# Electromagnetic momentum and the energy–momentum tensor in a linear medium
with magnetic and dielectric properties

Michael E. Crenshaw  US Army Aviation and Missile Research, Development, and
Engineering Center, Redstone Arsenal, AL 35898, USA

###### Abstract

The well-defined total energy and total momentum in a thermodynamically closed
system with complete equations of motion are used to construct the total
energy–momentum tensor for a stationary simple linear material with both
magnetic and dielectric properties illuminated by a quasimonochromatic pulse
of light through a gradient-index antireflection coating. The perplexing
issues surrounding the Abraham and Minkowski momenta are bypassed by working
entirely with conservation principles, the total energy, and the total
momentum. We derive electromagnetic continuity equations and equations of
motion for the macroscopic fields based on the material four-divergence of the
traceless, symmetric total energy–momentum tensor. The energy–momentum
formalism is consistent with the derivation of equations of motion for
macroscopic fields from a Lagrangian.

##  I Introduction

The resolution of the Abraham–Minkowski momentum controversy for the
electromagnetic momentum and the energy–momentum tensor in a linear medium is
multifaceted, complex, and nuanced. It is not sufficient to simply derive an
electromagnetic momentum or energy–momentum tensor from the macroscopic
Maxwell equations. That much is apparent from any examination of the
scientific literature. After a century of study BIPfei ; BIMilBoy ; BIKemp ;
BIBaxL ; BIBarL , almost all aspects of the Abraham–Minkowski momentum
controversy have been carefully scrutinized, yet there is scarce mention in
the literature of conservation of linear momentum. None of the well-known
forms of continuum electrodynamic momentum, Minkowski BIMin , Abraham BIAbr ,
Einstein–Laub BIEinLau , Peierls BIPei , etc., are conserved and each one is
generally regarded as the momentum of some unspecified or arbitrary portion of
the whole system BIPfei ; BIMilBoy ; BIKemp ; BIBaxL ; BIBarL ; BIPei ; BIKran
; BIPenHau ; BIMikura ; BIGord ; BIBPRL . A number of composite momentums have
been constructed from the Abraham, Minkowski, or other momentum for the field
subsystem with a momentum for the material subsystem BIPfei ; BIMilBoy ;
BIKemp ; BIBaxL ; BIBarL ; BIPei ; BIKran ; BIPenHau ; BIMikura ; BIGord ;
BIBPRL . With the exception of the Gordon BIGord ; BICB ; BICB2 ; BISPIE
momentum in a dielectric, conservation of the composite momentum in a
thermodynamically closed system has not been explicitly demonstrated.

It is important to work with the conserved forms of energy and momentum in
order to avoid the ambiguous definitions of field and material momentums in
unclosed physical systems with incomplete equations of motion BIKran ; BICB ;
BICB2 ; BISPIE . In this article, we carefully define a thermodynamically
closed continuum electrodynamic system with complete equations of motion
containing a stationary simple linear medium illuminated by a plane
quasimonochromatic electromagnetic pulse through a gradient-index
antireflection coating and identify the conserved total energy and the
conserved total momentum, thereby extending prior work BICB ; BICB2 ; BISPIE
on dielectrics to a medium with both magnetic and dielectric properties.
Assuming the validity of the macroscopic Maxwell–Heaviside equations, the
total momentum in a simple linear medium is found using the law of
conservation of linear momentum. We then populate the total energy–momentum
tensor with the densities of the conserved energy and momentum using the
uniqueness property that certain elements of the tensor correspond to elements
of the four-momentum. We prove that the uniqueness property directly
contradicts the property that the four-divergence of the energy–momentum
tensor generates continuity equations BISPIE . We retain the uniqueness
property and recast the four divergence of the energy–momentum tensor in terms
of a material four-divergence operator whose time-like coordinate depends on
the refractive index BICB ; BICB2 ; BISPIE ; BIFinn . Once the elements and
properties of the energy–momentum tensor are defined, we derive the continuity
equations for the total energy. The total energy continuity equation is a
mixed second-order differential equation that we write as first-order
equations of motion for the macroscopic fields. These equations are
mathematically equivalent to the macroscopic Maxwell–Heaviside equations in
that they can be transformed into one another using vector identities. It is
not that simple, though, because the momentum continuity equation that is
derived from the equations of motion for the the fields is not consistent with
an energy–momentum formalism. We propose a resolution to this contradiction by
a reformulation of continuum electrodynamics in which the permittivity and
permeability are eliminated in favor of a linear refractive index that is
defined by the electric and magnetic properties of the material.

##  II The Total Energy and the Total Momentum

A physical theory is an abstract mathematical description of some portion of a
real-world physical process. The theoretical description is useful to the
extent that there are correspondences between a subset of the theory and a
subset of the behavior of the real-world system BIRind . Real materials are
really complicated and we must carefully define the material properties and
boundary conditions in order to insure that the mathematical description
contains the most important characteristics while excluding the less
significant details. We define a simple linear material as an isotropic and
homogeneous medium that has a linear magnetic and a linear dielectric response
to an electromagnetic field that is tuned sufficiently far from any material
resonances that absorption and dispersion are negligible. Electrostrictive
effects and magnetostrictive effects are also taken as being negligible
BIMikura . Clearly, we are not dealing with any real material, but the
idealization of a simple linear material is necessary in order to develop the
basic concepts in a clear, concise, convincing, and incontrovertible manner.

A rectangular prism of the simple linear material in free space is illuminated
by a quasimonochromatic pulse of radiation at normal incidence in the plane-
wave limit. This system is thermodynamically closed with a conserved total
energy and a conserved total momentum but we also require a complete set of
equations of motion in order to obtain well-defined quantities for the total
energy and total momentum. There are various formulations of the
electrodynamics of uniformly moving media, but a complete treatment of
continuum electrodynamics in moving media is far from settled BIPenHau . We
also have to recognize that the material is accelerating due to a radiation
surface pressure that can be attributed to the partial reflection of the
incident radiation. In this work, the prism of material is covered with a thin
gradient-index antireflection coating, not an interference antireflection
coating, that makes reflections and the acceleration from radiation surface
pressure negligible allowing the material to be regarded as stationary in the
laboratory frame of reference. The assumption of a rare vapor, by Gordon
BIGord  for example, accomplishes the same purpose of making reflections
negligible, but then requires justification to apply the results to a material
with a non-perturbative index of refraction. There is nothing unusual,
mysterious, nefarious, or confusing about the conditions described above
because the use of a gradient-index antireflection coating on a homogeneous
linear material is required, and has always been required, for a rigorous
application of Maxwell’s equations to solids. Still, Maxwell’s equations are
almost always applied to moving and accelerating materials without adequate
justification. That s not to say that the use of the macroscopic Maxwell
equations cannot be justified in most cases, but it is especially important to
justify the use of the macroscopic Maxwell equations when invoking
conservation properties.

Classical continuum electrodynamics is founded on the macroscopic Maxwell
equations, so that is where we start our work. The common macroscopic
Maxwell–Heaviside equations

$\nabla\times{\bf E}=-\frac{\mu}{c}\frac{\partial{\bf H}}{\partial t}$ (1)
$\nabla\times{\bf H}=\frac{\varepsilon}{c}\frac{\partial{\bf E}}{\partial t}$
(2) $\nabla\cdot{\bf B}=0$ (3) $\nabla\cdot\varepsilon{\bf E}=0$ (4)

are the complete equations of motion for the macroscopic fields in a
stationary simple linear medium. Here, ${\bf E}$ is the electric field, ${\bf
B}$ is the magnetic field, ${\bf H}={\bf B}/\mu$ is the auxiliary magnetic
field, $\varepsilon$ is the electric permittivity, $\mu$ is the magnetic
permeability, and $c$ is the speed of light in the vacuum. The medium is
explicitly required to be stationary because the macroscopic Maxwell–Heaviside
equations, Eqs. (1)–(4), are not complete for moving or accelerating
materials. The electric and magnetic fields can be defined in terms of the
vector potential in the usual manner as

${\bf E}=-\frac{1}{c}\frac{\partial{\bf A}}{\partial t}$ (5) ${\bf
B}=\nabla\times{\bf A}\,.$ (6)

Then, the propagation of an electromagnetic field through free-space and into
a simple linear medium is described by the wave equation,

$\nabla\times(\nabla\times{\bf A})+\frac{n^{2}}{c^{2}}\frac{\partial^{2}{\bf
A}}{\partial t^{2}}=\frac{\nabla\mu}{\mu}\times(\nabla\times{\bf A})\,,$ (7)

assuming the validity of the macroscopic Maxwell–Heaviside equations, Eqs.
(1)–(4). Here, $n({\bf r})=\sqrt{\mu\varepsilon}$ is the spatially slowly
varying linear refractive index. The spatial variation of the index and
permeability is limited to a narrow transition region in which these
quantities change gradually from the vacuum values to the nominal material
properties. We write the vector potential in terms of a slowly varying
envelope function and a carrier wave as

${\bf A}({\bf r},t)=\frac{1}{2}\left({\bf\tilde{A}}({\bf
r},t)e^{-i(\omega_{d}t-{\bf k}_{d}\cdot{\bf r})}+{\bf\tilde{A}}^{*}({\bf
r},t)e^{i(\omega_{d}t-{\bf k}_{d}\cdot{\bf r})}\right)\,,$ (8)

where $\tilde{A}$ is a slowly varying function of ${\bf r}$ and $t$, ${\bf
k}_{d}=(n\omega_{d}/c){\bf e}_{\bf k}$ is the wave vector that is associated
with the center frequency of the field $\omega_{d}$, and ${\bf e}_{\bf k}$ is
a unit vector in the direction of propagation.

Figure 1 shows a one-dimensional representation of the slowly varying
amplitude of the plane incident field
$\tilde{A}_{i}(z)=({\bf\tilde{A}}(z,t_{0})\cdot{\bf\tilde{A}}^{*}(z,t_{0}))^{1/2}$
about to enter the simple linear medium with index $n=1.386$ and permeability
$\mu=1.2$ through a gradient-index antireflection coating.

Figure 1: Incident field (amplitude arbitrary) just before it enters the
linear medium.

The gradient that has been applied to the index has also been applied to the
permeability as a matter of convenience. Figure 2 presents a time-domain
numerical solution of the wave equation at a later time $t_{1}$ when the
refracted field
$\tilde{A}_{t}(z)=({\bf\tilde{A}}(z,t_{1})\cdot{\bf\tilde{A}}^{*}(z,t_{1}))^{1/2}$
is entirely inside the medium.

Figure 2: Refracted field entirely within the linear medium.

The pulse has not propagated as far as it would have propagated in the vacuum
due to the reduced speed of light $c/n$ in the material. In addition, the
spatial extent of the refracted pulse in the medium is $w_{t}=w_{i}/n$ in
terms of the width $w_{i}$ of the incidence pulse due to the reduced speed of
light. As shown in Fig. 2, the amplitude of the refracted field is
${\tilde{A}}_{t}=\sqrt{{\mu}/{n}}{\tilde{A}}_{i}$, that is, the amplitude of
the incident field scaled by $\sqrt{\mu/n}$. Using these relations we can
construct the temporally invariant quantity

$U_{total}=\int_{\sigma}\frac{n}{2}\left(\sqrt{\frac{n}{\mu}}\frac{\omega_{d}}{c}\tilde{A}\right)\left(\sqrt{\frac{n}{\mu}}\frac{\omega_{d}}{c}\tilde{A}\right)^{*}dv$
(9)

and note that $U_{total}$ is conserved when the integration is performed over
a region $V$ that contains all of the electromagnetic field that is present in
the system. In that case, the region of integration $V$ is extended to all
space, $\sigma$. The pulse that is used in Fig. 1 has a generally rectangular
shape in order to facilitate a graphical interpretation of pulse width and
integration under the field envelope. It can be shown by additional numerical
solutions of the wave equation that the relations described above are quite
general in terms of the permeability and the refractive index, as well as the
shape and amplitude of a quasimonochromatic field.

For a stationary linear medium, the total energy is the electromagnetic
energy. Although a material may possess many forms of energy, we intend total
energy to mean all of the energy that impacts the dynamics or electrodynamics
of the model system with the specified conditions. In particular, kinetic
energy is excepted by the requirement for the material to remain stationary.
We assume that the electromagnetic energy density for a stationary simple
linear medium is

$\rho_{e}=\frac{1}{2}\frac{1}{\mu}\left(n^{2}{\bf E}^{2}+{\bf
B}^{2}\right)dv\,,$ (10)

as inferred from the Poynting theorem. Again extending the region of
integration $V$ that contains all fields present to integration over all space
$\sigma$, the electromagnetic energy,

$U_{e}=\int_{\sigma}\rho_{e}dv=\int_{\sigma}\frac{1}{2}\frac{1}{\mu}\left(n^{2}{\bf
E}^{2}+{\bf B}^{2}\right)dv\,,$ (11)

is the total energy of the closed system containing a stationary simple linear
medium illuminated by a quasimonochromatic pulse of light. Using Eqs. (5) and
(6) to eliminate the electric and magnetic fields in Eq. (11), we obtain the
energy formula

$U_{e}=\int_{\sigma}\frac{1}{2}\frac{1}{\mu}\left(\left(\frac{n}{c}\frac{\partial{\bf
A}}{\partial t}\right)^{2}+(\nabla\times{\bf A})^{2}\right)dv\,.$ (12)

Employing the expression for the vector potential in terms of slowly varying
envelope functions and carrier waves, Eq. (8), results in

$U_{e}=\int_{\sigma}\frac{\omega_{d}^{2}n^{2}}{4\mu
c^{2}}\left(\tilde{A}\tilde{A}^{*}-\tilde{A}^{2}e^{-2i(\omega_{d}-k_{d}z)t}+c.c.\right)dv\,.$
(13)

Applying a time average allows the double frequency terms to be neglected and
shows the equality of Eqs. (9) and (12),

$U_{e}=\int_{\sigma}\frac{\omega_{d}^{2}n^{2}}{2\mu
c^{2}}|\tilde{A}|^{2}dv=U_{total}\,.$ (14)

We have theoretical confirmation that our numerical $U_{total}$, Eq. (9), is
the total energy of the system and that it is conserved. More importantly, we
have an interpretation of the continuum electrodynamic energy formula, Eq.
(11), in term of what happens to the shape and amplitude of the
electromagnetic field as it propagates into the medium.

In the first half of the last century, there was an ongoing theoretical and
experimental effort devoted to determining the correct description of the
momentum of light propagating through a linear medium BIPfei ; BIMilBoy ;
BIKemp ; BIBaxL ; BIBarL . At various times, the issue was resolved in favor
of the Abraham momentum

${\bf G}_{A}=\int_{\sigma}\frac{{\bf E}\times{\bf H}}{c}dv$ (15)

and at other times the momentum was found to be the Minkowski momentum

${\bf G}_{M}=\int_{\sigma}\frac{{\bf D}\times{\bf B}}{c}dv\,.$ (16)

Since Penfield and Haus BIPenHau  showed, in 1967, that neither the Abraham
momentum nor the Minkowski momentum is the total momentum, the resolution of
the Abraham–Minkowski controversy has been that the momentum for an arbitrary
or unspecified field subsystem must be supplemented by an appropriate material
momentum to obtain the total momentum BIPfei ; BIMilBoy ; BIKemp ; BIBaxL ;
BIBarL ; BIPei ; BIKran ; BIPenHau ; BIMikura ; BIGord ; BIBPRL . Then, it
must be demonstrated that the total momentum, so constructed, is actually
conserved in a thermodynamically closed system, as that is usually not the
case. The existence of a continuity equation, also known as a conservation
law, is not sufficient to prove conservation, as evidenced by the non-
conservation of Poynting’s vector ${\bf S}=c{\bf E}\times{\bf H}$.

Given the arbitrariness of the field momentum and the model dependence of the
material momentum that go into a construction of the total momentum, the only
convincing way to determine the total momentum is by the application of
conservation principles. Then conservation of the total energy is all that is
needed to prove that the total momentum BIGord ; BICB ; BICB2 ; BISPIE

${\bf G}_{total}=\int_{\sigma}\frac{1}{\mu}\frac{n{\bf E}\times{\bf
B}}{c}dv=\int_{\sigma}\frac{n{\bf E}\times{\bf H}}{c}dv$ (17)

is conserved in our closed system. In the limit of slowly varying plane waves,
we have the conserved vector quantity

${\bf G}_{total}=\int_{\sigma}\frac{\omega_{d}^{2}n^{2}}{2\mu
c^{3}}|\tilde{A}|^{2}dv{\bf e}_{\bf k}=\frac{U_{total}}{c}{\bf e}_{\bf k}$
(18)

that is equal to the conserved total energy, Eq. (14), divided by a constant
factor of $c$ and multiplied by a unit vector ${\bf e}_{\bf k}$. In addition
to being conserved, the total momentum, Eq. (17), is unique. There can be no
other non-trivial conserved macroscopic momentum in terms of the vector
product of electric and magnetic fields in our model system because these
would contain different combinations of the material properties. The momentum
formula, Eq. (17), was originally derived for a dielectric in 1973 by Gordon
BIGord  who combined the Abraham momentum for the field with a material
momentum that was obtained from integrating the Lorentz force on the
microscopic constituents of matter. At the time, Gordon’s momentum was
unremarkable among a large group of momentum formulas that had been derived by
various methods. In fact, Gordon viewed it as a portion of the Minkowski
momentum, along with a pseudomomentum. What is extraordinary about the Gordon
momentum is that it is uniquely conserved in a thermodynamically closed
continuum electrodynamic system containing a stationary simple linear medium.

##  III Total Energy–Momentum Tensor

A tensor is a very special mathematical object and a total energy–momentum
tensor is subject to even more stringent conditions. Any four continuity
equations can be combined using linear algebra to form a matrix differential
equation. Hence, the proliferation of matrices that have been purported to be
the energy–momentum tensor for a system containing an electromagnetic pulse
and a linear medium. In this section, we construct the unique total
energy–momentum tensor and the tensor continuity equation for a stationary
simple linear medium using the conservation, symmetry, trace, and divergence
conditions that must be satisfied.

For an unimpeded (force-free) flow, conservation of the components of the
total four-momentum $(U,{\bf G})$ BILL

$U=\int_{\sigma}T^{00}dv$ (19) $G^{i}=\frac{1}{c}\int_{\sigma}T^{i0}dv$ (20)

uniquely determines the first column of the total energy–momentum tensor. We
use the convention that Roman indices from the middle of the alphabet, like
$i$, run from 1 to 3 and Greek indices belong to $(0,1,2,3)$. As in Section
II, the region of integration has been extended to all-space, $\sigma$.
Conservation of angular momentum in a closed system imposes the diagonal
symmetry condition BILL

$T^{\alpha\beta}=T^{\beta\alpha}$ (21)

and uniquely determines the first row of the total energy–momentum tensor
based on the uniqueness of the first column. Applying the conservation of the
total energy and total momentum, we can construct the total energy–momentum
tensor

$T^{\alpha\beta}=$ $\left[\begin{matrix}(n^{2}{\bf E}^{2}+{\bf
B}^{2})/(2\mu)&(n{\bf E}\times{\bf H})_{x}&(n{\bf E}\times{\bf H})_{y}&(n{\bf
E}\times{\bf H})_{z}\cr(n{\bf E}\times{\bf
H})_{x}&W_{11}&W_{12}&W_{13}\cr(n{\bf E}\times{\bf
H})_{y}&W_{21}&W_{22}&W_{23}\cr(n{\bf E}\times{\bf
H})_{z}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right]$ (22)

where the elements of the Maxwell stress-tensor $W$ are yet to be specified.

Our approach here is very different from the usual technique of constructing
the energy–momentum tensor from the electromagnetic continuity equations. The
energy continuity equation is known to be given by Poynting’s theorem

$\frac{1}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left({\bf
E}\times{\bf H}\right)=0\,.$ (23)

However if we use Poynting’s theorem and the tensor energy continuity law

$\partial_{\beta}T^{0\beta}=0$ (24)

to populate the first row of the energy–momentum tensor, then Eq. (20) becomes

$G^{i}=\frac{1}{c}\int_{\sigma}({\bf E}\times{\bf H})_{i}dv$ (25)

by symmetry, Eq. (21). This result is contraindicated because

${\bf G}=\frac{1}{c}\int_{\sigma}\frac{\omega_{d}^{2}n}{2\mu
c^{2}}|\tilde{A}|^{2}{\bf e}_{\bf k}dv=\frac{U_{total}}{nc}{\bf e}_{\bf k}$
(26)

is not temporally invariant as the pulse travels from the vacuum into the
medium. The practice has been to ignore the condition that the components of
the four-momentum, Eqs. (19) and (20), be conserved and use the components of
the Poynting vector, along with the electromagnetic energy density, to
populate the first row of the energy–momentum tensor using the tensor energy
continuity law, Eq. (24).

We regard the conservation properties of the total energy–momentum tensor as
fundamental and these conservation properties conclusively establish Eq. (22)
as the form of the energy–momentum tensor. Applying the tensor continuity
equation, Eq. (24), to the energy–momentum tensor, Eq. (22), we obtain an
energy continuity equation

$\frac{1}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left(n{\bf
E}\times{\bf H}\right)=0$ (27)

that is inconsistent with the Poynting theorem, Eq. (23). In the limit of
slowly varying plane waves, we obtain an obvious contradiction

$\frac{n^{2}\omega_{d}^{2}}{2\mu
c^{3}}\left({\bf\tilde{A}}\frac{\partial{\bf\tilde{A}}^{*}}{\partial
t}+c.c.\right)-\frac{n^{3}\omega_{d}^{2}}{2\mu
c^{3}}\left({\bf\tilde{A}}\frac{\partial{\bf\tilde{A}}^{*}}{\partial
t}+c.c.\right)=0$ (28)

from the energy continuity equation, Eq. (27).

We have identified the essential contradiction of the Abraham–Minkowski
controversy: Absent the uniqueness that accompanies conservation of total
energy and total momentum, the components of the first row and column of the
energy–momentum tensor are essentially arbitrary thereby rendering the
energy–momentum tensor meaningless. Yet, the energy continuity equation that
is obtained from the four-divergence of an energy–momentum tensor that is
populated with the densities of the conserved energy and momentum quantities
is demonstrably false. We are at an impasse with the existing theory that
requires two contradictory conditions to be satisfied BISPIE .

The resolution of this contradiction cannot be found within the formal system
of continuum electrodynamics. Based on the reduced speed of light in the
medium, $c/n$, we make the $ansatz$ that the continuity equations are
generated by

$\bar{\partial}_{\beta}T^{\alpha\beta}=f_{\alpha}\,,$ (29)

where $\bar{\partial}_{\beta}$ is the material four-divergence operator BICB ;
BICB2 ; BISPIE ; BIFinn

$\bar{\partial}_{\beta}=\left(\frac{n}{c}\frac{\partial}{\partial
t},\frac{\partial}{\partial x},\frac{\partial}{\partial
y},\frac{\partial}{\partial z}\right)\,.$ (30)

The continuity equation, Eq. (29), is generalized in a form that allows a
source/sink $f_{\alpha}$ of energy and the components of momentum BICB2 .
Reference BIxxx  provides a solid theoretical justification for the ansatz of
Eqs. (29) and (30). Applying the tensor continuity law, Eq. (29), to the
energy–momentum tensor, Eq. (22), we obtain the energy continuity equation

$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left(n{\bf
E}\times{\bf H}\right)=f_{0}\,.$ (31)

Now, let us re-consider the Poynting theorem. We multiply Poynting’s theorem,
Eq. (23), by $n$ and use a vector identity to commute the index of refraction
with the divergence operator to obtain

$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left(n{\bf
E}\times{\bf H}\right)=\frac{\nabla n}{n}\cdot(n{\bf E}\times{\bf H})\,.$ (32)

The two energy continuity equations, Eqs. (31) and (32), are equal if we
identify the source/sink of energy as

$f_{0}=\frac{\nabla n}{n}\cdot(n{\bf E}\times{\bf H})\,.$ (33)

Then an inhomogeneity in the index of refraction, $\nabla n\neq 0$, is
associated with work done by the field on the material or work done on the
field by the interaction with the material BICB2 . Obviously, the gradient of
the index of refraction in our anti-reflection coating must be sufficiently
small that the work done is perturbative.

The Poynting’s theorem that is found in Eq. (32) is a mixed second-order
vector differential equation. It can be separated into two mixed first-order
vector differential equations for the macroscopic electric and magnetic fields

$\frac{n}{c}\frac{\partial{\bf B}}{\partial t}=-\nabla\times n{\bf
E}+\frac{\nabla n}{n}\times n{\bf E}$ (34) $\frac{n}{c}\frac{\partial n{\bf
E}}{\partial t}=\mu\nabla\times{\bf H}.$ (35)

We provide confirmation of this result in the Appendix, deriving the equations
of motion for the macroscopic fields from a Lagrangian.

The remaining task is to show that the material four-divergence of the
energy–momentum tensor is a faithful representation of the electromagnetic
continuity equations. We multiply Eq. (34) by ${\bf H}$ and multiply Eq. (35)
by $n{\bf E}$. The resulting equations are summed to produce the energy
continuity equation

$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot\left(n{\bf
E}\times{\bf H}\right)=\frac{\nabla n}{n}\cdot(n{\bf E}\times{\bf H})\,.$ (36)

To obtain the total momentum continuity equation, we substitute Eqs. (34) and
(35) into the material timelike derivative of the total momentum density

$\frac{n}{c}\frac{\partial{\bf g}_{total}}{\partial
t}=\frac{1}{\mu}\frac{n}{c}\frac{\partial n{\bf E}}{\partial t}\times{\bf
B}+n{\bf E}\times\frac{1}{\mu}\frac{n}{c}\frac{\partial{\bf B}}{\partial t}\,$
(37)

where the momentum density ${\bf g}_{total}$ is the integrand of Eq. (17). The
momentum continuity equation

$\frac{n}{c}\frac{\partial{\bf g}_{total}}{\partial t}=$
$\frac{1}{c\mu}\left[(\nabla\times n{\bf E})\times n{\bf
E}+\mu^{2}(\nabla\times{\bf H})\times{\bf H}+\frac{\nabla n}{n}(n{\bf
E})^{2}\right]$ (38)

is not expressible using the tensor continuity equation, Eq. (29), without
additional transformations.

There is a scientific record of algebraic transformations of the energy and
momentum continuity equations. Abraham BIAbr  was apparently the first to
pursue this approach, deriving a variant form of the continuity equation for
the Minkowski momentum in which the time-derivative of the Minkowski momentum
is split into the temporal derivative of the Abraham momentum and a fictitious
Abraham force. Kinsler, Favaro, and McCall BIKins  discuss various
transformations of Poynting’s theorem and the resulting differences in the way
various physical processes are expressed. Frias and Smolyakov BIFrias  did the
same for transformations of the momentum continuity equation. It appears to
have gone unrecognized, however, that the different expression of physical
processes in the electromagnetic continuity equations has to carry over into
the Maxwell equations of motion for the macroscopic fields, Eqs. (34) and
(35). As a consequence, the energy and momentum continuity equations cannot be
separately transformed. We could, for example, commute the magnetic
permeability with the curl operator in the Maxwell–Ampère Law, Eq. (35) in
order to allow the momentum continuity law, Eq. (38) to be expressed in the
form of a continuity equation. However, the resulting energy continuity law
would run afoul of the uniqueness condition on the elements of the first row
of the energy–momentum tensor.

The energy–momentum tensor stands at the center of theoretical continuum
electrodynamics. The total energy–momentum tensor, Eq. (22), was constructed
using conservation principles and it is inarguably correct as long as the
formula for the energy, Eq. (11), is correct. However, there is a
contradiction between the continuity equations of energy and momentum for a
linear medium with a full or partial magnetic response. Again, we are
confronted with a situation of contradiction that we cannot derive our way out
of and we must justify a correction based on an analysis of the physical
model.

##  IV The Linear Index of Refraction

Consider a rectangular volume of space $V=Ad$, where $A$ is the cross-
sectional area perpendicular to the direction of an incident electromagnetic
pulse of radiation in the plane-wave limit. The volume is separated into
halves by a barrier at $d/2$ and half of the volume is filled with a vapor
with refractive index $n_{1}$. The other half of the volume is filled with a
vapor with refractive index $n_{2}$. We define a material variable
$\xi_{i}=n_{i}-1$. Then the time that it takes a light pulse to traverse the
volume is $(\xi_{1}+\xi_{2})d/(2c)$ greater than the time it takes light to
travel the same distance in a vacuum. Note that reflections do not affect the
time-of-flight of the pulse, only the amount of light that is transmitted. The
time-of-flight of the light pulse does not change when the barrier is removed.
The atoms of each vapor are noninteracting and there is no physical process
that would affect the time-of-flight of the light pulse as the two species of
atoms diffuse into each other. Generalizing to an arbitrary number of
components, the linear refractive index of a mixture of vapors obeys a
superposition principle

$n=1+\sum_{i=1}^{q}\xi_{i}$ (39)

for $q$ components. If we take $\xi_{0}=1$ for the vacuum, then

$n=\sum_{i=0}^{q}\xi_{i}$ (40)

As long as the atoms are noninteracting, the superposition principle holds
whether the vapors are composed of atoms that behave as electric dipoles,
magnetic dipoles, or as a mixture of both. Near-dipole–dipole interactions,
and other interactions between atoms, require a more detailed and model-
dependent treatment that is outside the scope of the current work.

Now that the linear refractive index is truly linear, we can regard the
permittivity and permeability as anachronisms. The refractive index carries
the material effects and we can eliminate the permeability and permittivity
from the equations of motion for the macroscopic fields, Eqs. (34) and (35),
such that

$\frac{n}{c}\frac{\partial{\bf B}}{\partial
t}=\nabla\times{\bf\Pi}-\frac{\nabla n}{n}\times{\bf\Pi}$ (41)
$\frac{n}{c}\frac{\partial{\bf\Pi}}{\partial t}=-\nabla\times{\bf B}$ (42)

where $\Pi=-n{\bf E}$. The equations of motion for the macroscopic fields,
Eqs. (41) and (42), can be combined in the usual manner to write an energy
continuity equation

$\frac{n}{c}\frac{\partial}{\partial
t}\left[\frac{1}{2}\left({\bf\Pi}^{2}+{\bf
B}^{2}\right)\right]+\nabla\cdot\left({\bf
B}\times{\bf\Pi}\right)=\frac{\nabla n}{n}\cdot({\bf B}\times{\bf\Pi})$ (43)

in terms of a total energy density

$\rho_{total}=\frac{1}{2}\left({\bf\Pi}^{2}+{\bf B}^{2}\right)\,,$ (44)

a total momentum density

${\bf g}_{total}=\frac{{\bf B}\times{\bf\Pi}}{c}\,,$ (45)

and a power density

$p_{total}=\frac{\nabla n}{n}\cdot({\bf B}\times{\bf\Pi})\,.$ (46)

Substituting Eqs. (49) and (50) into the material timelike derivative of the
total momentum, Eq. (45), the momentum continuity equation becomes

$\frac{n}{c}\frac{\partial{\bf g}_{total}}{\partial
t}+\frac{1}{c}\nabla\cdot{\bf W}=-\frac{\nabla n}{nc}(n{\bf E})^{2}$ (47)

where the Maxwell stress-tensor $W$ is

$W_{ij}=-\Pi_{i}\Pi_{j}-B_{i}B_{j}+\frac{1}{2}({\bf\Pi}^{2}+{\bf
B}^{2})\delta_{ij}\,.$ (48)

We can construct the total energy–momentum tensor

$T^{\alpha\beta}=\left[\begin{matrix}({\bf\Pi}^{2}+{\bf B}^{2})/2&({\bf
B}\times{\bf\Pi})_{x}&({\bf B}\times{\bf\Pi})_{y}&({\bf
B}\times{\bf\Pi})_{z}\cr({\bf
B}\times{\bf\Pi})_{x}&W_{11}&W_{12}&W_{13}\cr({\bf
B}\times{\bf\Pi})_{y}&W_{21}&W_{22}&W_{23}\cr({\bf
B}\times{\bf\Pi})_{z}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right]$ (49)

from the homogeneous part of the new electromagnetic continuity equations,
Eqs. (43) and (47). The total energy–momentum tensor, Eq. (49), is entirely
electromagnetic in character BIyyy  and there is no need for a supplemental
dust energy–momentum tensor for the movement of the material BIPfei . The
Maxwell–Ampère Law can be written in terms of the vector potential as a wave
equation

$\nabla\times(\nabla\times{\bf A})+\frac{n^{2}}{c^{2}}\frac{\partial^{2}{\bf
A}}{\partial t^{2}}=0\,,$ (50)

without the inhomogeneous part found in Eq. (7). We also restate the
electromagnetic, or total, energy, Eq. (11),

$U_{e}=\int_{\sigma}\rho_{e}dv=\int_{\sigma}\frac{1}{2}\left(n^{2}{\bf
E}^{2}+{\bf B}^{2}\right)dv\,,$ (51)

and the total, or Gordon, momentum, Eq. (17),

${\bf G}_{total}=\int_{\sigma}\frac{n{\bf E}\times{\bf B}}{c}dv\,.$ (52)

The ${\\{\bf E}$, ${\bf D}$, ${\bf B}$, ${\bf H}\\}$, paradigm of classical
continuum electrodynamics is altogether broken. That much was obvious from the
conservation of the Gordon form of momentum in Refs. BIGord  and BICB , also
in Eq. (17). The Abraham and Minkowski momentums are expressible solely in
terms of the ${\\{\bf E}$, ${\bf D}$, ${\bf B}$, ${\bf H}$} fields, while the
Gordon momentum is not. This is likely a contributing factor to the longevity
of the Abraham–Minkowski controversy. The Gordon momentum and the macroscopic
equations of motion depend on the fields ${\bf\Pi}$ and ${\bf B}$.
Consequently, continuum electrodynamics can be reformulated in terms of a
single pair of fields $\\{{\bf\Pi},{\bf B}\\}$ and a single field tensor BIxxx

$F^{\alpha\beta}=\left[\begin{matrix}0&\Pi_{x}&\Pi_{y}&\Pi_{z}\cr-\Pi_{x}&0&-B_{z}&B_{y}\cr-\Pi_{y}&B_{z}&0&-B_{x}\cr-\Pi_{z}&-B_{y}&B_{x}&0\cr\end{matrix}\right]\,.$
(53)

The reduction to a single pair of fields and a single field tensor is an
exquisite simplification of continuum electrodynamics.

The changes to the theoretical treatment of electromagnetic fields in linear
media that have been presented here are stunning and it is customary to
propose experiments that would validate unordinary theoretical results.
However, the existing experimental record related to the Abraham–Minkowski
controversy is an indication of serious technical and conceptual difficulties
in the measurement of continuum electrodynamic phenomena related to
macroscopic fields inside materials, fields that cannot be measured directly.
On the other hand, the dilation of time $t^{\prime}=t/n$ in a linear medium is
relatively easy to measure. An open optical cavity, or free-space etalon, is a
clock that ticks once for every round-trip of a light pulse and that ticks
more slowly when it is immersed in a dielectric fluid.

##  V Conclusion

The Abraham–Minkowski momentum controversy has been treated as an ignorable
curiosity of classical continuum electrodynamics for a very long time.
However, we should never close our minds to opportunities to gain a more
complete understanding of the physical world. The Abraham–Minkowski
controversy is readily resolved by conservation of the total momentum of a
system consisting of a quasimonochromatic pulse of radiation passing through a
stationary simple linear medium with a gradient-index antireflection coating.
Unlike previous resolutions of the Abraham–Minkowski momentum dilemma, this
one is well-defined because it is based on exact conservation laws in a
thermodynamically closed system with complete equations of motion.

*

##  Appendix A Equations of Motion for Macroscopic Fields

In Ref. BIxxx , we considered a simple linear medium in which the speed of
light is $c/n$. The generalization of the Lagrange equations for fields in a
linear medium is

$\frac{d}{d\bar{x}_{0}}\frac{\partial{\cal L}}{\partial(\partial
A_{j}/\partial\bar{x}_{0})}=\frac{\partial{\cal L}}{\partial
A_{j}}-\sum_{i}\partial_{i}\frac{\partial{\cal
L}}{\partial(\partial_{i}A_{j})},$ (54)

where $\bar{x}_{0}=ct/n$ is the time-like coordinate in the dielectric and
$x_{1}$, $x_{2}$, and $x_{3}$ correspond to the respective $x$, $y$ and $z$
coordinates. The conjugate momentum field

$\Pi_{j}=\frac{\partial{\cal L}}{\partial(\partial
A_{j}/\partial\bar{x}_{0})}$ (55)

is used to construct the Hamiltonian density

${\cal H}=\sum_{j}\Pi_{j}\frac{\partial A_{j}}{\partial\bar{x}_{0}}-{\cal L}$
(56)

from which Hamilton’s equations of motion

$\frac{\partial A_{j}}{\partial\bar{x}_{0}}=\frac{\partial{\cal
H}}{\partial\Pi_{j}}$ (57)
$\frac{\partial\Pi_{j}}{\partial\bar{x}_{0}}=-\frac{\partial{\cal H}}{\partial
A_{j}}+\sum_{i}\partial_{i}\frac{\partial{\cal
H}}{\partial(\partial_{i}A_{j})}$ (58)

are derived.

We take the Lagrangian density of the electromagnetic field in the medium to
be

${\cal L}=\frac{1}{2}\left(\frac{1}{\mu}\left(\frac{\partial{\bf
A}}{\partial\bar{x}_{0}}\right)^{2}-\frac{(\nabla\times{\bf
A})^{2}}{\mu}\right)$ (59)

in the absence of charges. Applying Eq. (55), the momentum field

${\bf\Pi}=\frac{1}{\mu}\frac{\partial{\bf A}}{\partial\bar{x}_{0}}$ (60)

is used to construct the Hamiltonian density

${\cal H}=\frac{1}{2}\left(\mu{\bf\Pi}^{2}+\frac{(\nabla\times{\bf
A})^{2}}{\mu}\right).$ (61)

Hamilton’s equations of motion in the linear medium

$\frac{\partial{\bf A}}{\partial\bar{x}_{0}}=\mu{\bf\Pi}$ (62)
$\frac{\partial{\bf\Pi}}{\partial\bar{x}_{0}}=-\nabla\times\frac{\nabla\times{\bf
A}}{\mu}$ (63)

are obtained from the Hamiltonian density, Eq. (61), using Eqs. (57) and (58).
Next, we use Eqs. (5) and (6) to eliminate the vector potential in favor of
the macroscopic electric and magnetic fields. From Eq. (62), we have
${\bf\Pi}=-n{\bf E}/\mu$. Making this substitution for ${\bf\Pi}$, we have
equations of motion for the macroscopic fields

$\frac{n}{c}\frac{\partial{\bf B}}{\partial t}=-\nabla\times n{\bf
E}+\frac{\nabla n}{n}\times n{\bf E}$ (64) $\mu\frac{n}{c}\frac{\partial n{\bf
E}}{\partial t}=\mu\nabla\times{\bf H}=\nabla\times{\bf
B}-\frac{\nabla\mu}{\mu}\times{\bf B}.$ (65)

The equations of motion for the macroscopic fields, Eqs. (64) and (65),
confirm the results, Eqs. (34) and (35), that were obtained from the
energy–momentum tensor through the material four-divergence and continuity
equations.

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