 1952).
  * (20) B. Mashhoon, “Nonlocal Special Relativity”, Ann. Phys. (Berlin) 520, 705-727 (2008). [arXiv:0805.2926 [gr-qc]]
  * (21) B. Mashhoon, _Nonlocal Gravity_ (Oxford University Press, Oxford, UK, 2017).
  * (22) L. D. Landau and E. M. Lifshitz, _Mechanics_ (Pergamon, Oxford, UK, 1988).
  * (23) B. Mashhoon, “Gravitomagnetic Stern–Gerlach Force”, Entropy 23, no.4, 445 (2021). [arXiv:2102.06433 [gr-qc]]
  * (24) J. Larmor, “On the theory of the magnetic influence on spectra; and on the radiation from moving ions”, Phil. Mag. 44, 503-512 (1897).
  * (25) B. Mashhoon, “Modification of the Doppler effect due to the helicity - rotation coupling”, Phys. Lett. A 306, 66-72 (2002). [arXiv:gr-qc/0209079 [gr-qc]]
  * (26) J. C. Hauck and B. Mashhoon, “Electromagnetic waves in a rotating frame of reference”, Ann. Phys. (Berlin) 515, 275-288 (2003). [arXiv:gr-qc/0304069 [gr-qc]]
  * (27) A. R. Edmonds, _Angular Momentum in Quantum Mechanics_ (Princeton University Press, Princeton, NJ, 1960).
  * (28) P. J. Allen, “A radiation torque experiment”, Am. J. Phys. 34, 1185-1192 (1966).
  * (29) B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave plate”, Opt. Commun. 31, 1-3 (1979).
  * (30) B. A. Garetz, “Angular Doppler Effect”, J. Opt. Soc. Am. 71, 609-611 (1981).
  * (31) R. Simon, H. J. Kimble and E. C. G. Sudarshan, “Evolving Geometric Phase and Its Dynamical Manifestation as a Frequency Shift: An Optical Experiment”, Phys. Rev. Lett. 61, 19-22 (1988).
  * (32) B. Mashhoon, R. Neutze, M. Hannam and G. E. Stedman, “Observable frequency shifts via spin-rotation coupling”, Phys. Lett. A 249, 161-166 (1998). [arXiv: gr-qc/9808077]
  * (33) J. Courtial, K. Dholakia, D. A. Robertson, L. Allen and M. J. Padgett, “Measurement of the Rotational Frequency Shift Imparted to a Rotating Light Beam Possessing Orbital Angular Momentum”, Phys. Rev. Lett. 80, 3217-3219 (1998).
  * (34) J. Courtial, D. A. Robertson, K. Dholakia, L. Allen and M. J. Padgett, “Rotational Frequency Shift of a Light Beam”, Phys. Rev. Lett. 81, 4828-4830 (1998).
  * (35) B. Mashhoon, “On the Coupling of Intrinsic Spin with the Rotation of the Earth”, Phys. Lett. A 198, 9-13 (1995).
  * (36) E. P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group”, Ann. Math. 40, 149-204 (1939).
  * (37) B. Mashhoon, “Mach’s Principle and the Origin of Inertia”, Fundam. Theor. Phys. 183, 177-187 (2016). [arXiv:1509.01869 [gr-qc]]
  * (38) N. Ashby, “Relativity in the Global Positioning System”, Living Rev. Relativ. 6, 1 (2003).
  * (39) B. Demirel, S. Sponar and Y. Hasegawa, “Measurement of the spin-rotation coupling in neutron polarimetry”, New J. Phys. 17, 023065 (2015).
  * (40) A. Danner, B. Demirel, W. Kersten, R. Wagner, H. Lemmel, S. Sponar and Y. Hasegawa, “Spin-rotation coupling observed in neutron interferometry”, npj Quantum Information 6, 23 (2020). [arXiv:1904.07085 [quant-ph]]
  * (41) T. Yu, Z. Luo and G. E. W. Bauer, “Chirality as generalized spin-orbit interaction in spintronics”, Phys. Rept. 1009, 1-115 (2023). [arXiv:2206.05535 [cond-mat.mes-hall]]
  * (42) A. Einstein, in _Albert Einstein: Philosopher-Scientist_ , edited by P. A. Schilpp (Library of Living Philosophers, Evanston, IL, USA).
  * (43) N. Bohr and L. Rosenfeld, “Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen”, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 12, no. 8 (1933). [English translation: On the question of measurability of electromagnetic field quantities. In _Quantum Theory and Measurement_ , edited by J. A. Wheeler and W. H. Zurek (Princeton University Press, Princeton, NJ, USA, 1983)].
  * (44) B. Mashhoon, “Nonlocal theory of accelerated observers”, Phys. Rev. A 47, 4498-4501 (1993).
  * (45) B. Mashhoon, “Toward Nonlocal Electrodynamics of Accelerated Systems”, Universe 6, no.12, 229 (2020). [arXiv:2011.09251 [gr-qc]]
  * (46) B. Mashhoon, “Nonlocal Electrodynamics of Rotating Systems”, Phys. Rev. A 72, 052105 (2005). [arXiv: hep-th/0503205]
  * (47) B. Mashhoon, “Optics of Rotating Systems”, Phys. Rev. A 79, 062111 (2009). [arXiv:0903.1315 [gr-qc]]
  * (48) D. F. Jackson Kimball, J. Dudley, Y. Li and D. Patel, “In-situ measurement of light polarization with ellipticity-induced nonlinear magneto-optical rotation”, Phys. Rev. A 96, 033823 (2017). [arXiv:1706.08149 [physics.optics]]
  * (49) B. Mashhoon, M. Molaei and Y. N. Obukhov, “Spin-Gravity Coupling in a Rotating Universe”, Symmetry 15, no.8, 1518 (2023). [arXiv:2304.08835 [gr-qc]]
  * (50) J. D. Jackson, _Classical Electrodynamics_ , 3rd edn (Wiley, Hoboken, NJ, 1999).
  * (51) L. D. Landau and E. M. Lifshitz, _Electrodynamics of Continuous Media_ (Pergamon, Oxford, UK, 1960).
  * (52) F. W. Hehl and Y. N. Obukhov, _Foundations of Classical Electrodynamics: Charge, Flux, and Metric_ (Birkhäuser, Boston, MA, USA, 2003).
  * (53) S. Poisson, “Mémoire sur la théorie du magnétisme en mouvement”, Mém. Acad. Sci. France 6, 441-570 (1823).
  * (54) J. Hopkinson, “Residual Charge of the Leyden Jar.—Dielectric Properties of different Glasses”, Phil. Trans. Roy. Soc. London 167, 599-626 (1877).
  * (55) F. W. Hehl and B. Mashhoon, “Nonlocal Gravity Simulates Dark Matter”, Phys. Lett. B 673, 279-282 (2009). [arXiv:0812.1059 [gr-qc]]
  * (56) F. W. Hehl and B. Mashhoon, “Formal framework for a nonlocal generalization of Einstein’s theory of gravitation”, Phys. Rev. D 79, 064028 (2009). [arXiv:0902.0560 [gr-qc]]
  * (57) R. J. van den Hoogen, “Towards a covariant smoothing procedure for gravitational theories”, J. Math. Phys. 58, no.12, 122501 (2017).
  * (58) B. Mashhoon, “Nonlocal electrodynamics of linearly accelerated systems”, Phys. Rev. A 70, 062103 (2004). [arXiv: hep-th/0407278]
  * (59) G. N. Bremm and F. T. Falciano, “Nonlocal Effects in Black Body Radiation,” Ann. Phys. (Berlin) 527, 265-277 (2015). [arXiv:1601.04997 [gr-qc]]
  * (60) B. Mashhoon, “Nonlocal Special Relativity: Amplitude Shift in Spin-Rotation Coupling”, in _Proceedings of Mário Novello’s 70th Anniversary Symposium_ , edited by N. Pinto Neto and S. E. Perez Bergliaffa (Editora Livraria da Fisica, Sao Paulo, 2012), pp. 177-189. [arXiv:1204.6069 [gr-qc]]