## Abstract

Applications of the covariant theory of drive-forms are considered for a class of perfectly insulating media. The distinction between the notions of ‘classical photons’ in homogeneous bounded and unbounded stationary media and in stationary unbounded magneto-electric media is pointed out in the context of the Abraham, Minkowski and symmetrized Minkowski electromagnetic stress–energy–momentum tensors. Such notions have led to intense debate about the role of these (and other) tensors in describing electromagnetic interactions in moving media. In order to address some of these issues for material subject to the Minkowski constitutive relations, the propagation of harmonic waves through homogeneous and inhomogeneous, isotropic plane-faced slabs at rest is first considered. To motivate the subsequent analysis on accelerating media, two classes of electromagnetic modes that solve Maxwell’s equations for uniformly rotating *homogeneous* polarizable media are enumerated. Finally it is shown that, under the influence of an incident monochromatic, circularly polarized, plane electromagnetic wave, the Abraham and symmetrized Minkowski tensors induce different time-averaged torques on a uniformly rotating *materially inhomogeneous* dielectric cylinder. We suggest that this observation may offer new avenues to explore experimentally the covariant electrodynamics of more general accelerating media.

## 1. Introduction

This is paper II of a series of two papers. In paper I (Goto *et al.* in press), it was shown how the notion of a Killing vector field on spacetime could be used, together with a divergence-less total stress–energy–momentum tensor for a material continuum interacting with the electromagnetic field, in order to establish the dynamical classical evolution of the medium. It was emphasized that the precise form of an electrodynamic force or torque depends on the nature of the decomposition of the total stress–energy–momentum into parts describing its electromagnetic interaction with the medium and the electromagnetic constitutive relations for the macroscopic Maxwell equations. The motivation for this approach was to establish whether such quantities could be used to discriminate by experiment between a number of electromagnetic stress–energy–momentum tensors that have been proposed in the past to describe the electromagnetic interaction in dielectric media and to provide a comprehensive framework for the analysis of electrodynamic problems in accelerating media. In this paper, we adopt the conventions of paper I and apply the formulation to the computation of a time-averaged electromagnetic torque produced in stationary and rotating media by an incident plane harmonic electromagnetic wave using the electromagnetic stress–energy–momentum tensors proposed by Abraham and Minkowski. The first sections demonstrate how the notion of the classical ‘photon’ in stationary media can be formulated in this framework. It is emphasized that the linear momentum of such photons not only depends on the choice of an electromagnetic stress–energy–momentum tensor in the medium, but also on whether the medium has an interface with the vacuum or anisotropic and dispersive properties. It is then shown how the existence of a family of transverse electromagnetic modes satisfying Maxwell’s macroscopic equations subject to the Minkowski constitutive relations for a uniformly rotating homogeneous insulating medium can be used to solve the boundary value problem for a plane harmonic electromagnetic wave incident on a homogeneous rotating slab. That the time-averaged electromagnetic torque on such a slab is the same for the Abraham and Minkowski stress–energy–momentum tensors motivates an analysis of torques on rotating *inhomogeneous* cylinders. In this case, we demonstrate that the time-averaged torques are significantly different.

## 2. Classical ‘photons’ in stationary homogeneous-bounded and unbounded isotropic media

Although the quantization of the electromagnetic field in a bounded polarizable medium is non-trivial, the notion of the classical ‘photon’ has been used to highlight the different predictions concerning the linear momentum of light obtained by adopting the different electromagnetic stress–energy–momentum tensors in such a medium. These notions can be defined in terms of time-harmonic classical electromagnetic field configurations in homogeneous isotropic stationary media and the time averages of their energy and linear momentum in a fixed spatial volume 𝒱.

If a scalar field *A*(** r**,

*t*) depends on time

*t*, its time average over any time interval

*T*isFurthermore, if

**(**

*A***,**

*r**t*) is a real spatial

*p*-form,

**(**

*B***,**

*r**t*) a real spatial

*q*-form in any frame

*U*andwhere 𝒜(

**) and ℬ(**

*r***) are complex spatial forms with complex conjugates , thenHence2.1if**

*r**T*=2

*π*/

*ω*. For any bounded spatial 3-form

*α*

_{t}in frame

*U*with

*arbitrary*time variation, its time average 〈

*α*〉 over any finite interval of time

*T*is the 3-form2.2It follows immediately that if

*α*

_{t}is time-periodic, but not necessarily harmonic, with period

*T*(i.e.

*α*

_{t}=

*α*

_{t+T}), then .

If *K* is a *space-like translational* Killing vector field, the time-averaged linear momentum associated with a drive-form having orthogonal components in a fixed volume 𝒱 relative to *U* is2.3Similarly, since *U* is a *time-like translational* Killing vector field in Minkowski spacetime, the time-averaged energy associated with the same drive-form in a volume 𝒱 is2.4This classical energy can be parcelled into *N* ‘energy quanta’, each of which corresponds to that carried by a harmonic plane-wave quantum with energy *in the vacuum*,2.5One then defines the time-averaged *K*-component of linear momentum associated with a classical ‘photon’ in a volume 𝒱 of the medium to be2.6Different choices of stress–energy–momentum tensor for the electromagnetic field with the *same* constitutive relation for the medium will in general give different values for . In particular, for *plane harmonic waves* propagating in a simple *unbounded* non-accelerating homogeneous, isotropic medium (described by relative permittivity *ϵ*_{r} and relative permeability *μ*_{r}) in a direction aligned with *K*, the non-symmetric Minkowski electromagnetic stress–energy–momentum tensor yields2.7where is the refractive index of the medium, while that calculated from the Abraham tensor yields2.8Furthermore, from the symmetrized Minkowski tensor one finds2.9The underlying origin for these distinctions is that (see tables 1 and 2 in appendix A of paper I), although the contributions to the energy density are the same for the non-symmetric Minkowski tensor *T*^{M}, the symmetrized Minkowski tensor *T*^{SM} and the Abraham tensor *T*^{AB} (since the medium acceleration *A*=0), the contributions to the corresponding momentum densities are different.

However, even if the medium is stationary, homogeneous and isotropic, the presence of boundaries can modify these results. Given that all physical media do have interfaces with other media including the vacuum, the role of these interfaces between media with different constitutive properties is relevant in discussing the physical phenomena. To illustrate this point, consider a simple medium composed of a stationary plane-faced slab of arbitrary finite thickness and constant relative permittivity *ϵ*_{r} and relative permeability *μ*_{r}. If a harmonic circularly polarized plane wave is incident from the left vacuum half-space normally on one plane face, it will be partially reflected and transmitted. Suppose it propagates through the slab into a half-space composed of a different stationary simple homogeneous isotropic medium but with constant relative permittivity and relative permeability . One can readily compute the electromagnetic fields in each region from the electromagnetic junction conditions and hence the different classical photon momenta in the three different media. If the slab has its plane normals parallel to the direction of propagation *K*=∂/∂*z*, one finds for the different classical photon momenta in the slab,2.10and2.11where . Each of these momenta in the bounded slab depends on the properties of the medium outside the slab. Furthermore, if the right-hand half space is taken to be the vacuum (), they are different from the momenta above for classical photons in homogeneous *unbounded* media.

Analogous results can be obtained by calculating the total time-averaged classical angular momentum about some point in 𝒱 using a space-like rotational Killing vector generating rotations about an arbitrary direction in space. Again, the momentum densities derived from *T*^{M} and *T*^{AB} are different so that the corresponding classical ‘photon’ helicities in a finite volume 𝒱 are different. We stress that such ‘photon’ momenta and ‘photon’ helicities are strictly classical notions calculated from time averages of harmonic plane waves in a stationary medium. As such, they are not subject to the same conservation laws as genuine electromagnetic quanta in the medium.

## 3. Classical ‘photons’ in stationary unbounded homogeneous magneto-electric media

In dispersive media, constitutive relations between the real spatial fields *e*^{U},*b*^{U},*d*^{U},*h*^{U} are, in general, non-local in spacetime. If the medium is *spatially homogeneous*, so that it has no preferred spatial origin, then, in Minkowski spacetime, it is possible to Fourier-transform the inertial components of these fields with respect to space and time, and work with transformed local constitutive relations.

For any spatial 1-form *α*^{U} on spacetime with inertial components , define their complex-valued Fourier transforms by3.1where , , and . Then the source-free macroscopic Maxwell system reduces to3.2where the real propagation wave 1-form ** K**≡

**⋅d**

*k***. The remaining transformed Maxwell equations and follow trivially from equation (3.2). It also follows trivially that (i.e. is perpendicular to ). Similarly, and .**

*r*Assume that, in any inertial frame *U*, the medium is described in terms of the real magneto-electric (1,1) spatial tensors *ζ*^{de}(** k**,

*ω*),

*ζ*^{hb}(

**,**

*k**ω*),

*ζ*^{db}(

**,**

*k**ω*),

*ζ*^{he}(

**,**

*k**ω*) satisfying the symmetry conditionswhere the adjoint

*T*

^{†}of any spacetime tensor

*T*, which maps

*p*-forms to

*p*-forms, is defined by3.3and the dispersive magneto-electric constitutive relations in the

*U*frame are defined by3.4and3.5These will (by convolution) give rise to non-local spacetime constitutive relations.

Substituting equations (3.4) and (3.5) in (3.2) yields a degenerate 1-form eigen-equation for ,3.6The field then follows from equation (3.2) (up to a scaling) and from equations (3.4) and (3.5), respectively. Equation (3.6) may be written3.7defining the (1,1) tensor 𝒟(** k**,

*ω*). For non-trivial solutions , the determinant of the matrix 𝒟(

**,**

*k**ω*) representing 𝒟(

**,**

*k**ω*) must vanish,3.8Note that, in general, the roots of this dispersion relation are not invariant under the transformation

**→ −**

*K***. If one writes in terms of the Euclidean norm |**

*K***|, and introduces the refractive index 𝒩=|**

*k***|(**

*k**c*

_{o}/

*ω*)>0 and in place of

**, then solutions propagating in the direction described by with angular frequency**

*k**ω*>0 correspond to roots of equation (3.8) (labelled

*r*), which may be expressed in the form . Thus, there can be a set of distinct characteristic waves each with its unique refractive index that depends on the propagation direction and frequency

*ω*. When the characteristic equation (3.8) is a quadratic polynomial in 𝒩

^{2}and has two distinct roots that describe two distinct propagating modes for a given

*ω*, the medium is termed

*birefringent*. Roots , such that , imply that harmonic plane waves propagating in the opposite directions have different wave speeds.

Each eigen-wave will have a uniquely defined polarization obtained by solving the independent equations in (3.7) for , up to normalization. Since is complex, it is convenient to introduce the eigen-wave normalization by writingin terms of the complex 0-form and complex polarization 1-form **n**^{r}(** k**,

*ω*), normalized to satisfy3.9for each

*r*. If one applies to equation (3.6), making use of the symmetries between the real magneto-electric tensors

*ζ*^{de}(

**,**

*k**ω*),

*ζ*^{db}(

**,**

*k**ω*),

*ζ*^{he}(

**,**

*k**ω*),

*ζ*^{hb}(

**,**

*k**ω*), and evaluates it with the eigen-wave , one obtains the

*real*0-form dispersion relation for the characteristic mode

*r*,where

**n**

^{r}≡

**n**

^{r}(

**,**

*k**ω*) and in terms of 𝒩 and .

For illustration, consider an *unbounded* magneto-electric material with3.10in terms of the scalars *ζ*^{de}(** k**,

*ω*),

*ζ*

^{hb}(

**,**

*k**ω*) and the rank-3 identity tensor

*Id*in space. The medium is oriented in the laboratory spatial basis {∂

_{x},∂

_{y},∂

_{z}}, so that

*ζ*^{db}(

**,**

*k**ω*) takes the particular form3.11in terms of the real scalars

*β*

_{1}(

**,**

*k**ω*),

*β*

_{2}(

**,**

*k**ω*). The matrix representing

*ζ*^{db}(

**,**

*k**ω*) in the laboratory basis takes the form3.12It follows that

^{1}3.13Consider a (complexified) harmonic plane wave propagating through the medium along the

*z*-axis and polarized in the

*y*-direction,3.14for some (real) constant ℰ, angular frequency

*ω*>0 and wavenumber

*k*. From equation (3.8), the dispersion relation associated with a polarized eigen-mode is3.15describing propagation in a direction determined by

*sgnk*∂

_{z}with phase speed |

*ω*/

*k*| depending on the values of

*ζ*

^{de}(

**,**

*k**ω*),

*ζ*

^{hb}(

**,**

*k**ω*) and

*β*

_{2}(

**,**

*k**ω*).

Using equations (3.2) and (3.14) enable one to construct the 1-forms , , , . Then from equation (2.6), the classical photon momenta associated with different electromagnetic stress–energy–momentum tensors (see appendix A of paper I) may be calculated. The different magneto-electric photon momenta associated with the direction *K*=∂_{z} areandwhere *k*_{±} are the roots ofwith ** k**=(0,0,

*k*). These momenta are independent of

*β*

_{1}(

**,**

*k**ω*) owing to the choice of wave polarization. For

*β*

_{2}(

**,**

*k**ω*)=0, one writes

*ζ*

^{de}(

**,**

*k**ω*)=

*ϵ*

_{0}

*ϵ*

_{r}(

**,**

*k**ω*), (

*ζ*

^{hb}(

**,**

*k**ω*))

^{−1}=

*μ*

_{0}

*μ*

_{r}(

**,**

*k**ω*) and the magneto-electric photon momenta reduce to the classical photon momenta in a stationary, but

*dispersive*, homogeneous, isotropic, unbounded polarizable medium (equations (2.7–2.9)) with a frequency-dependent refractive index.

## 4. Transverse (TDB) modes in uniformly rotating homogeneous media

In §2, one used the result that plane harmonic electromagnetic waves can freely propagate in a simple homogeneous isotropic non-dispersive non-accelerating unbounded polarizable medium. For bounded media, one expects that boundary conditions will put constraints on the propagation characteristics. For inhomogeneous bounded media, single harmonic plane waves are no longer supported and, if the medium is accelerated (whether bounded or not), finding solutions to Maxwell’s equations in the medium becomes non-trivial in general.

A significant difference between the electromagnetic stress–energy–momentum tensors advocated by Minkowski and Abraham is that irrespective of the electromagnetic constitutive relation describing the medium the Abraham tensor depends *explicitly* on the bulk 4-velocity field of the medium. This in turn implies that its divergence will depend *explicitly* on the bulk 4-acceleration of the medium. This characteristic feature is frame-independent and gives rise, in general, to a non-trivial coupling between the medium acceleration and the electromagnetic field. This interaction is of course absent for media at rest or moving with constant linear velocity in any inertial frame. By contrast, a uniformly rotating medium should be sensitive in principle to such an interaction. In the following, we will attempt to calculate the significance of this effect. If it is possible to measure the torque on a uniformly rotating medium as a function of rotation speed this should in principle discriminate between the two electromagnetic stress–energy–momentum tensors.

To expedite this programme it proves necessary to excite, by some means, electromagnetic fields in an electrically neutral rotating polarizable medium. Since in practice one must deal with finite media and fields outside the body, one is confronted with a difficult problem of electromagnetic scattering from a bounded moving polarizable medium. To circumvent this we shall approach the problem in terms of the transmission of incident plane harmonic waves through a thin cylindrical slab uniformly rotating about its axis of symmetry. If the radius of the circular cylinder greatly exceeds its thickness, it is reasonable to neglect the boundary conditions on its rim.

Player (1976) was one of the first to explore the propagation of waves in a uniformly rotating (unbounded) medium using the non-relativistic Minkowski constitutive relations. He assumed a rigidly rotating medium with angular speed *rΩ*≪*c*_{0} for all points a distance *r* from the axis of rotation and concluded that in this approximation a certain type of plane harmonic wave in the medium was subject to a dispersion relation dependent on *Ω*. Later, Götte *et al.* (2007) argued that there should be a spectrum of such modes.

First, we look for harmonic plane-wave modes in a uniformly rotating homogeneous uncharged medium by supposing that the magnetic induction field *b*^{U} and electric displacement field *d*^{U} are transverse to the propagation of a wave (with wavenumber *k* and angular frequency *ω*) along the axis of rotation. In an inertial frame *U*, introduce the spatial cylindrical coframe {*e*^{1}=*dr*,*e*^{2}=*rdθ* *and* *e*^{3}=*dz*} in cylindrical polar coordinates (*r*,*θ*,*z*) centred at an interface of the medium and, for some complex amplitudes *B*(*r*),*D*(*r*), assume that a pair of the complexified interior spatial fields take the form4.1and4.2where and *ω*>0. Such fields will be referred to as circularly polarized TDB modes. For constant *Ω*, the spatial vector describes a rigid rotation in *U* about the *z*-axis in these coordinates and will be adopted to describe the bulk motion of the medium.^{2} Then to first-order (in *ν*/*c*_{0}) the constitutive relations derived (see the electronic supplementary material for further details of this calculation) from equation (2.23) of paper I for a simple rotating homogeneous medium determine the interior electric and magnetic fields asThese fields must be compatible with the source-free Maxwell system for time harmonic fields^{3}Substituting into the first Maxwell equation above requires a 2-form to be zero (i.e. each component to vanish),4.3This system has the solutionfor some constant *A*. Similarly, the second Maxwell equation above gives4.4with solutionfor some constant *C*. The solutions (4.3) and (4.4) are compatible provided4.5For real *ω*, a root *k* of this dispersion relation may become complex for certain values of *Ω* (and describe evanescent waves). Restricting to real *k* roots, with real *Ω* and real 𝒩>0, such thatyields propagating solutions with wavenumbers as real roots of the dispersion relation (4.5). These determine the two possible directions of propagation of each TDB mode labelled by *m*. Let4.6where *η*_{r}=1 and *η*_{L}=−1. Then, *sgnk*_{r}>0 and *k*_{r} denotes a right-moving wave, while *sgnk*_{L}<0 and *k*_{L} denotes a left-moving wave. Thus, for propagating TDB waves satisfying the dispersion relation (4.5)andfor the arbitrary constant . In this notation, the left and right propagating, left and right circularly polarized TDB modes in a uniformly rotating homogeneous dielectric cylinder can be writtenIf the axis *r*=0 is in the medium and there are no axial sources, then the integer *m*≥0. All integers *m* are permitted if the medium is a shell with the rotation axis excluded. Fields with *m*>0 are clearly unbounded if *r* extends indefinitely. All modes are eigen-forms of the rotation operator with eigenvalue *m*+1. The interesting physical modes in an unbounded medium are the harmonic plane waves with *m*=0. These are the waves first explored by Player (1976). One expects that these modes could be excited by a harmonic plane wave normally incident on a rotating slab from the vacuum. In a simple medium its refractive index 𝒩 is dispersion-free and independent of the harmonic wave polarization.

There also exists a set of circular polarized TEH modes that can be calculated using an initial ansatz, where the magnetic field *h*^{U} and electric field *e*^{U} are transverse to the direction of propagation along the rotation axis,4.7and4.8with as before. However, if *Ω*≠0 it appears that there are no harmonic plane-wave TEH modes among this set, even for *m*=0, so we will not discuss them further here.

If a plane circularly polarized harmonic wave in the vacuum is incident normally on a homogeneous, uniformly rotating thin cylindrical slab consisting of material satisfying a simple constitutive relation (with *ϵ*_{r},*μ*_{r} constant), then it is straightforward to satisfy the electromagnetic jump conditions at the slab–plane interfaces with left- and right-moving *m*=0, circularly polarized, TDB modes in the medium (and a reflected and transmitted circularly polarized, harmonic plane wave in the vacuum). From equation (3.2) of paper I, one can use such a solution to calculate the real instantaneous electromagnetic *K*-drive on the slab. Such a drive will fluctuate harmonically in time with angular frequency, a multiple of the incident frequency. For any , the field in the slab is time-harmonic in the laboratory frame *U* and the time-average of is zero. Furthermore, since is a real quadratic function of the time-harmonic fields *e*^{U},*b*^{U},*d*^{U},*h*^{U} in the medium, the time average of is independent of the propagation coordinate *z*. Hence, from equation (3.2) of paper I, the time-averaged total *K*-drive on a fixed volume 𝒱 of the rotating medium is4.9The surface ∂𝒱 consists of the two opposite interior circular discs and the interior cylindrical rim. For all the tensors in appendix A of paper I, the contribution to the integral from the latter is zero since the integrand in equation (4.9) is zero when pulled back to the interior rim. Furthermore, the contributions to the integral from the two opposite interior faces cancel since the time-averaged is independent of *z*. Hence, one concludes that, if a normally incident plane harmonic vacuum wave excites a plane harmonic TDB mode in such a simple (uniformly rotating or stationary) medium, the induced time-averaged *K*-drive will be zero. Thus one cannot discriminate between Minkowski and Abraham stress–energy–momentum tensors by exciting plane TDB modes in a simple medium in this manner. One alternative is to excite harmonic fields in a medium in which the time-averaged is not independent of *z*. Then, the contribution from opposite interior faces will not necessarily vanish. The simplest strategy to implement this requirement is to employ fields in an inhomogeneous dispersion-free medium in which the permittivity or permeability varies with *z* along the axis of the slab. Unfortunately, the simple TDB plane-wave modes above are no longer solutions to the Maxwell equations in such an inhomogeneous medium. Nevertheless, it will be shown below how the computation of the Maxwell fields in such cases can lead to a method of discrimination between different electromagnetic stress–energy–momentum tensors in uniformly rotating inhomogeneous media.

## 5. Plane waves incident on a stationary inhomogeneous dielectric slab

The theory of Killing forms was exploited above in order to calculate the time-averaged harmonic electromagnetic forces and torques on perfectly insulating homogeneous polarizable media described by the Minkowski constitutive relations. For stationary or non-relativistically uniformly rotating media, it was found that the induced electromagnetic drives did not discriminate between the electromagnetic stress–energy–momentum tensors proposed by Minkowski and Abraham. If one stays with slabs excited by incident plane harmonic vacuum electromagnetic waves, it is natural to enquire about the influence of material inhomogeneities on this result. However, a polarized plane harmonic wave incident in the vacuum on a *stationary inhomogeneous slab* of polarizable material will not in general propagate in the medium as a polarized plane harmonic wave. To facilitate a discussion in the next section of the behaviour of such an incident wave on a *rotating slab*, a solution to the problem of fields in a stationary inhomogeneous slab will be considered in this section.

The stationary slab is oriented in a Cartesian frame with coordinates {*x*,*y*,*z*} as indicated in figure 1. Suppose a linear-polarized plane monochromatic wave with angular frequency *ω*>0, propagating in the *z*-direction, is incident on a slab of thickness ℓ and infinite extent in the *x* and *y* directions.

Let the slab have relative permittivity *ϵ*_{r}(*z*) and constant relative permeability *μ*_{r}. The slab has parallel interfaces (with the vacuum) at *z*=0 and *z*=ℓ, thereby distinguishing the spatial region *z*<0 denoted I, 0<*z*<ℓ denoted II and *z*>ℓ denoted III.

Let *F*^{II}=d *A*^{II} in region II withfor some (complex) function 𝒜(*z*) and (real) constant 𝒜_{0} with physical dimensions (so that 𝒜(*z*) is dimensionless). ThenThe medium is at rest relative to the inertial frame *U* so is assigned the 4-velocity *V* =(1/*c*_{0})∂_{t}. Thus, from the covariant dielectric constitutive relation (2.23) of paper I,For an uncharged medium (*j*=0), the inhomogeneous Maxwell equation (2.21) of paper I yieldsIf the relative permittivity is taken to be the function *ϵ*_{r}(*z*)=*α*+(*βz*/ℓ) for some (dimensionless) constants *α*,*β*, this Schrödinger-like differential equation can be solved in terms of a basis of Airy functions of the first and second kind (Ai and Bi, respectively) to givewhere andThus, for this inhomogeneous dielectric with linearly varying permittivity,5.1The dimensionless complex constants *C*_{1},*C*_{2} can be determined from two interface conditions. Consider vacuum electromagnetic plane waves in region I with (real) amplitude 𝒜_{0} and complex amplitude , propagating in the (positive and negative) *z*-directions, respectively,5.2where *k*_{R} and *k*_{L} denote distinct roots of the vacuum dispersion relation5.3with and . Similarly, a transmitted field in region III is written as5.4for some complex amplitude *E*^{III}_{r} and *sgnk*^{III}_{r}>0. At each interface, one must satisfy the interface conditions on a spacetime hypersurface (see the electronic supplementary material for their formulation on spacetime). On the left interface, *f*=*z*, and on the right interface, *f*=*z*−ℓ. If () denote the pull-back of forms to *z*=0 (*z*=ℓ), respectively, the interface boundary conditions becomeThese yield a linear system of equations for the dimensionless complex variables ,whereand similarly for Bi′[*κ*(*a*)]. The system admits a solution provided the 4×4 complex matrix above is non-singular.

The solution is readily obtained but its complicated structure will not be displayed here. It replaces the plane-wave dispersion relation in a homogeneous medium. Suffice to add that, with the aid of equations (5.1–5.4), one may now explicitly construct the Maxwell and excitation 2-forms in all three regions, and hence the electromagnetic field 1-forms {*e*^{U},*b*^{U},*d*^{U},*h*^{U}} in all three regions. The schematic behaviour of the real electric field amplitude in all regions is shown in figure 2. Solutions with different incident polarizations follow in a similar manner.

Once one has a complete Maxwell solution for a given incident wave, this can be substituted into any *K*-drive in order to calculate time-averaged force and torque pressures on any area of the slab in terms of the incident wave parameters 𝒜_{0},*ω* and the medium characteristics *ϵ*_{r}(*z*),ℓ. For the linear polarized plane waves (5.2) and (5.4), one finds non-zero time-averaged integrated electromagnetic torques in any finite volume of the medium associated with the Minkowski, symmetrized Minkowski and Abraham drive forms. However, they are all the same. This is to be expected, since if one observes from the tables in appendix A of paper I that with the medium at rest (*U*=*V*), although the medium is inhomogeneous, the associated current 2-forms for *U*(*K*)=0 are the same and for harmonic waves. Moreover, if one considers incident circularly polarized harmonic plane waves, the total time-averaged torques are each zero. Hence, harmonic plane waves cannot distinguish the effects of the Minkowski, symmetrized Minkowski and Abraham electromagnetic stress–energy–momentum tensors when incident on non-accelerating inhomogeneous slabs of simple media. It is natural to enquire how this result may change when the medium accelerates. This is explored in the next section for a simple planar inhomogeneous dielectric slab in uniform rotation about its axis of symmetry.

## 6. Circularly polarized plane waves incident on a simple rotating inhomogeneous dielectric cylinder

In the last section, the fields in a stationary linearly inhomogeneous slab excited by an incident plane harmonic wave were found. Such fields yield a non-zero time-averaged net force (and torque) on such a slab, which is the same for both the Abraham and symmetrized Minkowski electromagnetic stress–energy–momentum tensors. One could attempt to find the fields excited by an incident electromagnetic pulse such as that produced by a laser. Since such fields are not harmonic, they may yield forces and torques that distinguish between such tensors particularly if one also employs a slab with anisotropic or magneto-electric properties (Tucker & Walton 2009). However, the precise nature of the fields in a laser pulse with a finite spot size is clearly more difficult to ascertain theoretically and this makes the problem of matching fields at the slab interfaces more difficult than with harmonic plane waves.

In this section, it is shown how circularly polarized *plane harmonic* waves can excite a transverse plane-wave field in a simple inhomogeneous *uniformly rotating* medium analogous to that found in the previous section. The form of this wave is determined by a single amplitude function of *z* in the cylindrical coordinates used in §4, satisfying a second-order *ordinary* differential equation with coefficients dependent on the frequency of the incident wave, the speed of rotation of the slab and its constitutive properties. Although this equation has no solution in terms of simple analytical functions, it is amenable to numerical analysis. Furthermore, one can match a basis of such solutions to the vacuum plane waves at the rotating plane–slab interfaces. If one accepts that the edge effects produced by the rim of the rotating slab are ignorable, then an approximate numerical estimate of the fields excited in the slab by normally incident harmonic plane waves can be made. From such a solution one may compute the net time-averaged torque on the slab for both the Abraham and symmetrized Minkowski electromagnetic stress–energy–momentum tensors. It will be shown that, for a simple medium with relative permeability 1 and a permittivity that varies linearly with *z* in the slab, these two torques are significantly different.

In cylindrical coordinates, suppose the cylindrical slab rotates with constant angular speed *Ω* about its axis of symmetry along the *z*-direction. Let the cylinder domain II have constant permeability *μ*_{r} but relative permittivity *ϵ*_{r}(*z*) and denote the left and right vacuum domains by I and III, respectively. Since the field in this medium is to be excited by a transverse circularly polarized plane harmonic vacuum wave, we look for monochromatic time-harmonic interior solutions with a TDB-type field ansatz in the cylindrical coframe used in §4,With the real constant ℰ assigned the physical dimensions of [*Q*]/[*L*^{2}][*ϵ*_{0}], the complex amplitudes *B*(*z*),*D*(*z*) to be determined are dimensionless. The electric field 1-form and the magnetic field 1-form are now determined from the constitutive relation for a rotating dielectric. Generalizing *m*=0, the TDB mode in homogeneous media found in §4 may be written in the formUsing the constitutive relations (2.23) of paper I with , it follows thatto leading order in *ν*/*c*_{0}. The fields *F*^{II} and *G*^{II} can now be constructed to satisfy *d* *F*^{II}=0 and *d* **G*^{II}=0. These equations requirewhere *f*′(*z*) denotes the derivative of *f*(*z*) with respect to *z*. Substituting *D*(*z*) into the first differential equation gives a second-order differential equation for *B*(*z*),6.1Note that with a real *ϵ*_{r}(*z*), the coefficients of this differential equation are real. Write the solution for *B*(*z*) as6.2in terms of complex dimensionless constants *C*_{1},*C*_{2} and a basis of *real* solutions {*σ*_{1}(*z*),*σ*_{2}(*z*)}. It follows thatThe forms *F*^{II} and *G*^{II} inside the rotating slab are now expressed in terms of a basis of solutions for equation (6.1) as6.3As we have seen in the last section, when *Ω*=0 and *ϵ*_{r}(*z*) varies linearly with *z*, the basis solutions can be expressed in terms of Airy functions. The complex constants in (6.3) are determined as before by matching *F*^{II} and *G*^{II} to harmonic plane waves at the interfaces of region II with regions I and III. Thus, with *F*^{I}=d *A*^{I}, *F*^{III}=d *A*^{III}, *G*^{I}=*ϵ*_{0}*F*^{I}, *G*^{III}=*ϵ*_{0}*F*^{III} andthe plane interface conditions yield the following equations for the dimensionless complex variables :where

This matrix equation admits a solution providedThus, one has a matched solution to the complete electromagnetic system in terms of a basis of solutions for equation (6.1) and *ϵ*_{r}(*z*). With *K*=∂_{θ}, one may now express the time-averaged total torques, and , on a thin rotating disc of volume 𝒱 in terms of this solution and, for each *Ω* and *ω*, numerically integrate equation (6.1) to find the needed values of *σ*_{1}(*z*) and *σ*_{2}(*z*) in the slab.

The results in figures 3 and 4 are for *ϵ*_{r}(*z*)=*α*+(*β*/ℓ)*z* with real constants *α*,*β* describing the variation of permittivity across the cylindrical disc of thickness ℓ. They clearly indicate that if one could devise an experiment to measure the mechanical torque needed to maintain a simple inhomogeneous dielectric medium in uniform rotation in both the absence and presence of a normally incident circularly polarized harmonic plane wave, then one may be able to discriminate between the effects induced by different electromagnetic stress–energy–momentum tensors.

## 7. Conclusions

Unlike the Minkowski tensor and its symmetrized version, the divergence of the Abraham tensor depends *explicitly* on the bulk acceleration field of the medium. Given the simple electromagnetic constitutive relations, this implies the existence of local acceleration-dependent Abraham forces and torques in the presence of electromagnetic fields. One could write the Abraham tensor as the sum of any other symmetric tensor (including the symmetrized Minkowski tensor) and the difference between the two. This of course makes no difference to the divergence, but *assigning* the difference in the tensors to the accompanying matter tensor appears to us ad hoc and unwarranted. As an illustration, we would prefer to identify the radiation reaction force experienced by a radiating point charge (which depends on the rate of change of the acceleration of the particle) with an intrinsic property of the charge’s interaction with the electromagnetic field rather than with the charge’s inertia (which conventionally depends only on the acceleration) and some additional force. Furthermore, from an action variational viewpoint, it is natural to associate particular stress–energy–momentum tensors with particular interacting systems since the extrema of the total action determine both the total stress–energy–momentum *and* the field equations for the fully coupled system.

The view advocated here is that it does make sense to try and pin down by theoretical modelling and experiment the most appropriate *form* of an electromagnetic stress–energy–momentum tensor for as large a range of electromagnetic constitutive relations as possible. As emphasized in Dereli *et al.* (2006, 2007), we believe that variational principles including gravitation are a valuable theoretical guide in this endeavour. On the experimental side, it is suggested here that experiments involving accelerating media might offer a sensitive means to discriminate between alternatives.

From the calculations for the macroscopic time-averaged specific torques on simple media shown in figures 3 and 4, it is clear that those corresponding to the symmetrized Minkowski tensor are much larger than those associated with the Abraham tensor. In each figure, the physical torque has been calculated for a cylinder of radius 1 m and thickness ℓ=1 mm as a function of the uniform rotation speed *Ω* in radians per second. The parameters defining the linear variation of permittivity across the thickness of the cylinder are *α*=1 and *β*=100. The torque curves are calculated for a range of frequencies (indicated in hertz in the legend to each figure) of the normally incident circularly polarized incident plane wave. The magnitude of each physical torque (in N m^{−1}) is obtained by multiplying the indicated specific torques by ℰ^{2}, where ℰ is the amplitude of the electric field of the incident plane wave in volts per metre. Recall that the time-averaged power in a harmonic plane-wave crossing unit normal area is ℰ^{2}/(2*Z*_{0}) watts per metre squared in terms of the characteristic impedance of free space. The detection of such a torque would require the measurement of an externally applied torque to maintain a non-deformed cylinder in uniform rotation. It is interesting to note the torque asymmetry about *Ω*=0 from the figures. This asymmetry can be shifted along the horizontal axes by using an oppositely polarized incident beam and is clearly a direct manifestation of its intrinsic angular momentum. The significant quantitative differences between the time-averaged Abraham and symmetrized Minkowski torques in simple inhomogeneous rotating dielectrics suggest that new opportunities exist for exploring the effects of classical electrodynamics in more general rotating media.

## Acknowledgements

The authors are grateful to the Cockcroft Institute, the Alpha-X project, STFC and EPSRC (EP/E001831/1) for financial support for this research.

## Footnotes

↵1 From the adjoint relation

*ζ*^{db †}(,*k**ω*)=−*ζ*^{he}(,*k**ω*).↵2 With

*Ω*constant, a real unconstrained medium would not remain unstressed in a strictly rigid state according to Newtonian continuum mechanics. Furthermore, we eschew all issues associated with notions of relativistic rigidity assuming that they are peripheral to the main discussion here.↵3 For

*homogeneous*media it is only necessary to solve these coupled equations since the other spatial Maxwell equations are then automatically satisfied.

- Received February 23, 2010.
- Accepted May 17, 2010.

- © 2010 The Royal Society