# A local model for the optical energy and momentum transfer in dielectric
media and the microscopic origin of Abraham’s force density

B. Anghinoni<EMAIL_ADDRESS>Department of Physics, Universidade
Estadual de Maringá, Maringá, PR 87020-900, Brazil  M. Partanen  Department of
Electronics and Nanoengineering, Aalto University, 00076 Aalto, Finland  N. G.
C. Astrath<EMAIL_ADDRESS>Department of Physics, Universidade Estadual de
Maringá, Maringá, PR 87020-900, Brazil

###### Abstract

We report on the continuity equations for linear momentum and energy
associated to a recently introduced electromagnetic formulation based on
classical dipolar sources [Eur. Phys. J. Plus 138, 1034 (2023)]. When
connected to the mass-polariton quasi-particle dynamics, these equations
provide a consistent microscopic description of the local optical energy and
momentum transfer inside dielectric media, called microscopic mass-polariton
formulation. This procedure also unveils the true microscopic origin of the
long-known Abraham optical force density as an interplay between induced
dipoles and mechanical stresses generated within the material.

##  I Introduction

Controlling opto-mechanical effects in matter is essential for numerous
applications in Physics, being especially important in optical manipulation
techniques [1, 2, 3, 4, 5, 6, 7, 8], photonics [9, 10, 11, 12] and
optofluidics [13, 14, 15]. Currently, however, the electromagnetic forces
acting inside matter when external fields are present do not have a definite
description. This problem is closely related to the Abraham-Minkowski
controversy [16, 17, 18, 19, 20, 21, 22, 23, 24], which treats the
electromagnetic momentum transfer inside dielectrics and appeared more than a
hundred years ago. Different electromagnetic formulations exist in the
literature to address this problem [25, 26, 27, 28], but no universal
agreement has been reached yet.

Recently, we introduced the microscopic Ampère (MA) formulation [29, 30],
which was developed for linear dielectric media under optical excitation. In
this formulation, the medium is described as a continuum of classical point
dipoles and the associated optical force density is

$\displaystyle\mathbf{f}_{\mathrm{MA}}=\frac{1}{2}\bm{\nabla}\left(\mathbf{P}\\!\cdot\\!\mathbf{E}\right)\\!+\\!\frac{1}{2}\bm{\nabla}\left(\mathbf{M}\\!\cdot\\!\mathbf{B}\right)\\!-\\!\frac{1}{2}|\mathbf{E}|^{2}\bm{\nabla}\varepsilon$
$\displaystyle\\!-\\!\frac{1}{2}|\mathbf{H}|^{2}\bm{\nabla}\mu\\!+\\!\frac{n^{2}\\!-\\!1}{c^{2}}\frac{\partial}{\partial
t}\left(\mathbf{E}\\!\times\\!\mathbf{H}\right),$ (1)

which is given in the laboratory frame and is valid for linear, isotropic,
lossless media with no dispersion, assuming the atoms’ velocities are much
smaller than the velocity of light $c$. In the last equation, $\mathbf{E}$ is
the electric field, $\mathbf{P}$ is the polarization, $\mathbf{H}$ is the
magnetic field, $\mathbf{M}$ is the magnetization,
$\mathbf{B}=\mu_{0}(\mathbf{M}+\mathbf{H})$ is the magnetic induction field,
$\varepsilon$ and $\mu$ are the medium’s permittivity and permeability,
respectively, $t$ is time, and
$n=\sqrt{\varepsilon\mu/\varepsilon_{0}\mu_{0}}$ is the medium’s refractive
index, with $\varepsilon_{0}$ and $\mu_{0}$ denoting the vacuum’s permittivity
and permeability, respectively.

The MA force density in Eq. (I) is capable of describing the vast majority of
experiments up to date involving optical forces and radiation pressure in
dielectric media as the main physical effects, as discussed in detail in Ref.
[30]. The formalism has the advantage of being derived from a well-established
microscopic model – the classical dipolar sources. On the other hand, it was
shown in Ref. [31] and subsequent works [32, 33, 34, 35] that, in order to
fulfill the covariance requirements from special relativity, there must be a
coupled state of field and matter propagating through the dielectric – the so-
called mass-polariton (MP) quasi-particle, composed of the electromagnetic
wave plus a mass density wave (MDW) generated by the disturbances in the
atomic positions due to the optical force. More specifically, the MP
formulation was built using the Abraham force density [25, 36], which has been
long adopted to describe external electromagnetic forces acting in dielectrics
[23, 37]. Despite its relatively wide range of application, Abraham’s force
density is known to not contemplate the electro- and magnetostriction effects
[16, 38], which are quadratic effects on the fields tending to compress the
material towards the region of higher field intensity and correspond to the
first and second terms in Eq. (I), respectively. This fact renders Abraham’s
force density unable to model experiments such as the ones reported in Refs.
[29, 39, 40, 41, 42] without further introducing (often phenomenologically)
extra contributions to the related continuity equations. In truth, to the best
of our knowledge, Eq. (I) is the only one in the literature in accordance with
the measurements of the optical electrostriction force density reported in
Refs. [29, 39, 42] while also being compatible with the classical model of
tiny current loops (also known as Ampèrian model) for the microscopic
magnetization mechanism in matter [43, 30]. Another very important fact is
that the few experiments that were truly able to measure the photon momentum
in matter showed a linear dependence on the medium’s refractive index [44, 45,
46, 47, 48], exactly as predicted by the coupled mass-polariton state [31].

In the context just presented, it is natural to consider the possibility of
properly incorporating the MA force density into the MP formulation – i.e., to
build the continuity equations for energy and momentum for the field+matter
coupled system by using the MP quasi-particle while retaining the force
density from Eq. (I). This will be developed in the present work, providing a
consistent model for the local optical energy and momentum transfer inside
dielectric media while also revealing the microscopic origin of the Abraham
optical force density in dielectrics as a byproduct. The results cover an
extensive variety of optomechanical applications and further elucidate the
centenary Abraham-Minkowski controversy on the photonic momentum inside
matter.

This work is organized as follows. In Sec. II, we build the total stress-
energy-momentum tensor of the coupled field and matter system. In Sec. III, we
derive the closed continuity equations for the system by using the MP quasi-
particle. In Sec. IV, the general validity of the theory is discussed in
detail and compared to the existing literature on the topic. In particular,
the long-known Abraham force density is shown to be originated from an
interplay between induced dipoles and mechanical stresses. At last,
conclusions and future perspectives are drawn in Sec. V.

##  II Stress-energy-momentum tensor

In flat space-time, the principles of conservation of energy and linear
momentum can be described simultaneously by the covariant four-continuity
equation [49] $\partial_{\nu}\mathcal{T}^{\mu\nu}=-f^{\mu}$, where
$\partial_{\nu}=(c^{-1}\partial_{t},\partial_{x},\partial_{y},\partial_{z})$,
the indices $\mu,\nu=0,1,2,3$ denote the space-time index and $\mathcal{T}$ is
the stress-energy-momentum (SEM) tensor, generically given as

$\mathcal{T}=\begin{pmatrix}W&\mathbf{S}/c\\\
c\mathbf{g}&\overleftrightarrow{\mathbf{T}}\\\ \end{pmatrix},$ (2)

where $\overleftrightarrow{\mathbf{T}}$ is the stress tensor, $\mathbf{g}$ is
the momentum density, $\mathbf{S}$ is the energy flux and $W$ is the energy
density. The four-force, on its turn, is $f^{\mu}=(\phi/c,\mathbf{f})$, where
$\phi$ is the power density and $\mathbf{f}$ is the force density. For a
closed system, we must always have $\partial_{\nu}\mathcal{T}^{\mu\nu}=0$.

The Lorentz force density is given as
$\mathbf{f}=\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}$, where $\rho$ and
$\mathbf{J}$ are the charge and current densities, respectively. The MA force
density in Eq. (I) was derived by introducing into the Lorentz force density
the four-current density associated to a single medium element (for example,
an atom or a molecule) endowed with electric and magnetic dipole moments,
namely [50, 51]

$J^{\nu}_{\mathrm{MA}}=(-c(\mathbf{p}\cdot\bm{\nabla})\delta^{3}(\mathbf{r}),\dot{\mathbf{p}}\delta^{3}(\mathbf{r})-(\mathbf{m}\times\bm{\nabla})\delta^{3}(\mathbf{r})),$
(3)

where $\mathbf{p}$ is the electric dipole moment, $\mathbf{m}$ is the magnetic
dipole moment, $\delta^{3}(\mathbf{r})$ the three-dimensional Dirac delta
distribution, the over-dot denotes the total time derivative and $\mathbf{r}$
is the position vector. As the force density from Eq. (I), the four-current
shown in Eq. (3) is valid for the laboratory frame and non-relativistic
velocities.

The procedure described above of defining an appropriate model for the
electromagnetic sources in matter and then inserting them into the Lorentz
force law is, of course, quite standard. For example, the conventional Ampère
formulation for macroscopic fields [36] is also derived in this fashion, where
the effective, bound four-current
$J^{\nu}_{\mathrm{b}}=(-c\bm{\nabla}\cdot\mathbf{P},\bm{\nabla}\times\mathbf{M}+\partial_{t}\mathbf{P})$
is employed. In fact, we notice that the widely known derivation of Maxwell’s
stress tensor starting from the Lorentz force density can be applied to any
four-current configuration. This happens because, excluding self-field
effects, it is always possible to substitute the electromagnetic sources by
the corresponding electromagnetic fields in the Lorentz equation through
Gauss’ and Ampère-Maxwell’s law, generating
$\mathbf{f}=\varepsilon_{0}(\bm{\nabla}\cdot\mathbf{E})\mathbf{E}+\mu_{0}^{-1}(\bm{\nabla}\times\mathbf{B})\times\mathbf{B}$
regardless of the microscopic sources’ details. We therefore conclude that
Maxwell’s SEM tensor, given by [49]

$\displaystyle\mathcal{T}_{\mathrm{Maxwell}}$
$\displaystyle=\begin{pmatrix}W_{\mathrm{Maxwell}}&\varepsilon_{0}c(\mathbf{E}\times\mathbf{B})^{\mathrm{T}}\\\
\varepsilon_{0}c\mathbf{E}\times\mathbf{B}&\quad
W_{\mathrm{Maxwell}}\overleftrightarrow{\mathbf{I}}-\varepsilon_{0}\mathbf{E}\\!\otimes\\!\mathbf{E}-\mu_{0}^{-1}\mathbf{B}\\!\otimes\\!\mathbf{B}\end{pmatrix},$
(4)

must be valid for any four-current configuration – even though it does not
contain the source terms explicitly. Here “T” denotes matrix transposition,
“$\otimes$” denotes outer product, $\overleftrightarrow{\mathbf{I}}$ is the
unit dyadic and
$W_{\mathrm{Maxwell}}=(\varepsilon_{0}|\mathbf{E}|^{2}+\mu_{0}^{-1}|\mathbf{B}|^{2})/2$.

The SEM tensor in Eq. (II) corresponds to the field part of our system.
Together with the SEM of the material (which will be shown in the next
section), they form a closed system, whose four-divergence must be zero.
Notice that in this SEM description there is no explicit interaction terms (as
there would generally exist in Lagrangian descriptions) – rather, the
interaction term is related to the fact that the four-divergences of the two
SEM tensor terms are nonzero four-force densities with opposite signs – whence
the four-divergence of the total SEM tensor is zero [33, 34, 35]. Thus,
interactions modify the values of the fields but they are not explicitly seen
in the SEM tensors. In our case, the four-divergence of Maxwell’s SEM
generates the force density $\mathbf{f}_{\mathrm{MA}}$ given in Eq. (I) and
the power density $\phi_{\mathrm{MA}}=\mathbf{E}\cdot\mathbf{J}_{\mathrm{MA}}$
see Appendix A), with $\mathbf{J}_{\mathrm{MA}}$ given by the spatial part of
the four-current in Eq. (3). As expected, contrary to Maxwell’s SEM tensor
these interactions do depend explicitly on the specific model adopted to
describe the electromagnetic sources within the system. To calculate
$\phi_{\mathrm{MA}}$, we start by writing the total power as

$\Phi_{\mathrm{MA}}=\int\mathbf{E}\cdot\left[\dot{\mathbf{p}}\delta^{3}(\mathbf{r})-\mathbf{m}\times\bm{\nabla}\delta^{3}(\mathbf{r})\right]\,\mathrm{d}^{3}\mathbf{r}.$
(5)

Equation. (5) corresponds to the contribution of a single medium element to
the total power. When a sum of similar terms from medium elements at different
positions is integrated over a small volume inside the dielectric (but still
microscopically large, i.e., containing a large number of dipoles), the
calculation yields the power density as (see Appendix B for the calculation)

$\phi_{\mathrm{MA}}=\frac{1}{2}\frac{\partial}{\partial
t}\left({\mathbf{P}}\cdot\mathbf{E}-\mathbf{M}\cdot\mathbf{B}\right),$ (6)

where $\phi_{\mathrm{MA}}$, $\mathbf{P}$ and $\mathbf{M}$ are given by
$\Phi_{\mathrm{MA}}$, $\mathbf{p}$ and $\mathbf{m}$ divided by the integration
volume and multiplied by the number of dipoles in this volume, respectively.

Assuming no external fields are initially present, if we integrate
$\phi_{\mathrm{MA}}$ in time we obtain the energy density
$W_{\mathrm{int}}=({\mathbf{P}}\cdot\mathbf{E}-\mathbf{M}\cdot\mathbf{B})/2$,
which represents the (negative of the) total work per unit volume that must be
supplied to assemble the fields+dipoles configuration and to establish the
dipoles within the fields. This is different from the well-known potential
energy density of linear, stationary dipoles in external fields,
$-({\mathbf{P}}\cdot\mathbf{E}+\mathbf{M}\cdot\mathbf{B})/2$, because to
assemble the configuration extra work must be supplied to overcome the effects
of the induced electric current, according to Lenz law. For a detailed
discussion on this subtlety, see, for example, Refs. [52, 50].

With Eq. (6), the four-continuity equation associated to the MA formulation,
$\partial_{\nu}\mathcal{T}_{\mathrm{Maxwell}}^{\mu\nu}=-f_{\mathrm{MA}}^{\mu}$,
is then completely defined. This equation contains the information of the
electromagnetic fields dynamics and how they interact with matter – therefore,
by finding the appropriate SEM tensor for matter we are able to delineate our
closed system. This tensor for matter will be discussed in the next section by
using the mass-polariton quasi-particle theory [31].

###  II.1 Incorporating the mass-polariton dynamics

The original MP formulation [31] adopts, in the laboratory frame, Abraham’s
force density
$\mathbf{f}_{\mathrm{Ab}}=-|\mathbf{E}|^{2}\bm{\nabla}\varepsilon/2-|\mathbf{H}|^{2}\bm{\nabla}\mu/2+c^{-2}(n^{2}-1)\partial_{t}(\mathbf{E}\times\mathbf{H})$
and the SEM tensor relation
$\mathcal{T}_{\mathrm{MP}}=\mathcal{T}_{\mathrm{Ab}}+\mathcal{T}_{\mathrm{MDW}}$.
In contrast to Maxwell’s SEM tensor in Eq. (II), the original MP formulation
used Abraham’s SEM tensor, given by [25]

$\mathcal{T}_{\mathrm{Ab}}=\begin{pmatrix}W_{\mathrm{Ab}}&(\mathbf{E}\times\mathbf{H})^{\mathrm{T}}/c\\\
\mathbf{E}\times\mathbf{H}/c&\quad
W_{\mathrm{Ab}}\overleftrightarrow{\mathbf{I}}-\mathbf{D}\otimes\mathbf{E}-\mathbf{B}\otimes\mathbf{H}\end{pmatrix},$
(7)

where
$W_{\mathrm{Ab}}=(\mathbf{D}\cdot\mathbf{E}+\mathbf{B}\cdot\mathbf{H})/2$, and
$\mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}$ is the electric displacement
field. The MDW SEM tensor $\mathcal{T}_{\mathrm{MDW}}$ is discussed later in
this section.

Before description of the MDW, we discuss the SEM tensor of the material. The
SEM tensor of the material mass is given by [49]

$\displaystyle\mathcal{T}_{\mathrm{mass}}^{\mu\nu}$
$\displaystyle=(\rho_{\mathrm{a}}/\gamma_{\mathbf{v}_{\mathrm{a}}}^{2})U_{\mathrm{a}}^{\mu}U_{\mathrm{a}}^{\nu}=\begin{pmatrix}\rho_{\mathrm{a}}c^{2}&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}^{\mathrm{T}}c\\\
\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}c&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}\otimes\mathbf{v}_{\mathrm{a}}\end{pmatrix},$
(8)

where
$\rho_{\mathrm{a}}=\gamma_{\mathbf{v}_{\mathrm{a}}}^{2}\rho_{\mathrm{ai0}}+W_{\mathrm{int}}/c^{2}$
is the mass density of the material, in which $\rho_{\mathrm{ai0}}$ is the
instantaneous rest mass density,
$\gamma_{\mathbf{v}_{\mathrm{a}}}=(1-|\mathbf{v}_{\mathrm{a}}|^{2}/c^{2})^{-1/2}$
is the Lorentz factor corresponding to the atomic velocity
$\mathbf{v}_{\mathrm{a}}$, and
$U_{\mathrm{a}}^{\mu}=\gamma_{\mathbf{v}_{\mathrm{a}}}(c,\mathbf{v}_{\mathrm{a}})$
is the four-velocity of the material. Under dynamical interactions, which
locally vary the positions of atoms, $\rho_{\mathrm{ai0}}$ can differ from the
rest mass density of the material at equilibrium with no interactions, which
we denote by $\rho_{\mathrm{a0}}$. As discussed in Ref. [36], the relation of
these two quantities is given by
$\rho_{\mathrm{ai0}}=\rho_{\mathrm{a0}}/(1+\bm{\nabla}\cdot\mathbf{r}_{\mathrm{a}})$,
where $\mathbf{r}_{\mathrm{a}}$ is the atomic displacement field averaged over
a small volume, which however contains many atoms. In the original MP
formulation [31], the energy of induced dipoles was considered as a part of
Abraham’s energy density $W_{\mathrm{Ab}}$, and consequently
$W_{\mathrm{int}}$ was set to zero in the expression of $\rho_{\mathrm{a}}$
above. In the present theory, $W_{\mathrm{int}}$ is associated with the
establishment of induced dipoles within the time-dependent electromagnetic
field by the power density in Eq. (6) as discussed above. Thus,
$W_{\mathrm{int}}$ is obtained by the time-integral of the power density of
Eq. (6), and it is equal to the difference of the Abraham energy density
$W_{\mathrm{Ab}}$ of the original MP formulation and the energy density
$W_{\mathrm{Maxwell}}$ of the present formulation as
$W_{\mathrm{int}}=\int_{-\infty}^{t}\phi_{\mathrm{MA}}dt^{\prime}=W_{\mathrm{Ab}}-W_{\mathrm{Maxwell}}$.

The SEM tensor of mechanical pressure is given by
$\mathcal{T}_{\mathrm{mech}}^{\mu\nu}=(\mathcal{P}/c^{2})U_{\mathrm{a}}^{\mu}U_{\mathrm{a}}^{\nu}+\mathcal{P}\eta^{\mu\nu}$
[49, 53], where $\mathcal{P}$ is the pressure. In the rest frame of the
material, where $\mathbf{v}_{\mathrm{a}}=0$, the SEM tensor of mechanical
pressure becomes

$\displaystyle\mathcal{T}_{\mathrm{mech}}=\begin{pmatrix}0&\mathbf{0}^{\mathrm{T}}\\\
\mathbf{0}&\mathcal{P}\overleftrightarrow{\mathbf{I}}\end{pmatrix}.$ (9)

Together with the SEM tensor of material mass in Eq. (8), the SEM tensor of
mechanical pressure in Eq. (9) makes the SEM tensor of a perfect fluid in flat
space-time, given in a general inertial frame by
$\mathcal{T}_{\mathrm{fluid}}^{\mu\nu}=\mathcal{T}_{\mathrm{mass}}^{\mu\nu}+\mathcal{T}_{\mathrm{mech}}^{\mu\nu}=(\rho_{\mathrm{a}}/\gamma_{\mathbf{v}_{\mathrm{a}}}^{2}+\mathcal{P}/c^{2})U_{\mathrm{a}}^{\mu}U_{\mathrm{a}}^{\nu}+\mathcal{P}\eta^{\mu\nu}$.
This SEM tensor is of the form known from previous literature [49] and its use
is not limited to the present theory.

The MDW SEM tensor is given for non-relativistic atomic velocities in the
laboratory frame as a difference of the SEM tensor of the material mass in Eq.
(8) disturbed from equilibrium by the electromagnetic force and power
densities and the same formula with $\mathbf{v}_{\mathrm{a}}=0$ and
$W_{\mathrm{int}}=0$, denoted by $\mathcal{T}_{\mathrm{mass,0}}^{\mu\nu}$, as
[31, 33, 34, 35]

$\displaystyle\mathcal{T}_{\mathrm{MDW}}$
$\displaystyle=\mathcal{T}_{\mathrm{mass}}^{\mu\nu}-\mathcal{T}_{\mathrm{mass,0}}^{\mu\nu}$
$\displaystyle=\begin{pmatrix}\rho_{\mathrm{MDW}}c^{2}&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}^{\mathrm{T}}c\\\
\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}c&\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}\otimes\mathbf{v}_{\mathrm{a}}\end{pmatrix}.$
(10)

In this equation, the mass density of the atomic MDW is defined as
$\rho_{\mathrm{MDW}}=\rho_{\mathrm{a}}-\rho_{\mathrm{a0}}=\gamma_{\mathbf{v}_{\mathrm{a}}}^{2}\rho_{\mathrm{ai0}}-\rho_{\mathrm{a0}}+W_{\mathrm{int}}/c^{2}$.
The kinetic energy of atoms, resulting from optical forces, is extremely small
due to its quadratic dependence on the small atomic velocity in the assumed
non-relativistic regime and hence can be neglected [35]. Thus, we can set
$\gamma_{\mathbf{v}_{\mathrm{a}}}^{2}=1$ in the equations above. Using the
relation
$\rho_{\mathrm{ai0}}=\rho_{\mathrm{a0}}/(1+\bm{\nabla}\cdot\mathbf{r}_{\mathrm{a}})$
in the limit of small $\bm{\nabla}\cdot\mathbf{r}_{\mathrm{a}}$, we then
obtain that the MDW mass density in the laboratory frame can be approximated
as

$\rho_{\mathrm{MDW}}=\rho_{\mathrm{a}}-\rho_{\mathrm{a0}}=-(\bm{\nabla}\cdot\mathbf{r}_{\mathrm{a}})\rho_{\mathrm{a0}}+W_{\mathrm{int}}/c^{2}.$
(11)

The first term is related to atomic displacements, and the second term, which
was absent in the original MP formulation, is the mass equivalent of the
interaction energy associated with the creation of induced dipoles by the
power density of Eq. (6).

On its turn, the MA force density is the sum of Abraham’s force density and
the electro- and magnetostriction effects, i.e.,

$\mathbf{f}_{\mathrm{MA}}=\mathbf{f}_{\mathrm{Ab}}+\bm{\nabla}(\mathbf{P}\cdot\mathbf{E})/2+\bm{\nabla}(\mathbf{M}\cdot\mathbf{B})/2.$
(12)

In terms of stress tensors and momentum densities, the last equation should be
equivalent to

$\displaystyle\overleftrightarrow{\bm{\nabla}}\cdot\left(\overleftrightarrow{\mathbf{T}}_{\mathrm{Ab}}-\overleftrightarrow{\mathbf{T}}_{\mathrm{Maxwell}}\right)+\frac{\partial}{\partial
t}\left(\mathbf{g}_{\mathrm{Ab}}-\mathbf{g}_{\mathrm{Maxwell}}\right)=$
$\displaystyle\frac{1}{2}\overleftrightarrow{\bm{\nabla}}\cdot\left[(\mathbf{P}\cdot\mathbf{E})\overleftrightarrow{\mathbf{I}}+(\mathbf{M}\cdot\mathbf{B})\overleftrightarrow{\mathbf{I}}\right].$
(13)

It can be verified that Eq. (II.1) does not hold as the magnetic part is
unable to satisfy the relation. The subtle issue to be noticed here is that
$\mathbf{f}_{\mathrm{MA}}$ and $\mathbf{f}_{\mathrm{Ab}}$ were actually
constructed for different physical systems. The force density
$\mathbf{f}_{\mathrm{MA}}$ is built from point dipoles and describes the
electromagnetic interactions locally, where by “locally” we mean in a small
volume inside the dielectric, which contains many dipoles (i.e., is
microscopically large) but is still smaller than the total volume of the
material. This feature is very important as it allows the MA formulation to
describe the spatio-temporal dependence of the optical forces in dielectric
media. The force density $\mathbf{f}_{\mathrm{Ab}}$, on its turn, is obtained
from the equation
$\mathbf{f}_{\mathrm{Ab}}=-\overleftrightarrow{\bm{\nabla}}\cdot\overleftrightarrow{\mathbf{T}}_{\mathrm{Ab}}-\partial_{t}\mathbf{g}_{\mathrm{Ab}}$
by employing Gauss’ and Ampère-Maxwell’s laws in their macroscopic forms [23],
i.e., $\bm{\nabla}\cdot\mathbf{D}=\rho_{\mathrm{f}}$ and
$\bm{\nabla}\times\mathbf{H}=\mathbf{J}_{\mathrm{f}}+\partial_{t}\mathbf{D}$,
respectively (where $\rho_{\mathrm{f}}$ and $\mathbf{J}_{\mathrm{f}}$ are the
free electric charge and current density, respectively). In these macroscopic
forms the effective bound four-current density
$J^{\nu}_{\mathrm{b}}=(-c\bm{\nabla}\cdot\mathbf{P},\bm{\nabla}\times\mathbf{M}+\partial_{t}\mathbf{P})$
is automatically assumed – therefore, formulations based on this
electromagnetic source model can not be applied to describe local optical
forces, being appropriate only if one is interested in the overall center-of-
mass motion of the material [16, 36]. Thus, in local descriptions of optical
forces in matter, Abraham’s force density can not be applied. In fact, the
same problem also happens to the so-called Einstein-Laub formulation [27],
which, although initially modelled through point dipoles as well, also employs
the macroscopic Maxwell’s equations in the derivation of its continuity
equations [16, 36]. To us, this is a very subtle and important fact for the
Abraham-Minkowski debate, which seems to have been overlooked in the
literature. Again, we stress that the only way to treat the electromagnetic
sources generically, be them bound and/or free, is by means of Maxwell’s SEM
tensor, as discussed at the beginning of this section.

In the scenario discussed, we propose an alternative form of the MP tensor,
called microscopic mass-polariton (MMP) tensor and given as

$\mathcal{T}_{\mathrm{MMP}}=\mathcal{T}_{\mathrm{Maxwell}}+\mathcal{T}_{\mathrm{MDW}},$
(14)

where the SEM tensors $\mathcal{T}_{\mathrm{Maxwell}}$ and
$\mathcal{T}_{\mathrm{MDW}}$ are given in Eqs. (II) and (II.1), respectively.
As stated earlier, $\mathcal{T}_{\mathrm{MDW}}$ encompasses the out-of-
equilibrium rest mass and kinetic energies of the system. The remaining
mechanical effects are described by the tensor $\mathcal{T}_{\mathrm{mech}}$,
which was explicitly written in the beginning of this section for a perfect
fluid as an example. For a solid elastic medium, the spatial part of
$\mathcal{T}_{\mathrm{mech}}$ is given in a more complicated way in terms of
the material’s strain – see Ref. [54] for example.

The total SEM tensor of the closed system in our model is therefore given by

$\displaystyle\mathcal{T}_{\mathrm{tot}}$
$\displaystyle=\mathcal{T}_{\mathrm{MMP}}+\mathcal{T}_{\mathrm{mass,0}}+\mathcal{T}_{\mathrm{mech}}$
$\displaystyle=\mathcal{T}_{\mathrm{Maxwell}}+\mathcal{T}_{\mathrm{mass}}+\mathcal{T}_{\mathrm{mech}}.$
(15)

In mechanical equilibrium in the presence of static electric and magnetic
fields, the pressure from $\mathcal{T}_{\mathrm{mech}}$ is counterbalanced
within the dielectric by the electro- and magnetostriction force densities, as
will be discussed in the following section. For time-varying electromagnetic
fields, on the other hand, the striction force densities act as sources in the
acoustic pressure wave equation [55], being therefore associated to the
transient acoustic relaxation of the medium.

##  III Continuity equations

As the electromagnetic field and the material in our case form a closed
system, we must have $\partial_{\nu}\mathcal{T}_{\mathrm{tot}}^{\mu\nu}=0$.
This equation can be conveniently split into its temporal ($\mu=0$) and
spatial ($\mu=1,2,3$) coordinates, providing explicitly the energy and
momentum continuity equations for our theory, respectively, which will be
shown next.

###  III.1 Energy continuity equation

Using the explicit forms of the SEM tensors and
$\partial_{\nu}=(c^{-1}\partial_{t},\partial_{x},\partial_{y},\partial_{z})$,
we obtain the energy continuity equation in the laboratory frame as

$\displaystyle\partial_{\nu}\mathcal{T}^{0\nu}_{\mathrm{tot}}=\frac{1}{c}\frac{\partial}{\partial
t}\left(\frac{1}{2}\varepsilon_{0}|\mathbf{E}|^{2}+\frac{1}{2\mu_{0}}|\mathbf{B}|^{2}+\rho_{\mathrm{a}}c^{2}\right)$
$\displaystyle+\frac{1}{c}\bm{\nabla}\cdot\left(\mathbf{E}\times\mathbf{H}+\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}c^{2}\right)=0.$
(16)

Here, the hidden energy contribution [56]
$\mathbf{S}_{\mathrm{h}}=\mathbf{M}\times\mathbf{E}$ (also known as Aharonov-
Casher interaction [57]) has been added to the electromagnetic part of
$\mathbf{S}$ because, as discussed in detail in Ref. [30], the derivations of
the MA force density are non-relativistic and therefore need the inclusion of
the hidden momentum and energy contributions in an _ad hoc_ manner, when
writing equations for the laboratory frame, to keep the correct relativistic
transformation properties [58] – had we employed a relativistic derivation
from the start, these terms would arise naturally when the center of mass-
energy of the dipole system is promoted to a dynamical variable, as shown in
Ref. [59] (see Appendix C for more details). By adding these contributions to
the four-continuity equation in the laboratory inertial reference frame it is
implied we are keeping contributions only to order
$|\mathbf{v}_{\mathrm{a}}|/c$ – therefore, the relatively small atomic kinetic
energy is neglected, as stated earlier. Notice that, as the motion of the
atoms is accelerated due to the external forces, their proper frames are not
inertial reference frames – the velocity $\mathbf{v}_{\mathrm{a}}$ is the
relative velocity between a local inertial reference frame, moving along with
the atoms, and the laboratory frame [34].

Notice that Eq. (III.1) has a distinct form from the original MP formulation
due to the field energy density
$W_{\mathrm{field}}=(\varepsilon_{0}|\mathbf{E}|^{2}+\mu_{0}^{-1}|\mathbf{B}|^{2})/2$
appearing in the time derivative term instead of $W_{\mathrm{Ab}}$ (see Eq.
(7)). To elucidate this discrepancy, we first note that, from the Sec. II, we
have
$\partial_{\nu}\mathcal{T}^{0\nu}_{\mathrm{Maxwell}}=-\phi_{\mathrm{MA}}/c$.
Using the last form of Eq. (II.1), we then have

$\displaystyle\partial_{\nu}\mathcal{T}^{0\nu}_{\mathrm{tot}}=-\phi_{\mathrm{MA}}+\partial_{t}(\rho_{\mathrm{a}}c^{2})+\bm{\nabla}\cdot(\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}c^{2})=0,$
(17)

which is an alternative form of Eq (III.1). As previously stated, in our
system the power density $\phi_{\mathrm{MA}}$ is related to the assembly and
establishment of the induced dipoles within the fields, with an associated
energy density equal to
$W_{\mathrm{int}}=(\mathbf{P}\cdot\mathbf{E}-\mathbf{M}\cdot\mathbf{B})/2$. As
discussed in Sec. II.1, this energy density becomes implicitly contained in
$\rho_{\mathrm{a}}c^{2}$. From the point of view of Eq. (17), this fact can be
seen as follows. First, we approximate
$\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}\approx\rho_{\mathrm{a0}}\mathbf{v}_{\mathrm{a}}=\rho_{\mathrm{a0}}\,\mathrm{d}\mathbf{r}_{\mathrm{a}}/\,\mathrm{d}t\approx\rho_{\mathrm{a0}}\partial_{t}\mathbf{r}_{\mathrm{a}}$.
Thus, Eq. (17) becomes
$\partial_{t}[\rho_{\mathrm{a}}c^{2}+(\bm{\nabla}\cdot\mathbf{r}_{\mathrm{a}})\rho_{\mathrm{a0}}c^{2}]=\phi_{\mathrm{MA}}$.
Using Eq. (11), this equation is further written as
$\partial_{t}(\rho_{\mathrm{a0}}c^{2}+W_{\mathrm{int}})=\phi_{\mathrm{MA}}$.
The time derivative of the equilibrium mass density $\rho_{\mathrm{a0}}$ is
zero, so we have $\partial_{t}W_{\mathrm{int}}=\phi_{\mathrm{MA}}$, which is
equivalent to the definition of the power density in Eq. (6). Thus, we see
that, regarding the energy continuity equation, the mathematical difference
between the MP and MMP formulations lies in how the interaction energy is
treated – explicitly in terms of the fields or implicitly as a part of the
time-dependent mass density of the material, respectively.

###  III.2 Momentum continuity equation

Keeping the same considerations and approximations employed in the energy
continuity equation, the momentum continuity equation, on its turn, is given
in the laboratory frame as

$\displaystyle\sum_{i=1}^{3}\hat{\mathbf{e}}_{i}\partial_{\nu}\mathcal{T}_{\mathrm{tot}}^{i\nu}=-\mathbf{f}_{\mathrm{MA}}-\mathbf{f}_{\mathrm{b}}+\overleftrightarrow{\bm{\nabla}}\cdot(\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}\otimes\mathbf{v}_{\mathrm{a}})$
$\displaystyle+\frac{\partial}{\partial
t}(\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}})=0,$ (18)

where $i=1,2,3$, $\hat{\mathbf{e}}_{i}$ is the unit vector in $i$-th direction
and $\mathbf{f}_{\mathrm{b}}$ denotes the internal body force density in the
medium, given by the spatial derivative of
$\mathcal{T}_{\mathrm{mech}}^{i\nu}$ – in our example of a perfect fluid given
in Eq. (9), we have $\mathbf{f}_{\mathrm{b}}=-\bm{\nabla}\mathcal{P}$. Besides
this term, the additional difference in Eq. (III.2) compared to the original
MP formulation is in the electro- and magnetostriction forces contained in
$\mathbf{f}_{\mathrm{MA}}$. Notice that, by using the material derivative
$\,\mathrm{d}/\,\mathrm{d}t=\partial_{t}+\mathbf{v}_{\mathrm{a}}\cdot\bm{\nabla}$
and the continuity equation of the atomic number density $n_{\mathrm{a}}$,
given by
$\partial_{t}n_{\mathrm{a}}+\bm{\nabla}\cdot(n_{\mathrm{a}}\mathbf{v}_{\mathrm{a}})=0$,
Eq. (III.2) reduces to Newton’s law of motion as
$n_{\mathrm{a}}\,\mathrm{d}\mathbf{p}_{\mathrm{a}}/\,\mathrm{d}t=\mathbf{f}_{\mathrm{MA}}+\mathbf{f}_{\mathrm{b}}$,
where
$\mathbf{p}_{\mathrm{a}}=\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}/n_{\mathrm{a}}$
is the momentum of a single atom.

At last, the coupling between field and matter expected to arise in
experimental observations, as already discussed for the energy case, is more
subtle for momentum and will be addressed in the next section.

##  IV Discussion

The general validity of Maxwell’s SEM tensor as the electromagnetic part in
coupled field plus matter systems with arbitrary sources, suggested in Sec.
II, is supported by the literature in works addressing electromagnetic force
and torque with diverse source models, such as dipoles [60, 61], multipoles
[62, 63], negative-index scatterers [64] and nanoparticles placed in
metamaterials [65] and plasmonic traps [66]. In these works, the total
electromagnetic force is calculated as appropriate surface integrals of
Maxwell’s SEM tensor as obtained by integrating the general relation
$\partial_{\nu}\mathcal{T}_{\mathrm{Maxwell}}^{\mu\nu}=-f^{\mu}$. There are,
however, works that employ Minkowski’s SEM tensor (which generates the same
force density as Abraham’s when time-averaged) – see, for example, Refs. [3,
6, 67]. These works, on their turn, are also subject to the problem discussed
in Sec. II of implicitly obtaining force densities that are valid only for
macroscopic, effective medium descriptions – nevertheless, for applications
where only the system’s center of mass-energy movement is important and the
total force is observed as a cycle-averaged quantity, such as in typical
optical traps, this approach is consistent as well and agrees with our
proposal [68, 20].

The essential coupling between field and matter in the context of the Abraham-
Minkowski problem has been discussed extensively in the literature, where
different subsystem’s separations (even arbitrary ones) were proposed [19, 69,
20, 23]. In Sec. III.1, the coupling in energy for the MMP formulation was
discussed in terms of the fields and interaction energy densities. A similar
but more subtle coupling is also present in the momentum continuity equation,
Eq. (III.2). By using the MP quasi-particle, it can be shown that in the
laboratory frame the coupling between field and matter leads to a momentum
transfer proportional to $n$ [31, 33] – i.e., a Minkowski-type momentum, in
accordance with the few existing measurements of photon momentum in matter
[44, 45, 48, 46, 47]. This result is unambiguously tied to the field momentum
carrying a momentum of Abraham type, proportional to $1/n$, while the matter
contribution is attributed to the MDW carrying the difference, proportional to
$(n-1/n)$. Therefore, as in the case of energy, our proposed theory is in
agreement with the experimental works observing the total, coupled linear
momentum of the field plus matter system. Indeed, as experimental observations
in this scenario will always capture the closed system, without further
argumentation it is reasonable to assume that the decomposition of the total
SEM tensor is arbitrary, as done for example in Ref. [20]; nevertheless, the
decompositions presented by us in Eqs. (14) and (II.1) have the advantage of
describing the subsystems in terms of very well-known tensors, providing,
therefore, a clearer physical interpretation. Lastly, recall that although
$\mathcal{T}_{\mathrm{Maxwell}}$, $\mathcal{T}_{\mathrm{MDW}}$ and
$\mathcal{T}_{\mathrm{mech}}$ are true tensors, the electromagnetic sources
and consequently the optical force and power densities, given is Eqs. (I), (3)
and (6) respectively, were given specifically for the laboratory frame and
non-relativistic velocities. Therefore, the presented theory is not covariant,
but this generalization is of technical interest and will be addressed in
future works.

From the definition of potential energy, the stress acting inside the material
due to the electromagnetic interaction with the dipoles can be found from the
derivative of the total (fields plus interaction) energy density,
$(\mathbf{D}\cdot\mathbf{E}+\mathbf{B}\cdot\mathbf{H})/2$, with respect to the
associated strain component (in solid media) [70]. It can be shown [38] that
such calculation leads to the force density
$\bm{\nabla}(\mathbf{P}\cdot\mathbf{E}+\mathbf{M}\cdot\mathbf{B})/2$, i.e.,
the work done by the electromagnetic field generates strain inside the
material, which is manifested through the electro- and magnetostriction
effects. The same occurs for fluid media, with strain being substituted by a
local pressure variation. We can then see that striction effects correspond to
pure stresses, and so they do not change the local momentum transfer from
light inside the material, which occurs exclusively in the pulse propagation
direction. Indeed, striction effects are long-known to not contribute to the
material’s center of mass-energy movement, having significant effects only in
local force considerations [16]. This interpretation is also in accordance
with the discrete picture of light, where the incident photons on an isotropic
dielectric have their average linear momentum in the light’s propagation
direction.

Eq. (III.2) is compatible with the opto-elastic simulations from Refs. [31,
12] – striction effects are balanced by the strain-induced elastic forces,
while the remaining force density is the dynamical part, equal to
$\mathbf{f}_{\mathrm{Ab}}$. In turn, this force density is responsible for the
MDW propagating along with the light pulse. This dynamics and the linear
momentum conservation can be visualized in detail in the two-dimensional
simulations presented for an elastic solid medium in Ref. [12]. Another
relevant fact is that in fluids under pulsed optical excitation an acoustic
pressure wave would develop to counterbalance the effects of the generated
striction force densities. This mechanism of acoustic relaxation was studied
experimentally for non-magnetic fluids in Refs. [29, 39, 42], providing
observations of the spatio-temporal dependence of the optical force density.
We recall that, among all the known formulations for optical forces up to
date, Eq. (I) is the only one capable of describing the experimental results
from Refs. [29, 39, 42] while being theoretically consistent with the observed
model for magnetization in matter [43, 71, 72, 30], as stated in Sec. I. This
fact together with the familiar energy continuity equations shown in Eqs.
(III.1) and (17) provide very good evidence for the validity of the presented
theory. Its complete experimental confirmation, however, requires the
measurement of the small mass transferred by the mass density wave [31]. While
experimental setups for this task have been suggested [73, 74], this
measurement is quite challenging and has not, to our knowledge, been performed
yet. For a detailed discussion on the different existing formulations for
electromagnetic energy and momentum transfer in matter, see for example Refs.
[16, 20, 36, 69].

A covariant theory of light propagating in dielectric media using the MP
formulation has already been reported [35, 38]. Specifically, in Ref. [38], a
decomposition of the total SEM tensor very similar to the one shown in Eq.
(14) was adopted – however, the field part was given by Abraham’s SEM tensor,
while this work adopts Maxwell’s SEM tensor. In truth, whenever effective
macroscopic treatments for the matter part are appropriate
$\mathbf{f}_{\mathrm{Ab}}$ does originate from Abraham’s SEM tensor (as
discussed in Sec. II.1) – in these cases, the MP formulation with the addition
of striction effects, as described in Ref. [38], yields the exact same results
as the MMP formulation. This occurs because the total force acting on the
medium is the same in both MP and MMP formulations, i.e.,
$\int\mathbf{f}_{\mathrm{Ab}}\mathrm{d^{3}}\mathbf{r}=\int\mathbf{f}_{\mathrm{MA}}\mathrm{d^{3}}\mathbf{r}$
since striction forces always integrate to zero [16, 68]. Therefore, the MMP
formulation shares all the momentum transfer properties and center of mass-
energy movement of the MP formulation, which have been extensively discussed
and shown to be mathematically consistent [31, 32, 75, 33, 34, 35]. As a
consequence, the MP and MMP formulations are not mutually exclusive: all the
force densities and simulations from Refs. [31, 38] are correct, but the
direct association of Abraham’s SEM to $\mathbf{f}_{\mathrm{Ab}}$ is
implicitly valid only for the conditions just cited. In this context, the MMP
formulation can be seen as a clarification of the MP formulation, where the
specific microscopic mechanisms of optical energy and momentum transfer
involved are described in terms of electric and magnetic dipoles, eliminating
the necessity of effective medium descriptions or phenomenological approaches.

Having treated energy and linear momentum conservation, it is natural to
ponder if the presented theory is also compatible with angular momentum
conservation. Indeed, this is a very interesting topic that has drawn much
attention in the past few decades, especially because it is now known that
structured fields can carry both orbital and spin angular momentum [76, 77,
78, 79, 80, 81]. In this scenario, simulations of optical angular momentum
transfer using the MP formulation were already reported and seen to be in
accordance with the conservation principle [75]. As striction forces are not
related to linear momentum transfer, it is expected that this behavior is also
shared by the MMP formulation.

The consideration of temporally dispersive media within the mass-polariton
dynamics has been treated earlier [32], where the momentum transfer was found
to depend on both the wave’s group and phase velocity. This is also expected
to occur in the MMP formulation, but more careful analyses are needed because,
for example, the power density shown in Eq. (6) is not valid for dispersive
media. Besides dispersion, the extension of the theory to include non-
conservative optical forces [82], anisotropy and nonlinearities is also of
interest. Additionally, a more fundamental formulation can be sought by
working in a quantum mechanics regime, as done for example in Refs. [83, 84,
85].

At last, recall that, as discussed, when electro- and magnetostriction force
densities are counter-balanced by stresses in the medium and the interaction
energy density is accounted for explicitly, the equations for energy density
and force density in the dielectric are equal to Abraham’s formulation –
however, we emphasize that in local descriptions of energy and momentum there
is no physically meaningful association of these conserved quantities to
Abraham’s SEM tensor as it necessarily employs macroscopic sources in its
application [36, 23]. Therefore, the argumentation developed here reveals the
true microscopic origin of Abraham’s force density in linear, isotropic,
lossless and non-dispersive dielectric media: it results from the optical
force acting on moving induced dipoles partially counter-balanced by local,
internal mechanical stresses. To the best of our knowledge, this simple
interpretation of the long-known and widely employed Abraham force density in
terms of microscopic dipoles had not been reported in the literature before.
Clarifying this subtlety contributes to our fundamental understanding of the
interaction of electromagnetic fields with matter and its numerous
applications.

##  V Conclusions

We have described a microscopic theory for the local electromagnetic energy
and momentum transfer inside linear dielectric media called microscopic mass-
polariton formulation. The theory arises when the recently introduced
microscopic Ampère formulation, based on classical dipolar electromagnetic
sources, is extended to include the theoretical mass-polariton quasi-particle.
This was made possible by noticing that the widely known Maxwell’s stress-
energy-momentum tensor is the only one capable of describing arbitrary
electromagnetic sources in matter. The field part of the system is then given
by this tensor, while the matter part is given by the mass density wave
tensor, which corresponds to the disturbances in atomic positions propagating
along with the electromagnetic wave, and a general mechanical tensor, which
encompasses all effects not related to rest mass or kinetic energies, such as
internal stresses. Additionally, in our model the electro- and
magnetostriction effects are counter-balanced by the medium’s local stresses,
with Abraham’s force density corresponding exactly to the remaining terms of
the optical force density – therefore, we have shown that whenever local
continuity considerations must take place Abraham’s force density does not
originate from Abraham’s stress-energy-momentum tensor. This significantly
improves our current knowledge of the field and matter interactions and also
sheds light on the centenary Abraham-Minkowski controversy on the photon’s
momentum.

Our proposed theory agrees with the majority of experiments reported so far
involving optical forces and radiation pressure in dielectric media, providing
a paradigm for general opto-mechanical applications. When compared to the
original mass-polariton formulation, our theory presents the advantage of
being directly related to the very well-established dipolar model for
electromagnetic sources in matter, which clarifies the microscopic dynamics of
the system. Besides, electro- and magnetostriction effects are also naturally
accounted for. The complete confirmation of our theory, however, still
requires the measurement of the small mass transferred by the propagating mass
density wave, which remains a very difficult experimental challenge and has
not, to our knowledge, been performed yet.

###### Acknowledgements.

The research leading to these results received funding from CNPq
(304738/2019-0), CAPES (Finance Code 001), Fundação Araucária, and FINEP. M.P.
acknowledges funding from the Research Council of Finland under Contract No.
349971.

##  Appendix A Continuity equation for $\mathcal{T}_{\mathrm{Maxwell}}$

In covariant form, Maxwell’s SEM tensor given in Eq. (II) is given as [53, 49]

$\displaystyle\mathcal{T}_{\mathrm{Maxwell}}^{\mu\nu}=-\frac{1}{\mu_{0}}\left(\eta_{\alpha\lambda}F^{\mu\alpha}F^{\lambda\nu}+\frac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right).$
(19)

Here, $\eta^{\mu\nu}$ is Minkowski’s metric in $(-,+,+,+)$ signature and
$F^{\mu\nu}$ is the usual electromagnetic field tensor, given in terms of the
electromagnetic four-potential $A^{\mu}$ as
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$. By taking the four-
divergence of the above equation, we get [53, 49]

$\partial_{\nu}\mathcal{T}^{\mu\nu}_{\mathrm{Maxwell}}=-J_{\alpha}F^{\mu\alpha}.$
(20)

In terms of electromagnetic fields and sources, the right-hand side of the
above equation is (apart from the minus sign) the four-force
$f^{\mu}=(\mathbf{J}\cdot\mathbf{E}/c,\rho\mathbf{E}+\mathbf{J}\times\mathbf{B})$,
from where we can recognize the Lorentz force density. By substituting into
this equation the four-current from Eq. (3) and properly integrating over the
volume, we obtain [30] Eqs. (I) and (6) for the force and power densities,
respectively, in the MA formulation.

##  Appendix B Calculation of $\phi_{\mathrm{MA}}$

The total power in the MA formulation is

$\Phi_{\mathrm{MA}}=\int\mathbf{E}\cdot\left[\dot{\mathbf{p}}\delta^{3}(\mathbf{r})-\mathbf{m}\times\bm{\nabla}\delta^{3}(\mathbf{r})\right]\,\mathrm{d}^{3}\mathbf{r}.$
(21)

Adopting index notation for the last term, this equation is given by

$\Phi_{\mathrm{MA}}=\dot{\mathbf{p}}\cdot\mathbf{E}-\int
E_{i}\epsilon_{ijk}m_{j}\partial_{k}\delta^{3}(\mathbf{r})\,\mathrm{d}^{3}\mathbf{r},$
(22)

where $\epsilon_{ijk}$ is the Levi-Civita symbol and the summations in
repeated indices are implied. Integrating the last term by parts we obtain

$\Phi_{\mathrm{MA}}=\partial_{t}\mathbf{p}\cdot\mathbf{E}+\epsilon_{ijk}m_{j}\partial_{k}E_{i},$
(23)

where we took $\dot{\mathbf{p}}\approx\partial_{t}\mathbf{p}$ as the dipole’s
velocity in the laboratory frame is much smaller than $c$. The second term in
the right hand side is, in vector notation, equal to
$\mathbf{m}\cdot\left(\bm{\nabla}\times\mathbf{E}\right)$. Invoking Faraday’s
law, we have
$\Phi_{\mathrm{MA}}=\partial_{t}\mathbf{p}\cdot\mathbf{E}-\mathbf{m}\cdot\partial_{t}\mathbf{B}$.
This result represents the contribution to the total power from one single
dipole inside the integration volume. By summing the contribution from all
dipoles in the integration volume, we obtain

$\phi_{\mathrm{MA}}=\partial_{t}\mathbf{P}\cdot\mathbf{E}-\mathbf{M}\cdot\partial_{t}\mathbf{B},$
(24)

where $\mathbf{P}$ and $\mathbf{M}$ correspond, respectively, to $\mathbf{p}$
and $\mathbf{m}$ divided by the integration volume and multiplied by the
number of dipoles in this volume.

At last, recalling the medium is assumed non-dispersive, we have then

$\phi_{\mathrm{MA}}=\frac{1}{2}\frac{\partial}{\partial
t}\left({\mathbf{P}}\cdot\mathbf{E}-\mathbf{M}\cdot\mathbf{B}\right),$ (25)

which is exactly Eq. (6).

##  Appendix C The MMP Lagrangian and hidden momentum contribution

To justify the inclusion of the hidden energy and momentum contributions in
Eqs. (III.1) and (III.2), respectively, we will initially follow the main
steps from the derivation of the Lagrangian given in Ref. [59]. The starting
point is the classical version of the well-known Quantum Electrodynamics (QED)
Lagrangian for spinless, charged point particles interacting within external
electromagnetic fields. As we are interested in electromagnetic dipolar
sources, we will consider two point particles with identical mass $m$ and
opposite elementary charges $e$. The QED Lagrangian in this case reads

$\displaystyle
L=-mc^{2}\left[\left(1-|\dot{\mathbf{r}}_{1}|^{2}/c^{2}\right)^{1/2}+\left(1-|\dot{\mathbf{r}}_{2}|^{2}/c^{2}\right)^{1/2}\right]$
$\displaystyle-\int\left[J_{\mu}A^{\mu}+\frac{\varepsilon_{0}}{4}F_{\mu\nu}F^{\mu\nu}\right]\mathrm{d}^{3}\mathbf{r}.$
(26)

Here, $\mathbf{r}_{\mathrm{i}}$ and $\dot{\mathbf{r}}_{\mathrm{i}}$, $i=1,2$,
are the position and velocity of the $i$-th particle, respectively, as given
in the laboratory inertial reference frame, and
$A^{\mu}=(\varphi/c,\mathbf{A})$ is the electromagnetic four-potential. The
electrical charge density is
$\rho(\mathbf{r})=e\delta^{3}(\mathbf{r}-\mathbf{r}_{1})-e\delta^{3}(\mathbf{r}-\mathbf{r}_{2})$
and the electrical current density is
$\mathbf{J}(\mathbf{r})=e\dot{\mathbf{r}}_{1}\delta^{3}(\mathbf{r}-\mathbf{r}_{1})-e\dot{\mathbf{r}}_{2}\delta^{3}(\mathbf{r}-\mathbf{r}_{2})$,
which are equivalent to the four-current in Eq. (3) for large observation
distances $|\mathbf{r}|$.

Next, the Power-Zienau-Woolley (PZW) gauge transformation [86, 87, 88] is
applied to the Lagrangian in Eq. (C), allowing us to write the modified
Lagrangian $L^{\prime}$ in terms of field quantities as

$\displaystyle L^{\prime}$ $\displaystyle=$ $\displaystyle-
mc^{2}\left[\left(1-|\dot{\mathbf{r}}_{1}|^{2}/c^{2}\right)^{1/2}+\left(1-|\dot{\mathbf{r}}_{2}|^{2}/c^{2}\right)^{1/2}\right]$
(27) $\displaystyle+$
$\displaystyle\int\left\\{\frac{\varepsilon_{0}}{2}\left[|\mathbf{E}(\mathbf{r})|^{2}-c^{2}|\mathbf{B}(\mathbf{r})|^{2}\right]+\frac{1}{2}\mathbf{P}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})\right.$
$\displaystyle\left.+\frac{1}{2}\mathbf{M}(\mathbf{r})\cdot\mathbf{B}(\mathbf{r})-\dot{\mathbf{R}}\cdot\left[\mathbf{P}(\mathbf{r})\times\mathbf{B}(\mathbf{r})\right]\right\\}\mathrm{d}^{3}\mathbf{r}.$

In the above equation, $\mathbf{R}$ is the center of mass-energy of the system
as measured in the laboratory frame. The first two terms on the right-hand
side are not affected by the PZW transformation. The first term inside the
integral is an invariant of the electromagnetic field and is related to the
field’s energy density. The second and third ones are associated to the
dipolar interaction and, consequently, to electro- and magnetostriction
effects (actually $\mathbf{P}$ and $\mathbf{M}$ are initially related to the
complete multipole expansions [59], but we have already truncated it to
account only for dipolar interactions). The 1/2 factor occurs due to the
dipoles being linearly induced by the fields. The last term is a linear
momentum contribution known as Röntgen interaction and appears naturally even
in non-relativistic derivations [89, 90, 30].

Now, we promote $\mathbf{R}$ to a dynamical variable of the system and
introduce the relative coordinate $\mathbf{q}=\mathbf{r}_{2}-\mathbf{r}_{1}$.
To leading order in the velocities divided by $c$, it can be shown the
Lagrangian in Eq. (27) becomes

$\displaystyle L^{\prime}$ $\displaystyle=$ $\displaystyle-
Mc^{2}+\frac{M|\dot{\mathbf{R}}|^{2}}{2}+\frac{m_{\mathrm{red}}|\dot{\mathbf{q}}|^{2}}{2}$
(28)
$\displaystyle+\int\biggr{[}\frac{\varepsilon_{0}}{2}\left(|\mathbf{E}(\mathbf{r})|^{2}-c^{2}|\mathbf{B}(\mathbf{r})|^{2}\right)+\frac{1}{2}\mathbf{P}(\mathbf{r})\cdot\mathbf{E}(\mathbf{r})$
$\displaystyle\left.+\frac{1}{2}\mathbf{M}(\mathbf{r})\cdot\mathbf{B}(\mathbf{r})-\dot{\mathbf{R}}\cdot\left(\mathbf{P}(\mathbf{r})\times\mathbf{B}(\mathbf{r})\right)\right.$
$\displaystyle+\frac{\dot{\mathbf{R}}}{c^{2}}\cdot\left(\mathbf{M}(\mathbf{r})\times\mathbf{E}(\mathbf{r})\right)\biggr{]}\mathrm{d}^{3}\mathbf{r},$

where $M=2m$ is the total mass of the system and $m_{\mathrm{red}}=m/2$ is the
reduced mass. The last term in the integral is directly related to the hidden
momentum and hidden energy contributions (which are also known as Aharonov-
Casher interaction in the QED context), and arises as a first order effect
($\gamma\approx 1$) of the relativistic Lagrangian.

In the MMP formulation, the dielectric medium is described under the continuum
approximation as a distribution of point dipoles – this way, we have
$\mathbf{q}\to 0$, and the third term on the right-hand side of Eq. (28) is
zero. By introducing the atomic mass density $\rho_{\mathrm{a}}$, the
remaining mechanical terms are transformed as

$\displaystyle-
Mc^{2}\to-\int\rho_{\mathrm{a}}(\mathbf{r})c^{2}\mathrm{d}^{3}\mathbf{r}$ (29)

and

$\displaystyle\frac{M|\dot{\mathbf{R}}|^{2}}{2}\to\frac{1}{2}\int\rho_{\mathrm{a}}(\mathbf{r})|\mathbf{v}_{\mathrm{a}}(\mathbf{r})|^{2}\mathrm{d}^{3}\mathbf{r},$
(30)

which are associated to the rest and kinetic energies of the medium,
respectively, with $\dot{\mathbf{R}}$ being identified as the atomic velocity
$\mathbf{v}_{\mathrm{a}}$. With this continuum description, and recalling the
kinetic energy term is negligible, we can then finally write the Lagrangian
density for the MMP formulation as

$\displaystyle\mathcal{L}_{\mathrm{MMP}}$ $\displaystyle=$
$\displaystyle-\rho_{\mathrm{a}}c^{2}+\frac{1}{2}\mathbf{D}\cdot\mathbf{E}-\frac{1}{2}\mathbf{B}\cdot\mathbf{H}-\dot{\mathbf{R}}\\!\cdot\\!\left(\mathbf{P}\\!\times\\!\mathbf{B}\right)$
(31)
$\displaystyle+\frac{\dot{\mathbf{R}}}{c^{2}}\cdot\left(\mathbf{M}\times\mathbf{E}\right),$

which is valid for the laboratory frame and approximated to leading order in
the velocities, with the dependence on $\mathbf{r}$ omitted for brevity. The
MDW mass density is explicitly obtained in the analysis when we study how much
the original mass density $\rho_{\mathrm{a0}}$ becomes disturbed by the
optical force as $\rho_{\mathrm{MDW}}=\rho_{\mathrm{a}}-\rho_{\mathrm{a0}}$.

The momentum density of the MDW can be obtained by varying the action integral
in Eq. (31) with respect to $\dot{\mathbf{R}}$ as
$\partial\mathcal{L}_{\mathrm{MMP}}/\partial\dot{\mathbf{R}}=-\mathbf{P}\times\mathbf{B}+\frac{1}{c^{2}}\mathbf{M}\times\mathbf{E}$,
in which the hidden momentum contribution is transparently present.
Alternatively, if the kinetic energy density term
$\rho_{\mathrm{a}}|\mathbf{v}_{\mathrm{a}}|^{2}/2$ is kept, the equations of
motion yield
$\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}-\mathbf{P}\times\mathbf{B}+\frac{1}{c^{2}}\mathbf{M}\times\mathbf{E}=0$,
which is an equation for the MDW momentum density as
$\rho_{\mathrm{MDW}}\mathbf{v}_{\mathrm{l}}=\rho_{\mathrm{a}}\mathbf{v}_{\mathrm{a}}=\mathbf{P}\times\mathbf{B}-\frac{1}{c^{2}}\mathbf{M}\times\mathbf{E}$.

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