# Toward Nonlocal Electrodynamics of Accelerated Systems

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), P.
O. Box 19395-5531, Tehran, Iran

###### Abstract

We revisit acceleration-induced nonlocal electrodynamics and the phenomenon of
photon spin-rotation coupling. The kernel of the theory for the
electromagnetic field tensor involves parity violation under the assumption of
linearity of the field kernel in the acceleration tensor. However, we show
that parity conservation can be maintained by extending the field kernel to
include quadratic terms in the acceleration tensor. The field kernel must
vanish in the absence of acceleration; otherwise, a general dependence of the
kernel on the acceleration tensor cannot be theoretically excluded. The
physical implications of the quadratic kernel are briefly discussed.

relativity, accelerated systems, nonlocal electrodynamics

###### pacs:

03.30+p, 11.10.Lm, 04.20.Cv

##  I Introduction

Consider an inertial observer $\mathcal{O}$ moving with constant velocity
$\mathbf{v}$ in an inertial frame of reference in Minkowski spacetime. The
observer’s 4-velocity $u^{\mu}=dx^{\mu}/d\tau$ can be expressed in Cartesian
coordinates $x^{\mu}=(t,x,y,z)$ as $u^{\mu}=\gamma(1,\mathbf{v})$, where
$\tau$ is the observer’s proper time, $d\tau=dt/\gamma$, and $\gamma$ is the
Lorentz factor. The observer measures the frequency $\omega$ of an incident
plane monochromatic electromagnetic wave with propagation vector
$k_{0}^{\mu}=(\omega_{0},\mathbf{k}_{0})$, where $\omega_{0}$ is the frequency
and $\mathbf{k}_{0}$ is the wave vector with $\omega_{0}=|\mathbf{k}_{0}|$, as
measured by inertial observers $\mathcal{O}_{0}$ at rest in the background
inertial frame. As is well known, $\omega$ is given by the relativistic
Doppler formula, namely,

$\omega=-\eta_{\mu\nu}\,k_{0}^{\mu}\,u^{\nu}=\gamma\,(\omega_{0}-\mathbf{v}\cdot\mathbf{k}_{0})\,.$
(1)

Throughout this paper, we use the convention that c = 1, unless specified
otherwise; moreover, Greek indices run from 0 to 3, while Latin indices run
from 1 to 3. The Minkowski metric tensor $\eta_{\mu\nu}$ is given by
diag$(-1,1,1,1)$.

Let us note that Eq. (1) can be expressed as
$-\omega_{0}\,dt+\mathbf{k}_{0}\cdot d\mathbf{x}=-\omega\,d\tau$, which
follows from the invariance of the phase of the wave under Lorentz
transformation. Indeed, if we consider the class of moving inertial observers
all at rest in the inertial frame of $\mathcal{O}$, we recover from the
Lorentz invariance of the phase the relativistic relations for both the
frequency $\omega$ and the corresponding wave vector $\mathbf{k}$, namely,

$\mathbf{k}=\mathbf{k}_{0}+\frac{1}{v^{2}}(\gamma-1)(\mathbf{v}\cdot\mathbf{k}_{0})\,\mathbf{v}-\gamma\,\mathbf{v}\,\omega_{0}\,,$
(2)

where $v=|\mathbf{v}|$.

The Lorentz invariance of the phase is a consequence of a more basic approach
that deals with the electromagnetic radiation field in the two Lorentz frames
under consideration here. Transforming the radiation field, either the gauge
field $A_{\mu}$ or the field tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, from the background
inertial frame to the moving inertial frame results in the invariance of the
phase of the radiation under Lorentz transformation.

We now wish to extend our treatment to accelerated observers, since actual
observers are all more or less accelerated. Let us for the moment extend the
theory of relativity to such observers in the standard manner by applying
Lorentz transformations point by point along their accelerated world lines AE
. The pointwise extension of Lorentz invariance is the content of the
hypothesis of locality that is at the foundation of the special and general
theories of relativity. The locality assumption implies that an accelerated
observer is, at each event along its world line, locally equivalent to an
otherwise identical momentarily comoving inertial observer. To avoid
unphysical situations when dealing with accelerated motion, we generally
assume that the acceleration is turned on at some initial time $t_{i}$ and,
after a finite interval of time, the acceleration is turned off at $t_{f}$;
moreover, for $t<t_{i}$ and $t>t_{f}$, the observer moves uniformly in an
inertial frame of reference. In most laboratory experiments, the accelerations
are small and the locality postulate is a good approximation; however, in
situations involving high accelerations the locality assumption may not be
adequate BM1 ; BM2 ; Maluf:2010fb ; Maluf:2011gt ; BMB .

By way of illustration, let us consider a thought experiment involving an
observer that for $t\geq 0$ rotates uniformly about the $z$ axis in the
$(x,y)$ plane of an inertial frame of reference. More specifically, for
$-\infty<t<0$, the observer under consideration moves uniformly along a
straight line parallel to the $y$ axis with speed $r\,\Omega_{0}$ in the
$(x,y)$ plane; that is, $x=r>0$ and $y=r\,\Omega_{0}\,t$. At $t=0$, the
observer is forced to move counterclockwise with uniform angular velocity
$\Omega_{0}>0$ on a circular orbit of radius $r$ about the $z$ axis. Thus, for
$t\geq 0$, $x=r\,\cos\varphi$ and $y=r\,\sin\varphi$, where
$\varphi=\Omega_{0}\,t$. The rotating observer with proper time
$\tau=t/\gamma$ measures the frequency of a plane monochromatic
electromagnetic wave that propagates along the $z$ axis. In this case of
normal incidence, the locality postulate leads to the local invariance of the
phase and we find from Eq. (1),

$\omega_{\rm D}=\gamma\,\omega_{0}\,,\qquad\gamma=(1-v^{2})^{-1/2}\,,\qquad
v=r\,\Omega_{0}\,,$ (3)

which is the expression for the transverse Doppler effect. This result can be
physically interpreted in terms of the phenomenon of time dilation. On the
other hand, transforming the radiation field to the rotating frame results in
Fourier components with frequencies

${\tilde{\omega}}_{\pm}=\gamma\,(\omega_{0}\mp\Omega_{0})\,.$ (4)

The actual calculation involved in this more general result can be carried out
using the explicit transformation of either $A_{\mu}$ or $F_{\mu\nu}$ to the
rotating frame BM3  or by means of projecting these fields on the natural
orthonormal tetrad frame of the rotating observer HaMa . An observer generally
carries an adapted orthonormal tetrad frame field
$\lambda^{\mu}{}_{\hat{\alpha}}$ such that

$\eta_{\mu\nu}\,\lambda^{\mu}{}_{\hat{\alpha}}\,\lambda^{\nu}{}_{\hat{\beta}}=\eta_{\hat{\alpha}\hat{\beta}}\,,\qquad\lambda^{\mu}{}_{\hat{0}}=\frac{dx^{\mu}}{d\tau}\,.$
(5)

For $t\geq 0$, the components of the rotating observer’s tetrad frame in
$(t,x,y,z)$ coordinates are given by

$\displaystyle\lambda^{\mu}{}_{\hat{0}}$
$\displaystyle=\gamma(1,-v\sin\varphi,v\cos\varphi,0)\,,$ (6)
$\displaystyle\lambda^{\mu}{}_{\hat{1}}$
$\displaystyle=(0,\cos\varphi,\sin\varphi,0)\,,$ (7)
$\displaystyle\lambda^{\mu}{}_{\hat{2}}$
$\displaystyle=\gamma(v,-\sin\varphi,\cos\varphi,0)\,,$ (8)
$\displaystyle\lambda^{\mu}{}_{\hat{3}}$ $\displaystyle=(0,0,0,1)\,.$ (9)

Let us note that the unit axes of the observer’s spatial frame
$\lambda^{\mu}{}_{\hat{i}}$, for $i=1,2,3$, point in the radial, tangential
and $z$ directions, respectively, in the cylindrical system of coordinates in
the background inertial frame.

The derivation of Eq. (4) is based on the application of the hypothesis of
locality to the radiation field together with the nonlocal operation of
Fourier analysis in time. Indeed, the measured frequency
${\tilde{\omega}}_{\pm}$ approaches the local Doppler result $\omega_{\rm D}$
as $\Omega_{0}/\omega_{0}\to 0$, since

${\tilde{\omega}}_{\pm}=\omega_{\rm
D}\,\left(1\mp\frac{\Omega_{0}}{\omega_{0}}\right)\,.$ (10)

Let us note that $\Omega_{0}/\omega_{0}$ is equal to the ratio of the reduced
wavelength of the incident radiation to $c/\Omega_{0}$, which is the
acceleration length of the rotating observer. Therefore, it follows from Eq.
(10) that we recover the pointwise transverse Doppler result in the eikonal
(or WKB) limit; moreover, phase invariance in inertial frames does not in
general extend to accelerated systems.

Ignoring time dilation for the moment,
${\tilde{\omega}}_{\pm}\approx\omega_{0}\mp\Omega_{0}$ comes about as a
consequence of the coupling of the helicity of the radiation with the rotation
of the observer. The incident radiation can be expressed in the circular
polarization basis as a linear superposition of positive and negative helicity
components. In the positive (negative) helicity case, static inertial
observers in the background global inertial frame observe the electric and
magnetic radiation fields rotate in the positive (negative) sense with
frequency $\omega_{0}$ about the direction of propagation of the wave. From
the viewpoint of the rotating observer, the electric and magnetic radiation
fields rotate with frequency $\omega_{0}-\Omega_{0}$ ($\omega_{0}+\Omega_{0}$)
in the positive (negative) helicity case. This “angular Doppler effect” has
been studied in classical optics; see BM4  and the references cited therein.
The helicity-rotation coupling is responsible for the phenomenon of phase
wrap-up in the GPS Ash . It is also the origin of a certain frequency shift
that occurs when circularly polarized radiation passes through a rotating spin
flipper; indeed, for $\Omega_{0}\ll\omega_{0}$, this phenomenon has been
observationally known for a long time PJA ; MNHS .

Imagine a uniformly rotating half-wave plate (HWP) instead of the rotating
observer in our thought experiment. At the lower edge of the plate the
frequency of an incident positive-helicity wave is
$\omega^{\prime}\approx\omega_{0}-\Omega_{0}$. Inside the plate, the frequency
remains constant and equal to $\omega^{\prime}$ as the rotation of the plate
is assumed to be uniform. At the upper edge of the plate, the connection of
$\omega^{\prime}$ with the frequency $\omega_{1}$ of the emerging negative-
helicity wave is $\omega^{\prime}\approx\omega_{1}+\Omega_{0}$, so that
$\omega_{1}-\omega_{0}\approx-2\,\Omega_{0}$. There is therefore a downshift
in the frequency of the radiation as it passes through the spin flipper. A
corresponding upshift would occur for incident negative-helicity radiation.
The general formula for the energy shift is $\mp 2\hbar\,s\,\Omega_{0}$, where
$s$ is the spin of the radiation field MK . Further discussion of helicity-
rotation coupling for photons is contained in Refs. Mash ; AnMa ; HaMa ; BMash
; Bliokh:2015yhi . The spin-rotation coupling is a general phenomenon that is
a manifestation of the inertia of intrinsic spin and has extensive
observational support; in fact, it has been recently observed directly in
neutron interferometry BM5 ; HN ; BM5a ; SoTi ; LR ; SP ; Pa ; ShHe ; LaPa ;
Ran ; Pan:2011zza ; Arminjon:2013kxa ; Werner ; HRW ; DSH ; DDSH ; DDKWLSH ;
MISM ; MISM2 ; MIHSM ; MIM ; CB ; IMM ; PAP ; KYM ; KMN ; NaTa ; KMMN .

Let us now return to Eq. (4) and note that this equation has observational
support only for $\Omega_{0}\ll\omega_{0}$. Are there higher-order terms in
Eq. (4)? For instance, could there be terms of order
$(\Omega_{0}/\omega_{0})^{2}$ or higher in the wave expansion (10)? The
existence of such terms is quite unlikely; otherwise, there would be
divergence as $\omega_{0}\to 0$. Various aspects of such problems have been
discussed in Ref. BM3 . There is, however, one important issue that we need to
address, namely, the possibility that for incident positive-helicity radiation
of frequency $\omega_{0}=\Omega_{0}$, we have ${\tilde{\omega}}_{+}=0$. By a
mere rotation of angular velocity $\Omega_{0}=\omega_{0}$, the whole radiation
field stands completely still with respect to the rotating observer. This is
not possible for inertial observers in Minkowski spacetime, since $\omega=0$
implies $\omega_{0}=0$ in Eq. (1). However, before the advent of special
relativity theory, an observer could in principle move at the speed of light
and hence stay completely at rest with a radiation field. The situation here
is therefore reminiscent of the pre-relativistic Doppler effect; indeed,
Einstein mentioned such a puzzling possibility in connection with the pre-
relativistic Doppler formula in his autobiographical notes Sch . This
conceptual difficulty was removed for inertial observers via the Lorentz
invariance of physical laws. We do the same via an appropriate nonlocal
extension of the locality hypothesis of the standard special relativity
theory.

The hypothesis of locality, as applied to the electromagnetic radiation field,
is the origin of the circumstance that a radiation field can in principle
stand completely still with respect to a rotating observer. The locality
postulate in turn originates from classical mechanics where the state of a
classical particle is determined by its position and velocity. On the other
hand, in accordance with the Huygens principle, wave phenomena are in general
nonlocal. In 1993, a nonlocal theory of accelerated systems was put forward in
which the past history of accelerated observers was taken into account via
kernels that would be so constructed as to mitigate the conceptual difficulty
of radiation standing completely still BM6 . Using electrons in
electromagnetic fields as accelerated systems in the correspondence limit, we
have shown that the nonlocal theory is in better agreement with quantum
mechanics than the standard theory based on the locality postulate BM6a . We
now turn to a brief description of the theory of acceleration-induced
nonlocality, which leads to nonlocal special relativity theory BM7 ; BF .

###  I.1 Nonlocality of Accelerated Systems

Imagine an accelerated observer following a path $x^{\mu}(\tau)$ in Minkowski
spacetime in the presence of a field $\psi(x)$, where $x$ here represents an
event $x^{\mu}$. What is the field that the accelerated observer measures?
According to the locality postulate of the standard relativity theory, Lorentz
invariance can be applied point by point along the path; in this way, the
observer measures a field

${\tilde{\psi}}(\tau)=\Lambda(\tau)\,\psi(x(\tau))\,,$ (11)

where $\Lambda$ is a matrix representation of the Lorentz group. On the other
hand, according to Bohr and Rosenfeld, it is not possible to measure even the
classical electromagnetic field at a spacetime event by inertial observers;
indeed, a spacetime averaging procedure is necessarily involved in such a
measurement process BR1 ; BR2 .

Following Bohr and Rosenfeld BR1 , consider for the sake of illustration an
extended body of volume $V$ and uniform charge density $\bar{\rho}$ that is
placed in an external electric field $\mathbf{E}(t,\mathbf{x})$. The Lorentz
force law is indispensable for the measurement of the electromagnetic field;
indeed, the electric and magnetic fields are defined via the Lorentz force
law. In this case, we have

$\frac{d\mathbf{P}}{dt}=\bar{\rho}\,\int_{V}\mathbf{E}(t,\mathbf{x})\,d^{3}x\,.$
(12)

Measuring the momentum of the extended charged body $\mathbf{P}$ at
$t^{\prime}$ and $t^{\prime\prime}>t^{\prime}$, we get

$\mathbf{P}(t^{\prime\prime})-\mathbf{P}(t^{\prime})=\bar{\rho}\,\int_{t^{\prime}}^{t^{\prime\prime}}\int_{V}\mathbf{E}(t,\mathbf{x})\,d^{3}x\,;$
(13)

thus, the field as measured by static inertial observers is given by

$<\mathbf{E}>~{}=\frac{1}{\Delta}\int_{\Delta}\mathbf{E}(x)\,d^{4}x\,,$ (14)

where $\Delta=(t^{\prime\prime}-t^{\prime})\,V$.

Extending the Bohr–Rosenfeld argument to accelerated systems is a daunting
task. Instead, we implement the averaging requirement as follows: Let
$\Psi(\tau)$ be the field actually measured by the accelerated observer; then,
the most general linear relationship linking $\Psi(\tau)$ to
${\tilde{\psi}}(\tau)=\Lambda(\tau)\,\psi(x(\tau))$ that is consistent with
causality is given by

$\Psi(\tau)={\tilde{\psi}}(\tau)+\int_{\tau_{0}}^{\tau}\mathcal{K}(\tau,\tau^{\prime})\,{\tilde{\psi}}(\tau^{\prime})\,d\tau^{\prime}\,$
(15)

for $\tau\geq\tau_{0}$, where $\tau_{0}$ is the instant at which the
acceleration is turned on. In this ansatz, which is consistent with the
superposition principle, the locality postulate is extended by a certain
average over the past world line of the observer.

The Volterra integral equation of the second kind (15) relates $\psi$ to
$\Psi$ via ${\tilde{\psi}}$ given by Eq. (11); indeed, the resulting
connection is unique in the space of continuous functions by Volterra’s
theorem, which has been extended to the space of square-integrable functions
by Tricomi Volt ; Tric . The Volterra–Tricomi uniqueness result is an
important feature of the nonlocal theory of accelerated systems.

The kernel of ansatz (15), $\mathcal{K}(\tau,\tau^{\prime})$, is expected to
vanish in the absence of acceleration; moreover, the kernel should be such
that a basic radiation field could never stand completely still with respect
to an accelerated observer. Detailed investigations have revealed that a
consistent theory can be developed based on

$\mathcal{K}(\tau,\tau^{\prime})=K(\tau^{\prime})=-\frac{d\Lambda(\tau^{\prime})}{d\tau^{\prime}}\,\Lambda^{-1}(\tau^{\prime});$
(16)

see Ref. BM7  and the references cited therein. The formula for $K(\tau)$,
which vanishes in the absence of the acceleration of the observer, simply
follows from the assumption that a constant field $\psi$ will lead to a
constant $\Psi$ in ansatz (15). The Volterra–Tricomi uniqueness theorem then
implies that an actual radiation field will never be constant as measured by
accelerated observers. Moreover, after the acceleration has been turned off at
$\tau_{f}$, the memory of past acceleration remains as a _constant_ addition
to the measured field; that is, for $\tau\geq\tau_{f}$,

$\Psi(\tau)={\tilde{\psi}}(\tau)+\int_{\tau_{0}}^{\tau_{f}}K(\tau^{\prime})\,{\tilde{\psi}}(\tau^{\prime})\,d\tau^{\prime}\,.$
(17)

The nonlocal theory of accelerated systems has been applied to the Dirac field
with satisfactory results BM8 . There is a problem, however, in the case of
the electromagnetic field $F_{\mu\nu}$ due to the existence of electrostatic
and magnetostatic fields. To see this, let us note that with the substitution
of kernel (16), ansatz (15) can be written as

$\Psi(\tau)={\tilde{\psi}}(\tau_{0})+\int_{\tau_{0}}^{\tau}\Lambda(\tau^{\prime})\,\frac{d\psi(\tau^{\prime})}{d\tau^{\prime}}\,d\tau^{\prime}\,.$
(18)

It follows that in a static background electromagnetic field, an accelerated
observer will always measure a constant field ${\tilde{\psi}}(\tau_{0})$,
which contradicts experimental results involving small accelerations that
appear to be in general agreement with the locality postulate of the special
relativity theory Ken ; Peg ; Swa .

To resolve this difficulty, one can assume that kernel $K$ given by Eq. (16)
is appropriate for the vector potential $A_{\mu}$, while a different kernel
should be chosen for the field tensor $F_{\mu\nu}$. A specific choice for such
a kernel involving parity violation has been put forward in Ref. BM9 . The
purpose of the present paper is to re-examine our previous proposal and
provide a deeper and more general analysis of the problem.

##  II Nonlocal Electrodynamics

According to the hypothesis of locality, the vector potential $A_{\mu}$ and
the field tensor $F_{\mu\nu}$ as measured by the accelerated observer are

$\tilde{A}_{\hat{\alpha}}=A_{\mu}\,\lambda^{\mu}{}_{\hat{\alpha}}\,,\qquad\tilde{F}_{\hat{\alpha}\hat{\beta}}=F_{\mu\nu}\,\lambda^{\mu}{}_{\hat{\alpha}}\,\lambda^{\nu}{}_{\hat{\beta}}\,.$
(19)

On the other hand, we claim that the actually measured fields after the
acceleration is turned on at $\tau_{0}$ are

$\displaystyle\mathcal{A}_{\hat{\alpha}}(\tau)$
$\displaystyle=\tilde{A}_{\hat{\alpha}}(\tau)+\int_{\tau_{0}}^{\tau}K_{\hat{\alpha}}{}^{\hat{\beta}}(\tau^{\prime})\,\tilde{A}_{\hat{\beta}}(\tau^{\prime})\,d\tau^{\prime}\,,$
(20) $\displaystyle\mathcal{F}^{\hat{\alpha}\hat{\beta}}(\tau)$
$\displaystyle=\tilde{F}^{\hat{\alpha}\hat{\beta}}(\tau)+\int_{\tau_{0}}^{\tau}K^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}(\tau^{\prime})\,\tilde{F}^{\hat{\gamma}\hat{\delta}}(\tau^{\prime})\,d\tau^{\prime}\,.$
(21)

Once the kernels are fixed, other aspects of the nonlocal electrodynamics of
accelerated systems can be determined. For the kernel of Eq. (20) involving
the vector potential $A_{\mu}$, the simplest possibility would be to choose
the kernel given by Eq. (16). If $A_{\mu}$ is constant, then $F_{\mu\nu}=0$
everywhere and via Eq. (18) we have that
$\mathcal{A}_{\hat{\alpha}}(\tau)=\tilde{A}_{\hat{\alpha}}(\tau_{0})$ is
constant as well. This circumstance does not appear to pose any difficulty;
moreover, it is clear that under the gauge transformation $A_{\mu}\mapsto
A_{\mu}+\partial_{\mu}\Sigma$,
$\tilde{A}_{\hat{\alpha}}\mapsto\tilde{A}_{\hat{\alpha}}+(\partial_{\mu}\Sigma)\,\lambda^{\mu}{}_{\hat{\alpha}}$
and, similarly, $\mathcal{A}_{\hat{\alpha}}$ will be subject to a nonlocal
gauge transformation, but Eq. (21) will remain invariant.

The kernels depend upon the acceleration of the observer; therefore, a
description of the acceleration tensor is required.

###  II.1 Acceleration Tensor

Along the observer’s world line, the motion of the adapted tetrad frame can be
expressed as

$\frac{d\lambda^{\mu}{}_{\hat{\alpha}}}{d\tau}=\phi_{\hat{\alpha}}{}^{\hat{\beta}}\,\lambda^{\mu}{}_{\hat{\beta}}\,,\qquad\phi_{\hat{\alpha}\hat{\beta}}=-\phi_{\hat{\beta}\hat{\alpha}}\,.$
(22)

The antisymmetry of the acceleration tensor $\phi_{\hat{\alpha}\hat{\beta}}$
is a simple consequence of the orthonormality of the adapted tetrad frame.

In general, the acceleration tensor has a decomposition,
$\phi_{\hat{\alpha}\hat{\beta}}\mapsto(-\mathbf{g},\bm{\Omega})$, in close
analogy with the electromagnetic field tensor, namely,
$F_{\mu\nu}\mapsto(\mathbf{E},\mathbf{B})$, where $F_{0i}=-E_{i}$ and
$F_{ij}=\epsilon_{ijk}\,B^{k}$ in our convention. The spacetime invariants
$\mathbf{g}$ and $\bm{\Omega}$ have local components with respect to the
adapted frame. The “electric” part of the acceleration tensor corresponds to
the translational acceleration of the observer,
$g_{\hat{i}}=\phi_{\hat{0}\hat{i}}$, while the “magnetic” part,
$\Omega_{\hat{i}}=\frac{1}{2}\epsilon_{\hat{i}\hat{j}\hat{k}}\,\phi^{\hat{j}\hat{k}}$,
corresponds to the angular velocity of the spatial frame of the observer
relative to a nonrotating frame that is Fermi-Walker transported along the
observer’s world line.

###  II.2 Acceleration Kernels

It follows from inspection of Eqs. (11) and (16) as well as (19) and (22) that
for the kernel of the vector potential we have

$K_{\hat{\alpha}}{}^{\hat{\beta}}=-\phi_{\hat{\alpha}}{}^{\hat{\beta}}\,,$
(23)

which we must use in Eq. (20).

Let us next turn to the evaluation of the field kernel in Eq. (21),
$K^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}$, which is
antisymmetric in its first and second pairs of indices and is the main subject
of the present paper. It should be constructed out of the Minkowski metric
tensor $\eta_{\hat{\alpha}\hat{\beta}}$, the Levi-Civita tensor
$\epsilon_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}$ with
$\epsilon_{\hat{0}\hat{1}\hat{2}\hat{3}}=1$ and the acceleration tensor
$\phi_{\hat{\alpha}\hat{\beta}}$. In this connection, the field kernel
corresponding to Eq. (16) is given by

$\kappa_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}=-\frac{1}{2}(\phi_{\hat{\alpha}\hat{\gamma}}\,\eta_{\hat{\beta}\hat{\delta}}+\phi_{\hat{\beta}\hat{\delta}}\,\eta_{\hat{\alpha}\hat{\gamma}}-\phi_{\hat{\beta}\hat{\gamma}}\,\eta_{\hat{\alpha}\hat{\delta}}-\phi_{\hat{\alpha}\hat{\delta}}\,\eta_{\hat{\beta}\hat{\gamma}})\,,$
(24)

which, as explained before, cannot be the electromagnetic field kernel by
itself. Indeed, in previous work BM9 , we assumed

$K_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}=p\,\kappa_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}+q\,^{*}\kappa_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}\,,$
(25)

where $p$ and $q$ are constant real dimensionless parameters and
${}^{*}\kappa$ is the dual of $\kappa$,

$^{*}\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}=\frac{1}{2}\,\epsilon^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}\,.$
(26)

The right and left duals of $\kappa$ turn out to be equal as a consequence of
the antisymmetry of the acceleration tensor. Clearly, the existence of $q\neq
0$ violates parity conservation. On the other hand, the adoption of kernel
(25) was based on the additional assumption that $K$ is linearly dependent
upon the acceleration tensor $\phi_{\hat{\alpha}\hat{\beta}}$.

Let us observe that acceleration-induced nonlocal electrodynamics shares
similarities with the nonlocal electrodynamics of media MuHM ; HeOb . Even in
the linear electrodynamics of media, there is evidence for parity violation;
see Refs. Hehl:2007jy ; Hehl:2007ut ; Hehl:2009eqa ; Essin:2008rq  and the
references cited therein. However, no corresponding observational evidence
regarding accelerated systems is available at present. The main purpose of the
present work is to point out that one can remove the linearity constraint on
the field kernel and thereby develop an acceleration-induced nonlocal
electrodynamics that preserves parity conservation. The mathematical form of
the various possible field kernels are explored in the last part of Section
IV.

Specifically, in the present work we assume a field kernel that is quadratic
in the acceleration tensor and is of the form

$K^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}=\mu\,\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}+\nu\,\phi^{\hat{\alpha}\hat{\beta}}\,\phi_{\hat{\gamma}\hat{\delta}}\,,$
(27)

where $\mu$ and $\nu$ are constant real parameters that carry dimensions of
length. The new field kernel is symmetric; that is,
$K_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}=K_{\hat{\gamma}\hat{\delta}\hat{\alpha}\hat{\beta}}$.
In this paper, we work out the physical consequences of kernel (27) for the
sake of illustration; in principle, a parity conserving kernel could be a sum
of linear, quadratic and higher-order terms in the acceleration tensor.

##  III Parity Violation

Soon after the discovery of parity violation in the weak interactions,
Stueckelberg pointed out the theoretical possibility of parity violation in
gravity due to the existence of the covariantly constant Levi-Civita totally
antisymmetric tensor field $\epsilon_{\mu\nu\rho\sigma}$ in general relativity
Stu . Indeed, this tensor leads to the definition of dual fields that could
help induce parity violation. An instance of this circumstance is given in Eq.
(25), the kernel of previously published nonlocal electrodynamics of
accelerated systems BM9 ; BM7 ; BMB . Observational consequences of the
possible breakdown of parity and time-reversal invariance in the gravitational
interaction have been considered by a number of authors; see, for instance,
Refs. KoOk ; LeOk ; WiRa ; BaMa1 ; BaMa2 ; Freidel:2005sn ; Ivanov:2016krk ;
Conroy:2019ibo ; ZZQ ; Iosifidis:2020dck ; Obukhov:2020zal  and the references
cited therein. On the other hand, there is no parity violation in standard
general relativity. Furthermore, parity conservation is maintained in the
standard electrodynamics of accelerated systems and gravitational fields; see
Appendix A for a detailed discussion.

In acceleration-induced nonlocal electrodynamics, when one assumes that the
kernel is only linear in acceleration, the existence of parity violating term
in Eq. (25) becomes absolutely necessary; otherwise, there would be
inconsistency in the physical results of the theory. A somewhat similar
situation occurs in the gravitational context involving the torsion
pseudovector in the constitutive relation of nonlocal gravity HM1 ; HM2 ; BMB
. A brief discussion of parity violation in nonlocal gravity is contained in
Section VIII.

There is no observational data regarding accelerated systems that could be
used to exclude completely parity violation in electrodynamics. The main
purpose of the present paper is to examine quadratic terms in acceleration
that could supplant parity violating terms in the kernel. The rest of this
paper is devoted to the derivation of some consequences of parity conserving
nonlocal electrodynamics of accelerated systems. To this end, it is convenient
to express the main integral relation (21) in matrix form.

##  IV Matrix Representation

For the sake of simplicity, we wish to represent our main field ansatz as a
matrix equation. We use the antisymmetry of the field tensor $F^{\mu\nu}$ and
assume that its pair of indices in effect takes the values
$\\{01,02,03,23,31,12\\}$ and we can then take care of the remaining
possibilities by means of a simple factor of $2$. Thus, we replace
$F^{\mu\nu}$ by a column 6-vector $F$ with components $\mathbf{E}$ and
$\mathbf{B}$, since $E_{i}=F^{0i}$ and
$B_{i}=\frac{1}{2}\,\epsilon_{ijk}\,F^{jk}$ in our convention,

$F=\left[\begin{array}[]{cc}\mathbf{E}\cr\mathbf{B}\cr\end{array}\right]\,.$
(28)

Similarly, we replace kernel
$K^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}$ by a $6\times 6$
matrix $\mathbb{K}$ , where the first and second pair of indices range over
the set $\\{01,02,03,23,31,12\\}$. Equation (21) then takes the form

$\mathcal{F}(\tau)=\tilde{F}(\tau)+2\int_{\tau_{0}}^{\tau}\mathbb{K}(\tau^{\prime})\,\tilde{F}(\tau^{\prime})\,d\tau^{\prime}\,,$
(29)

where $\tilde{F}=\Lambda\,F$ and $\Lambda$ is a $6\times 6$ matrix that can be
determined from the adapted tetrad frame field.

###  IV.1 Relations Involving Kernels

We collect here a number of relations that will be useful in explicit
applications. It follows from Eq. (24) that for the special kernel $\kappa$ we
have

$\kappa_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}=-\kappa_{\hat{\gamma}\hat{\delta}\hat{\alpha}\hat{\beta}}\,,\qquad\kappa^{\mu}{}_{\hat{\alpha}\hat{\mu}\hat{\beta}}=-\phi_{\hat{\alpha}\hat{\beta}}\,,\qquad\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\alpha}\hat{\beta}}=0\,.$
(30)

Moreover, ${}^{*}\kappa=\kappa^{*}$ with

$\kappa^{*}_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}=\frac{1}{2}\,\kappa_{\hat{\alpha}\hat{\beta}}{}^{\hat{\rho}\hat{\sigma}}\,\epsilon_{\hat{\rho}\hat{\sigma}}{}^{\hat{\gamma}\hat{\delta}}\,=-\frac{1}{2}\,(\phi_{\hat{\alpha}}{}^{\hat{\rho}}\,\epsilon_{\hat{\rho}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}-\phi_{\hat{\beta}}{}^{\hat{\rho}}\,\epsilon_{\hat{\rho}\hat{\alpha}}{}^{\hat{\gamma}\hat{\delta}})\,.$
(31)

The mixed dual

$\zeta_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}=\frac{1}{2}\,\kappa_{\hat{\alpha}\hat{\rho}\hat{\sigma}\hat{\beta}}\,\epsilon^{\hat{\rho}\hat{\sigma}\hat{\gamma}\hat{\delta}}\,$
(32)

is symmetric in its first pair of indices

$\zeta_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}=\frac{1}{4}\,(\kappa_{\hat{\alpha}\hat{\rho}\hat{\sigma}\hat{\beta}}-\kappa_{\hat{\alpha}\hat{\sigma}\hat{\rho}\hat{\beta}})\,\epsilon^{\hat{\rho}\hat{\sigma}\hat{\gamma}\hat{\delta}}=\frac{1}{4}\,(\kappa_{\hat{\alpha}\hat{\rho}\hat{\sigma}\hat{\beta}}+\kappa_{\hat{\beta}\hat{\rho}\hat{\sigma}\hat{\alpha}})\,\epsilon^{\hat{\rho}\hat{\sigma}\hat{\gamma}\hat{\delta}}\,.$
(33)

It is therefore not suitable as a field kernel in any case; here, we correct
the erroneous assertion that the mixed dual vanishes BM9 ; BM7 .

In expression (24) for the special kernel $\kappa$, we can replace the
acceleration tensor $\phi_{\hat{\alpha}\hat{\beta}}$ by its dual
$\phi^{*}_{\hat{\alpha}\hat{\beta}}$ and thereby obtain
$\chi_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\delta}}$; then,

$\chi_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}=\kappa^{*}_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}\,,\qquad\chi^{*}_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}=-\kappa_{\hat{\alpha}\hat{\beta}}{}^{\hat{\gamma}\hat{\delta}}\,,$
(34)

so that the double duality operation equals minus the identity operation in
this case.

Regarding quadratic terms in the acceleration tensor, we recall that

$\phi_{\hat{\alpha}\hat{\beta}}\,\phi^{\hat{\alpha}\hat{\beta}}=-2\,(g^{2}-\Omega^{2})\,,\qquad\phi^{*}_{\hat{\alpha}\hat{\beta}}\,\phi^{\hat{\alpha}\hat{\beta}}=-2\,\mathbf{g}\cdot\bm{\Omega}\,.$
(35)

Using a function $W$ of these invariants, we can form field kernels of the
form $W\,\epsilon^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}$.
Moreover, we find

$\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}\,\kappa^{\hat{\gamma}\hat{\delta}}{}_{\hat{\rho}\hat{\sigma}}=\phi^{\hat{\alpha}}{}_{\hat{\rho}}\,\phi^{\hat{\beta}}{}_{\hat{\sigma}}-\phi^{\hat{\alpha}}{}_{\hat{\sigma}}\,\phi^{\hat{\beta}}{}_{\hat{\rho}}+\frac{1}{2}\,(\phi^{\hat{\alpha}}{}_{\hat{\gamma}}\,\phi^{\hat{\gamma}}{}_{\hat{\rho}}\,\delta^{\hat{\beta}}_{\hat{\sigma}}-\phi^{\hat{\alpha}}{}_{\hat{\gamma}}\,\phi^{\hat{\gamma}}{}_{\hat{\sigma}}\,\delta^{\hat{\beta}}_{\hat{\rho}}-\phi^{\hat{\beta}}{}_{\hat{\gamma}}\,\phi^{\hat{\gamma}}{}_{\hat{\rho}}\,\delta^{\hat{\alpha}}_{\hat{\sigma}}+\phi^{\hat{\beta}}{}_{\hat{\gamma}}\,\phi^{\hat{\gamma}}{}_{\hat{\sigma}}\,\delta^{\hat{\alpha}}_{\hat{\rho}})\,$
(36)

and

$\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}\,\kappa^{\hat{\gamma}\hat{\delta}}{}_{\hat{\alpha}\hat{\beta}}=-2\,\phi_{\hat{\rho}\hat{\sigma}}\,\phi^{\hat{\rho}\hat{\sigma}}\,.$
(37)

In connection with the matrix representation, a detailed treatment reveals
that for the special kernel $\kappa$ we have

$\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}\mapsto\kappa=\frac{1}{2}\begin{bmatrix}\kappa_{1}&-\kappa_{2}\\\
\kappa_{2}&\kappa_{1}\end{bmatrix}\,,$ (38)

where the $3\times 3$ matrices $\kappa_{1}$ and $\kappa_{2}$ are given by

$\kappa_{1}=\bm{\Omega}\cdot\mathbf{I}\,,\qquad\kappa_{2}=\mathbf{g}\cdot\mathbf{I}\,.$
(39)

Here, $I_{i}$ is proportional to the operator of infinitesimal rotations about
the $x^{i}$ axis, so that $I_{1}$, $I_{2}$ and $I_{3}$ form a basis for the
Lie algebra of the rotation group as a 3-dimensional vector space; that is,

$(I_{i})_{jk}=-\epsilon_{ijk}\,,\qquad[I_{i},I_{j}]=\epsilon_{ijk}\,I_{k}\,.$
(40)

Let us note that in our matrix representation

$\epsilon^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}\mapsto\begin{bmatrix}0&-I\\\
I&0\end{bmatrix}\,,$ (41)

where $I=\,$diag$(1,1,1)$ is the identity matrix; therefore, Eq. (26) results
in

$^{*}\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}\mapsto{{}^{\ast}\kappa}=\frac{1}{2}\begin{bmatrix}-\kappa_{2}&-\kappa_{1}\\\
\kappa_{1}&-\kappa_{2}\end{bmatrix}\,.$ (42)

Using the matrix representation, it is straightforward to show that

$\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}=-^{*}\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,^{*}\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}\,,\quad^{*}\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}=\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,^{*}\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}\,.$
(43)

Furthermore,

$\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}\mapsto\frac{1}{2}\begin{bmatrix}\kappa_{1}^{2}-\kappa_{2}^{2}&-(\kappa_{1}\,\kappa_{2}+\kappa_{2}\,\kappa_{1})\\\
\kappa_{1}\,\kappa_{2}+\kappa_{2}\,\kappa_{1}&\kappa_{1}^{2}-\kappa_{2}^{2}\end{bmatrix}\,,$
(44)

where

$\mathrm{tr}(\kappa_{1}^{2})=-2\,\Omega^{2}\,,\qquad\mathrm{tr}(\kappa_{2}^{2})=-2\,g^{2}\,.$
(45)

For the other quadratic kernel in Eq. (27), we find the $6\times 6$
representation

$\phi^{\hat{\alpha}\hat{\beta}}\,\phi_{\hat{\gamma}\hat{\delta}}\mapsto\begin{bmatrix}-g_{i}\,g_{j}&-g_{i}\,\Omega_{j}\\\
\Omega_{i}\,g_{j}&\Omega_{i}\,\Omega_{j}\end{bmatrix}\,;$ (46)

indeed, similar expressions exist for
${}^{*}\phi^{\hat{\alpha}\hat{\beta}}\,\phi_{\hat{\gamma}\hat{\delta}}$ as
well as
$\phi^{\hat{\alpha}\hat{\beta}}\,{}^{*}\phi_{\hat{\gamma}\hat{\delta}}$.

Assuming parity conservation, our quadratic kernel $\mathbb{K}$ given by Eq.
(27) has a matrix representation that is given by $\mu$ times the matrix in
Eq. (44) plus $\nu$ times the matrix in Eq. (46); that is,

$\mathbb{K}=\frac{\mu}{2}\,\begin{bmatrix}\kappa_{1}^{2}-\kappa_{2}^{2}&-(\kappa_{1}\,\kappa_{2}+\kappa_{2}\,\kappa_{1})\\\
\kappa_{1}\,\kappa_{2}+\kappa_{2}\,\kappa_{1}&\kappa_{1}^{2}-\kappa_{2}^{2}\end{bmatrix}+\nu\,\begin{bmatrix}-g_{i}\,g_{j}&-g_{i}\,\Omega_{j}\\\
\Omega_{i}\,g_{j}&\Omega_{i}\,\Omega_{j}\end{bmatrix}\,.$ (47)

We examine the physical consequences of the quadratic kernel (27) in the next
three sections. In Section V, we use the matrix formalism to explore helicity-
rotation coupling, while in Sections VI and VII we study the case of constant
background electromagnetic fields. We will find that of the two terms in our
proposed quadratic kernel (47), only the first term gives reasonable results,
while the second term must vanish (i.e., $\nu=0$). Section VIII contains a
general discussion of our results.

##  V Influence of the Quadratic Kernel on Helicity-Rotation Coupling

The purpose of this section is to examine the implications of kernel (47) for
the phenomenon of spin-rotation coupling. We therefore imagine an incident
plane monochromatic electromagnetic wave that is normally incident on the
rotating observer under consideration in this work. Equation (29) is linear by
assumption; hence, we can employ complex fields in our calculations and assign
physical significance to their real parts in our convention. We represent the
incident wave in the circular polarization basis by

$F_{\pm}(t,\mathbf{x})=i\,\omega_{0}\,a_{\pm}\begin{bmatrix}\mathbf{e}_{\pm}\\\
\mathbf{b}_{\pm}\end{bmatrix}e^{-i\,\omega_{0}\,(t-z)},$ (48)

where

$\mathbf{e_{\pm}}=\frac{1}{\sqrt{2}}\,(\widehat{\mathbf{x}}\pm
i\,\widehat{\mathbf{y}})\,,\qquad\mathbf{b}_{\pm}=\mp\,i\,\mathbf{e}_{\pm}\,,\qquad\mathbf{e}_{\pm}\cdot\mathbf{e}^{\ast}_{\pm}=1\,$
(49)

and the upper (lower) sign represents positive (negative) helicity radiation.
Here, $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are unit vectors in
the $x$ and $y$ directions, respectively, while $\omega_{0}$ is the frequency
and $a_{+}$ ($a_{-}$) is the constant amplitude of positive (negative)
helicity incident radiation.

Next, we imagine the instantaneous projection of this field on the tetrad
frame (6)–(9) of the rotating observer. The acceleration tensor for the
rotating observer has a translational part that corresponds to the centripetal
acceleration $\mathbf{g}$ and a rotational part that corresponds to the
angular velocity $\bm{\Omega}$ with respect to a locally nonrotating frame.
These have components $\mathbf{g}=-v\,\gamma^{2}\,\Omega_{0}(1,0,0)$ and
$\bm{\Omega}=\gamma^{2}\,\Omega_{0}(0,0,1)$ with respect to the local adapted
spatial frame $\lambda^{\mu}{}_{\hat{i}}$, $i=1,2,3$, that indicate the
radial, tangential and $z$ directions, respectively. The projected field
$\tilde{F}$ is given by $\tilde{F}=\Lambda\,F$, where the $6\times 6$ matrix
$\Lambda$ is obtained from Eq. (19) and can be expressed as

$\Lambda=\begin{bmatrix}\Lambda_{1}&-\Lambda_{2}\\\
\Lambda_{2}&\Lambda_{1}\end{bmatrix}\,,$ (50)

where

$\Lambda_{1}=\begin{bmatrix}\gamma\cos\varphi&\gamma\sin\varphi&0\\\
-\sin\varphi&\cos\varphi&0\\\
0&0&\gamma\end{bmatrix}\,,\qquad\Lambda_{2}=v\,\gamma\begin{bmatrix}0&0&-1\\\
0&0&0\\\ \cos\varphi&\sin\varphi&0\end{bmatrix}\,.$ (51)

Let us recall here that $\varphi=\Omega_{0}\,t=\gamma\,\Omega_{0}\,\tau$.

According to the locality postulate, the projected field $\tilde{F}$ measured
by the momentarily comoving inertial observers along the world line of the
rotating observer is given by

$\tilde{F}_{\pm}(\tau)=i\,\gamma\,\omega_{0}\,a_{\pm}\begin{bmatrix}\tilde{\mathbf{e}}_{\pm}\\\
\tilde{\mathbf{b}}_{\pm}\end{bmatrix}e^{-i\,\tilde{\omega}_{\pm}\tau}\,,$ (52)

where

$\tilde{\mathbf{e}}_{\pm}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\ \pm
i\,\gamma^{-1}\\\ \pm
i\,v\end{bmatrix}\,,\qquad\tilde{\mathbf{b}}_{\pm}=\mp\,i\,\tilde{\mathbf{e}}_{\pm}\,,\qquad\tilde{\mathbf{e}}_{\pm}\cdot\tilde{\mathbf{e}}_{\pm}^{\ast}=1\,.$
(53)

As expected, we recover Eq. (4),
$\tilde{\omega}_{\pm}=\gamma(\omega_{0}\mp\Omega_{0})$, which is the frequency
of the wave as measured by the momentarily tangent inertial observer;
furthermore, the relative strength of the measured amplitudes of the helicity
states in Eq. (52) is still $(a_{+}/a_{-})$ and is therefore not affected by
the observer’s rotation.

Finally, we must compute the integral of $\mathbb{K}\,\tilde{F}$ over the
proper time of the rotating observer in accordance with Eq. (29). The second
term in the quadratic kernel (47), which is represented by matrix (46), does
not contribute to the integral in Eq. (29) because
$\phi_{\hat{\alpha}\hat{\beta}}\,\tilde{F}^{\hat{\alpha}\hat{\beta}}=0$ in
this case based on the circumstance that $g_{1}+v\,\Omega_{3}=0$ for the
uniformly rotating observer. Regarding the first term in Eq. (47), we have

$\kappa_{1}^{2}=-\gamma^{4}\,\Omega_{0}^{2}\,\,\mathrm{diag}(1,1,0)\,,\qquad\kappa_{2}^{2}=-v^{2}\,\gamma^{4}\,\Omega_{0}^{2}\,\,\mathrm{diag}(0,1,1)\,.$
(54)

Moreover, the elements of the matrix (44) are given by

$\kappa_{1}^{2}-\kappa_{2}^{2}=-\gamma^{4}\,\Omega_{0}^{2}\,\,\mathrm{diag}(1,\gamma^{-2},-v^{2})\,,\qquad\kappa_{1}\,\kappa_{2}+\kappa_{2}\,\kappa_{1}=-v\,\gamma^{4}\,\Omega_{0}^{2}\,\begin{bmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{bmatrix}\,.$ (55)

It follows from Eq. (29) that for $\tau\geq 0$, we have

$\mathcal{F}_{\pm}(\tau)=\left[1+i\,\mu\,\gamma^{2}\,\Omega_{0}^{2}\,\frac{e^{i\,\tilde{\omega}_{\pm}\,\tau}-1}{\tilde{\omega}_{\pm}}\right]\,\tilde{F}_{\pm}(\tau)\,.$
(56)

It is important to note that $\mathcal{F}_{\pm}(\tau)$ can never become
constant; moreover, for incident positive-helicity radiation with resonance
frequency $\omega_{0}=\Omega_{0}$, we have $\tilde{\omega}_{+}=0$ and

$\mathcal{F}_{+}(\tau)=(1-\mu\,\gamma^{2}\,\Omega_{0}^{2}\,\tau)\,\tilde{F}_{+}\,,$
(57)

where $\tilde{F}_{+}$ is constant. In the measured field (56), the ratio of
the positive-helicity wave amplitude to the negative-helicity wave amplitude
is given by $(a_{+}/a_{-})\,\rho$, where

$\rho=\frac{\omega_{0}+\Omega_{0}}{\omega_{0}-\Omega_{0}}~{}\frac{\omega_{0}-\Omega_{0}-i\,\mu\,\gamma\,\Omega_{0}^{2}}{\omega_{0}+\Omega_{0}-i\,\mu\,\gamma\,\Omega_{0}^{2}}\,.$
(58)

If $\mu=0$, then $\rho=1$, in accordance with the observation that the
locality postulate does not affect the relative strength of the helicity
amplitudes BM6a ; BM7 . However, when $\mu\neq 0$, $|\rho|>1$, as expected;
that is, for an observer that is rotating uniformly in the positive sense
about the $z$ axis, the measured amplitude of the incident positive-helicity
radiation is enhanced, while the measured amplitude of the negative-helicity
radiation is diminished BM6a ; BM7 ; BM9 . This result is independent of the
sign and magnitude of $\mu\neq 0$.

##  VI Rotating Observer in Constant Electromagnetic Field

Consider a constant electromagnetic field in the background global inertial
frame. Suppose that for $\tau>0$, the uniformly rotating observer measures the
electromagnetic field. Thus in our nonlocal ansatz, Eq. (29), we use the
quadratic kernel (47) together with $\tilde{F}=\Lambda\,F$, where $F$ given in
Eq. (28) is constant and $\Lambda$ is given by Eq. (50).

For the sake of simplicity, it is useful to define

$C:=\cos\varphi\,,\qquad
S:=\sin\varphi\,,\qquad\varphi=\Omega_{0}\,t=\gamma\,\Omega_{0}\,\tau\,$ (59)

and auxiliary dimensionless parameters $\mu^{\prime}$ and $\nu^{\prime}$ given
by

$\mu^{\prime}:=\gamma^{2}\,\Omega_{0}\,\mu\,,\qquad\nu^{\prime}=2\,\gamma\,\Omega_{0}\,\nu\,.$
(60)

A detailed but straightforward calculation reveals that

$\mathcal{E}_{1}=\gamma\,(C\,E_{1}+S\,E_{2})-\mu^{\prime}\,[S\,E_{1}+(1-C)\,E_{2}]+v\gamma\,(1+\nu^{\prime}\,\varphi)B_{3}\,,$
(61)
$\mathcal{E}_{2}=-S\,E_{1}+C\,E_{2}+\gamma^{-1}\,\mu^{\prime}\,[(1-C)\,E_{1}-S\,E_{2}]\,,$
(62)
$\mathcal{E}_{3}=\gamma\,E_{3}-v\gamma\,(C\,B_{1}+S\,B_{2})+v\mu^{\prime}\,[S\,B_{1}+(1-C)\,B_{2}]\,,$
(63)
$\mathcal{B}_{1}=\gamma\,(C\,B_{1}+S\,B_{2})-\mu^{\prime}\,[S\,B_{1}+(1-C)\,B_{2}]-v\gamma\,E_{3}\,,$
(64)
$\mathcal{B}_{2}=-S\,B_{1}+C\,B_{2}+\gamma^{-1}\,\mu^{\prime}\,[(1-C)\,B_{1}-S\,B_{2}]\,,$
(65)
$\mathcal{B}_{3}=v\gamma\,(C\,E_{1}+S\,E_{2})-v\mu^{\prime}\,[S\,E_{1}+(1-C)\,E_{2}]+\gamma\,(1+\nu^{\prime}\,\varphi)B_{3}\,.$
(66)

It is important to observe that while terms proportional to $\mu^{\prime}$ are
harmonic, terms proportional to $\nu^{\prime}$ are secular. That is, as the
observer rotates uniformly, the $\mu^{\prime}$ terms are simply periodic with
proper period $2\,\pi/(\gamma\,\Omega_{0})$, but the $\nu^{\prime}$ terms are
cumulative. If $B_{3}\neq 0$, then $\mathcal{E}_{1}$ and $\mathcal{B}_{3}$ can
grow linearly with time. This is not physically acceptable and indicates that
we should eventually drop the second term in the quadratic field kernel (i.e.,
$\nu=0$).

##  VII Linearly Accelerated Observer in Constant Field

Consider next an observer moving with uniform acceleration along the $z$ axis
for $t>0$. For $-\infty<t<0$, the observer is at rest at the spatial origin of
a Cartesian coordinate system $(x,y,z)$ in an inertial reference frame. The
acceleration is turned on at $t=\tau_{0}=0$ and will be turned off at
$\tau_{f}>0$. For $0\leq\tau\leq\tau_{f}$, the tetrad frame of the observer in
$(t,x,y,z)$ coordinates is given by

$\displaystyle\lambda^{\mu}{}_{\hat{0}}$
$\displaystyle=(\mathbb{C},0,0,\mathbb{S})\,,$ (67)
$\displaystyle\lambda^{\mu}{}_{\hat{1}}$ $\displaystyle=(0,1,0,0)\,,$ (68)
$\displaystyle\lambda^{\mu}{}_{\hat{2}}$ $\displaystyle=(0,0,1,0)\,,$ (69)
$\displaystyle\lambda^{\mu}{}_{\hat{3}}$
$\displaystyle=(\mathbb{S},0,0,\mathbb{C})\,,$ (70)

where

$\mathbb{C}=\cosh(g\,\tau)\,,\qquad\mathbb{S}=\sinh(g\,\tau)\,.$ (71)

Here, $g$ is the constant invariant translational acceleration of the
observer; that is, the acceleration tensor in this case is given by
$\mathbf{g}=g\,(0,0,1)$ and $\bm{\Omega}=0$ with respect to the local adapted
spatial frame $\lambda^{\mu}{}_{\hat{i}}$, $i=1,2,3$, that indicate the
Cartesian axes $x$, $y$ and $z$ directions, respectively. The observer’s
Lorentz factor and speed are given by $\mathbb{C}$ and
$\mathbb{S}/\mathbb{C}$, respectively.

The background field $F$ given by Eq. (28) is again constant. The field as
measured by the instantaneously tangent inertial observer is
$\tilde{F}=\Lambda\,F$, where $\Lambda$ has the same form as in Eq. (50), but
with $\Lambda_{1}$ and $\Lambda_{2}$ given by

$\Lambda_{1}=\mathrm{diag}(\mathbb{C},\mathbb{C},1)\,,\qquad\Lambda_{2}=-\mathbb{S}\,I_{3}\,.$
(72)

The special kernel $\kappa$ given by Eq. (38) reduces in this case to

$\kappa_{1}=0\,,\qquad\kappa_{2}=g\,I_{3}\,,\qquad
I_{3}=\begin{bmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\end{bmatrix}\,.$ (73)

In the formula for $\mathbb{K}$, we need
$-\kappa_{2}^{2}=g^{2}\,\mathrm{diag}(1,1,0)$; hence,

$2\,\mathbb{K}=g^{2}\,\mathrm{diag}(\mu,\mu,-2\,\nu,\mu,\mu,0)\,.$ (74)

As before, a detailed but straightforward calculation based on Eq. (29)
reveals that

$\mathcal{E}_{1}=\mathbb{C}\,E_{1}-\mathbb{S}\,B_{2}+\mu\,g\,[\mathbb{S}\,E_{1}-(\mathbb{C}-1)\,B_{2}]\,,$
(75)
$\mathcal{E}_{2}=\mathbb{C}\,E_{2}+\mathbb{S}\,B_{1}+\mu\,g\,[\mathbb{S}\,E_{2}+(\mathbb{C}-1)\,B_{1}]\,,$
(76) $\mathcal{E}_{3}=(1-2\,\nu\,g^{2}\,\tau)E_{3}\,,$ (77)
$\mathcal{B}_{1}=\mathbb{C}\,B_{1}+\mathbb{S}\,E_{2}+\mu\,g\,[\mathbb{S}\,B_{1}+(\mathbb{C}-1)\,E_{2}]\,,$
(78)
$\mathcal{B}_{2}=\mathbb{C}\,B_{2}-\mathbb{S}\,E_{1}+\mu\,g\,[\mathbb{S}\,B_{2}-(\mathbb{C}-1)\,E_{1}]\,,$
(79) $\mathcal{B}_{3}=B_{3}\,.$ (80)

The only contribution from the second term in the quadratic kernel (47) occurs
in $\mathcal{E}_{3}=(1-2\,\nu\,g^{2}\tau)E_{3}$, which is again cumulative. If
$E_{3}\neq 0$, then the measured magnitude of the electric field along the
direction of motion varies linearly with proper time. As before, it seems
appropriate to set $\nu=0$.

##  VIII Discussion

It is noteworthy that parity violating terms had to be introduced for the sake
of consistency in the theories of acceleration-induced nonlocal
electrodynamics as well as nonlocal gravity BMB . On the other hand, it is
important to understand theoretically why their presence was deemed necessary;
that is, one has to find out whether these were genuine or somehow spurious
and a consequence of some assumption that had no firm physical foundation. It
is simpler to look at the problem in nonlocal electrodynamics, where the
kernel was originally assumed to be linear in the acceleration tensor.
Removing this restriction has been the main objective of the present work. We
find that the kernel could in principle contain linear, quadratic and possibly
higher-order parity conserving terms in the acceleration tensor. Concentrating
on quadratic kernels, we have worked out some of the physical consequences of
an acceleration-induced nonlocal field theory of electrodynamics that involves
a kernel given by

$K^{\hat{\alpha}\hat{\beta}}{}_{\hat{\gamma}\hat{\delta}}=\mu\,\kappa^{\hat{\alpha}\hat{\beta}}{}_{\hat{\rho}\hat{\sigma}}\,\kappa^{\hat{\rho}\hat{\sigma}}{}_{\hat{\gamma}\hat{\delta}}\,,$
(81)

where $\mu$ is a constant parameter with dimensions of length that must be
determined from comparison of the theory with observational data.

What are the implications of the present investigation for the problem of
parity violation in nonlocal gravity? Let us note that general relativity (GR)
has an equivalent tetrad formulation as a teleparallel theory of gravity
(GR||). Nonlocal gravity is indeed the nonlocal version of GR||, the
teleparallel equivalent of GR. Specifically, consider a tetrad field
$e^{\mu}{}_{\hat{\alpha}}(x)$ adapted to a set of preferred observers in
spacetime. The tetrad system is orthonormal, i.e.,

$g^{\mu\nu}=\eta^{\hat{\alpha}\hat{\beta}}\,e^{\mu}{}_{\hat{\alpha}}\,e^{\nu}{}_{\hat{\beta}}\,,$
(82)

which defines the spacetime metric tensor $g_{\mu\nu}(x)$. The gravitational
field in this framework is given by the torsion tensor

$C_{\mu\nu}{}^{\hat{\alpha}}=\partial_{\mu}e_{\nu}{}^{\hat{\alpha}}-\partial_{\nu}e_{\mu}{}^{\hat{\alpha}}\,,$
(83)

which, for each $\alpha=0,1,2,3$, bears a strong resemblance to the
electromagnetic field tensor. Using the torsion tensor, one can define three
independent algebraic invariants given by

$\mathcal{I}_{1}=C_{\alpha\beta\gamma}C^{\alpha\beta\gamma},\quad\mathcal{I}_{2}=C_{\alpha\beta\gamma}C^{\gamma\beta\alpha},\quad\mathcal{I}_{3}=C_{\alpha}C^{\alpha}\,,$
(84)

where $C_{\alpha}$ is the torsion vector:
$C_{\alpha}=C_{\beta\alpha}{}^{\beta}=-C_{\alpha}{}^{\beta}{}_{\beta}$. With a
Lagrangian density

$\mathcal{L}_{g}=-\frac{c^{3}}{32\,\pi\,G}\,\sqrt{-g}\,\left(\frac{1}{2}\,\mathcal{I}_{1}+\mathcal{I}_{2}-2\,\mathcal{I}_{3}\right),$
(85)

we recover the field equations of GR||, which exhibit formal similarities with
the Maxwell field equations in a material medium. In the nonlocal
electrodynamics of media, nonlocality enters the theory via constitutive
relations. The same idea can be implemented in nonlocal gravity; that is, the
local constitutive relation of GR|| can be rendered nonlocal via the
introduction of a scalar kernel. The resulting theory has only one known
trivial exact solution, namely, the Minkowski spacetime in the absence of
gravitation. It has not been possible thus far to find any nontrivial exact
solution of the theory Bini:2016phe ; instead, linearized GR|| has been
studied in detail BMB . As elucidated in Ref. BMB , the equations of
linearized nonlocal gravity exhibit a certain inconsistency that can be
eliminated by adding to the constitutive relation a parity violating term
proportional to

$\check{C}_{\mu}\,g_{\nu\rho}-\check{C}_{\nu}\,g_{\mu\rho}\,,$ (86)

where $\check{C}_{\mu}$ is the torsion pseudovector given by

$\check{C}_{\mu}=\frac{1}{3!}\,C^{\alpha\beta\gamma}\,\epsilon_{\alpha\beta\gamma\mu}\,.$
(87)

General discussions of parity violation in tetrad theories of gravitation are
contained in Refs. Baekler:2011jt ; Itin:2018dru . In connection with nonlocal
GR||, various theoretical aspects of parity violating term (86) have been
explored in Ref. Itin:2018dru . Can one avoid parity violation in nonlocal
gravity as in the nonlocal electrodynamics of accelerated systems? The
resolution of this issue requires further investigation.

##  Appendix A $P$ and $T$ Invariance in Standard Electrodynamics of
Accelerated Systems and Gravitational Fields

It can be shown that the electromagnetic field equations in an accelerated
system or a gravitational field can be reformulated as Maxwell’s equations in
flat Minkowski spacetime in Cartesian coordinates but in the presence of a
certain hypothetical “medium”. To this end, imagine Maxwell’s equations

$F_{[\mu\nu,\rho]}=0\,,\qquad(\sqrt{-g}\,F^{\mu\nu})_{,\nu}=4\,\pi\,j^{\mu}\,$
(88)

on a background spacetime metric $-ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}$
expressed in standard Cartesian coordinates $x^{\mu}=(t,x,y,z)$. We use the
natural decompositions

$F_{\mu\nu}\mapsto(\mathbf{E},\mathbf{B})\,,\qquad\sqrt{-g}\,F^{\mu\nu}\mapsto(-\mathbf{D},\mathbf{H})\,,$
(89)

where $\sqrt{-g}\,F^{0i}=D_{i}$ and $\sqrt{-g}\,F^{ij}=\epsilon^{ijk}H_{k}$ in
our convention. In this way, we recover the original form of Maxwell’s
equations in Minkowski spacetime but in a background gyrotropic medium with
constitutive relations Sk ; Pl ; Fe ; VIS

$D_{i}=\epsilon_{ij}\,E_{j}-(\mathbf{G}\times\mathbf{H})_{i}\,,\qquad
B_{i}=\mu_{ij}\,H_{j}+(\mathbf{G}\times\mathbf{E})_{i}\,,$ (90)

where the conformally invariant properties of the optical medium are given by
equal dielectric and permeability tensors as well as a gyration vector which
has the interpretation of the gravitational vector potential,

$\epsilon_{ij}=\mu_{ij}=-\sqrt{-g}\,\frac{g^{ij}}{g_{00}}\,,\qquad
G_{i}=-\frac{g_{0i}}{g_{00}}\,.$ (91)

We briefly discuss the derivation of the constitutive relations (90) at the
end of this appendix. There is no _double refraction_ when the electric
permittivity and magnetic permeability tensors coincide, which is a special
property of the medium that simulates acceleration and gravitation in this
case VIS .

Under the parity transformation $P:(x,y,z)\mapsto(-x,-y,-z)$ or time reversal
$T:t\mapsto-t$, the background spacetime interval remains invariant, while the
components of the metric tensor transform as
$(g_{00},g_{0i},g_{ij})\mapsto(g_{00},-g_{0i},g_{ij})$. Similarly, we find for
the inverse metric $(g^{00},g^{0i},g^{ij})\mapsto(g^{00},-g^{0i},g^{ij})$.
Furthermore, we can write

$-ds^{2}=g_{00}\,(dt-G_{k}\,dx^{k})^{2}+\gamma_{ij}\,dx^{i}\,dx^{j}\,,$ (92)

where $(\gamma_{ij})$, defined by

$\gamma_{ij}:=g_{ij}-\frac{g_{0i}\,g_{0j}}{g_{00}}\,,$ (93)

is the inverse of $(g^{ij})$; that is, $g^{ik}\,\gamma_{kj}=\delta^{i}_{j}$.
For the determinant of the metric tensor $g:=\det(g_{\mu\nu})$, we have the
general relation $g=g_{00}\,\det(\gamma_{ij})$. Moreover,

$g_{00}=g\,\det(g^{ij})\,,\qquad g^{00}=\frac{\det(g_{ij})}{g}\,,$ (94)

which imply that $g$ does not change under parity and time reversal. These
results are all consistent with the fact that under parity and time reversal,
we have $(\mu_{ij},\mathbf{G}_{i})\mapsto(\mu_{ij},-\mathbf{G}_{i})$.
Maxwell’s equations in an inertial frame in Minkowski spacetime are such that
under parity $\mathbf{E}$ and $\mathbf{D}$ transform as vectors, while
$\mathbf{B}$ and $\mathbf{H}$ transform as pseudovectors. Moreover, under time
reversal we have $(\mathbf{E},\mathbf{B})\mapsto(\mathbf{E},-\mathbf{B})$ and
$(\mathbf{D},\mathbf{H})\mapsto(\mathbf{D},-\mathbf{H})$. The constitutive
relations (90) remain unchanged separately under these parity and time
reversal transformations; therefore, parity is conserved and time reversal
invariance is maintained in the standard electrodynamics of accelerated
systems and gravitational fields.

It is interesting to write the general Maxwell equations in Dirac form. For
this purpose, we introduce the Riemann-Silberstein vectors

$\mathbf{F}^{\pm}=\mathbf{E}\pm
i\,\mathbf{H}\,,\qquad\mathbf{S}^{\pm}=\mathbf{D}\pm i\,\mathbf{B}\,$ (95)

in terms of complex fields, so that Maxwell’s equations can now be written as
BMa ; BMb ; BMc ; BMd

$\frac{1}{i}\,\nabla\times\mathbf{F}^{\pm}=\pm\,\frac{\partial\mathbf{S}^{\pm}}{\partial
t}\pm 4\,\pi\,\mathbf{J}\,,\qquad\nabla\cdot\mathbf{S}^{\pm}=4\,\pi\,\rho\,,$
(96)

where $j^{\mu}=(\rho,\mathbf{J})$ and

$S^{\pm}_{p}=\mu_{pq}\,F^{\pm}_{q}\pm
i\,(\mathbf{G}\times\mathbf{F^{\pm}})_{p}\,.$ (97)

This formalism has been explicitly employed in Ref. HaMa  to study
electromagnetic wave propagation in a rotating frame of reference.

The parity operation can also be applied to the Dirac form of Maxwell’s
equations given by Eqs. (96) and (97). For this purpose, we must think of
$\pm\,i\,\mathbf{B}$ and $\pm\,i\,\mathbf{H}$ as vectors and
$-\,i\,\mathbf{E}$ as a pseudovector even though $\mathbf{B}$ and $\mathbf{H}$
are axial vectors and $\mathbf{E}$ is a vector. Similarly, under time
reversal, we must assume that $\pm\,i\,\mathbf{B}$ and $\pm\,i\,\mathbf{H}$
remain invariant, while $-\,i\,\mathbf{E}$ changes sign. In this way, the
fields become consistent with sources under parity and time reversal
operations; that is, we have $\rho\mapsto\rho$ and $J\mapsto-J$ separately
under $P$ and $T$.

An interesting consequence of $\epsilon_{ij}=\mu_{ij}$ should be noted here:
Under duality rotations with constant $\alpha$, namely,

$\mathbf{E}^{\prime}=\mathbf{E}\,\cos\alpha+\mathbf{H}\,\sin\alpha\,,\qquad\mathbf{D}^{\prime}=\mathbf{D}\,\cos\alpha+\mathbf{B}\,\sin\alpha\,$
(98)

and

$\mathbf{H}^{\prime}=-\mathbf{E}\,\sin\alpha+\mathbf{H}\,\cos\alpha\,,\qquad\mathbf{B}^{\prime}=-\mathbf{D}\,\sin\alpha+\mathbf{B}\,\cos\alpha\,,$
(99)

constitutive relations (90) and Maxwell’s equations in the absence of external
sources remain invariant; in this connection, let us observe that
$\mathbf{F}^{\prime\,\pm}=\mathbf{F}^{\pm}\,e^{\mp\,i\,\alpha}$ and
$\mathbf{S}^{\prime\,\pm}=\mathbf{S}^{\pm}\,e^{\mp\,i\,\alpha}$ under the
duality rotations.

###  A.1 Derivation of the Constitutive Relations

For the sake of completeness, we briefly indicate here the derivation of
constitutive relations (90), which follow from the electric and magnetic parts
of

$\sqrt{-g}\,F_{\mu\nu}=g_{\mu\alpha}\,g_{\nu\beta}\,\sqrt{-g}\,F^{\alpha\beta}\,,\qquad\sqrt{-g}\,F^{\mu\nu}=\sqrt{-g}\,g^{\mu\alpha}\,g^{\nu\beta}\,F_{\alpha\beta}\,,$
(100)

respectively. The electric part of the first relation in Eq. (100) implies

$-\sqrt{-g}\,E_{j}=g_{00}\,\gamma_{jk}\,D_{k}-g_{jp}\,g_{0q}\,\epsilon^{pqn}\,H_{n}\,.$
(101)

We multiply both sides of this equation by $g^{ij}/g_{00}$ and use
$g^{ij}\,\gamma_{jk}=\delta^{i}_{k}$ as well as
$g^{ij}\,g_{jp}=\delta^{i}_{p}-g^{0i}\,g_{0p}$ to obtain the first
constitutive relation.

Next, the magnetic part of the second relation in Eq. (100) can be written as

$H_{j}=\sqrt{-g}\,\epsilon_{jmn}\,[g^{mp}\,g^{0n}\,E_{p}+\frac{1}{2}\,\epsilon_{abc}\,g^{ma}\,g^{nb}\,B_{c}]\,.$
(102)

Starting with $(g^{\mu\nu})$, one can determine the inverse matrix element
$g_{0k}$ via

$g\,\epsilon_{abc}\,g^{ai}\,g^{bj}\,g^{0c}=-\epsilon^{ijk}\,g_{0k}\,.$ (103)

Furthermore, from the definition of the determinant and Eq. (94) we find

$\epsilon_{abc}\,g^{ai}\,g^{bj}\,g^{ck}=\epsilon^{ijk}\,\det(g^{pq})=\epsilon^{ijk}\,\frac{g_{00}}{g}\,.$
(104)

Multiplying both sides of Eq. (102) by $-\sqrt{-g}\,g^{ij}/g_{00}$ and using
Eqs. (103) and (104), we get the second constitutive relation.

## Acknowledgments

Discussions initiated by Friedrich W. Hehl about parity violation in nonlocal
gravity motivated the present investigation. I am also grateful to him for his
insightful and critical comments.

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