Reply to "Comment on 'Analysis of Recent Interpretations of the Abraham-Minkowski Problem'"

TL;DR

This paper defends previous criticisms of interpretations of the Abraham-Minkowski problem, clarifying the physical meaning of radiation pressure experiments, the nature of electromagnetic momentum versus energy, and the conditions for relativistic formulations in media.

Contribution

It clarifies misconceptions about radiation pressure experiments, emphasizes the distinction between electromagnetic momentum and energy, and discusses conditions for relativistic electromagnetic theory in media.

Findings

Radiation pressure experiments relate to boundary forces, not the Abraham-Minkowski problem.

Electromagnetic waves carry momentum but not mechanical energy density.

The Minkowski energy-momentum tensor satisfies relativistic divergence-free conditions, unlike Abraham's.

Abstract

The Comment of M. Partanen and J. Tulkki [Phys. Rev. A {\bf 100}, 017801 (2019)] claims that my criticism expressed in Phys. Rev. A {\bf 98}, 043847 (2018) of the earlier paper of Partanen {\it et al.} [Phys. Rev. A {\bf 95}, 063850 (2017)] was incorrect. There are essentially three points involved here: (1) the first one regards the physical interpretation of the radiation pressure experiment of A. Kundu {\it et al.} [Sci. Rep. {\bf 7}, 42538 (2017)]. My mathematical analysis of this situation, although simple, was able to illustrate the main property of the experiment, namely that it showed the action from the radiation forces on the dielectric boundaries, and from the Lorentz force in the interior, but it had not any relationship to the Abraham-Minkowski momentum problem as was originally stated by the investigators. (2) The second point was my emphasis on the fact that in an electromagnetic pulse in a medium there cannot be an accompanying mechanical energy density of the same order of magnitude as the electromagnetic energy density itself. The electromagnetic wave carries with it a mechanical {\it momentum}, but not a mechanical {\it energy}. Here I illustrate this point by a simple numerical analysis. (3) When going to a relativistic formulation of the electromagnetic theory in matter, care must be taken to secure that fundamental conditions from field theory are satisfied. In particular, one cannot in general take the electromagnetic total energy and momentum of a radiation pulse to constitute a four-vector; such a property holds only if the electromagnetic energy-momentum tensor is divergence-free. For the Minkowski tensor this condition is satisfied, whereas for the Abraham tensor it is not.

Equations

$$W=\frac{1}{2}(\mathbf{E}\cdot \mathbf{D}+\mathbf{H}\cdot \mathbf{B}),$$

$${\mathbf{g}}^{\mathrm{mech}}=\frac{n^2-1}{c^2}(\mathbf{E}\times \mathbf{H})=\frac{n^2-1}{c^2}\mathbf{S},$$

$$E_{\mathrm{kin}}=\frac{1}{2}\rho v^2=3.6\times 10^{-25}J/{\mathrm{m}}^3,$$

$$W_{\mathrm{MDW}}^{(L)}=(n^2-1)W,$$

Full text

Reply to ”Comment on ”Analysis of recent interpretations of the Abraham-Minkowski problem””

Abstract

The Comment of M. Partanen and J. Tulkki [Phys. Rev. A 100, 017801 (2019)] claims that my criticism expressed in Phys. Rev. A 98, 043847 (2018) of the earlier paper of Partanen et al. [Phys. Rev. A 95, 063850 (2017)] was incorrect. Now, there are essentially three points involved here: (1) the first one regards the physical interpretation of the radiation pressure experiment of A. Kundu et al. [Sci. Rep. 7, 42538 (2017)]. My mathematical analysis of this situation was certainly simplistic, but yet it was able to illustrate the main property of the experiment, namely that it showed the action from the radiation forces on the dielectric boundaries, and from the Lorentz force in the interior, but it had not any relationship to the Abraham-Minkowski momentum problem as was originally stated by the investigators. (2) The second and most important point was my emphasis on the fact that in an electromagnetic pulse in a medium there cannot be an accompanying mechanical energy density of the same order of magnitude as the electromagnetic energy density itself. The electromagnetic wave carries with it a mechanical momentum, but not a mechanical energy (the latter being of second order in the particle velocity). In the present note I illustrate this point by a simple numerical analysis. (3) When going to a relativistic formulation of the electromagnetic theory in matter, care must be taken to secure that fundamental conditions from field theory are satisfied. In particular, one cannot in general take the electromagnetic total energy and momentum of a radiation pulse to constitute a four-vector; such a property holds only if the electromagnetic energy-momentum tensor is divergence-free. For the Minkowski tensor this condition is satisfied, whereas for the Abraham tensor it is not. As a conclusion, my earlier standpoint on the Abraham-Minkowski problem remains unchanged.

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As mentioned in the Abstract above, the Comment by Partanen and Tulkki (PT) partanen19 on my recent paper brevik18 consists essentially of the following points:

1. First, my analysis of the radiation pressure experiment of Kundu et al. kundu17 was considered. Kundu et al. claimed this experiment to show the correctness of the Abraham photon momentum in matter, in contrast to the Minkowski momentum. One of my purposes in the analysis in brevik18 was to point out that this claim was not right. What the experiment does, is to show the combined effect from the surface forces (common for the Abraham and Minkowski alternatives) and the Lorentz forces on charges in the interior. The Abraham momentum does not come into play here at all. PT claim that my elastic model was simplistic, not able to describe the irreversible character of the experiment. That statement is in itself right. My intention was however not to describe the real experiment in detail, but to adopt a simple elastic model that was sufficient to explain the basic physics. The physical effect is solely due to the combined action of electromagnetic forces in the boundary regions, and in the interior.

2. The second, and a more important point, was the analysis of an optical pulse traveling in a resting medium. The core of the problem is shown in the Comment’s expression (10) for the energy-momentum tensor. In this context it is convenient to consider an example. Let a cw laser beam $E_{inc}=E_{0}e^{i(kz-\omega t)}$ be propagating in the $z$ direction in water. Assume power $P=1~{}$watt. With a cross section of $1~{}$mm2 this corresponds to a Poynting vector ${\bf S}={\bf E\times H}$ of magnitude $S=1~{}$MW/m2. The electromagnetic field energy density is

[TABLE]

which amounts to $W=Sn/c=4.44~{}\rm{mJ/m}^{3}$ when $n=1.33$ is the refractive index. The mechanical momentum density given to the medium from the leading edge of the pulse, here called ${\bf g}^{\rm mech}$, is

[TABLE]

in accordance with Eq. (8) in PT, and is correct. (The existence of this momentum density was demonstrated experimentally by R. V. Jones and others a long time ago jones54 ; jones78 .) Now put the expression (2) equal to $\Delta p\cdot N$, with $\Delta p$ the impulse given to each molecule and $N$ the particle density. Taking the molecule mass $m$ to be $18u$ with $u=1.66\times 10^{-27}~{}$kg the atomic mass unit, we get $N=3.35\times 10^{28}~{}$m*-3*, and $\Delta p=2.55\times 10^{-40}~{}$N s. With $\Delta p=mv$ we thus obtain for the particle velocity $v=0.85\times 10^{-14}~{}\rm{m/s}$. It corresponds to a kinetic mechanical energy density

[TABLE]

with $\rho$ the density of water.

Now consider Eq. (7) in PT, where the energy density for a propagating wave, called $W_{\rm MDW}^{(L)}$, is given as

[TABLE]

where $W$ is the same as in Eq. (1) above. Expression (4) means that there should be an extra energy term of order $W$, in addition to the electromagnetic energy density $W$, in the propagating wave. Where can this extra term come from? Obviously not from the immensely smaller kinetic energy density $E_{\rm kin}$ in Eq. (3). I cannot see that the PT energy density expression, Eq. (4) above, can be correct.

There is in addition another special property of the PT formalism that causes concern: according to their energy-momentum tensor (10) the energy flux density should be equal to $n^{2}\bf({E\times H})$, instead of the usual Poynting vector $\bf({E\times H})$. Such a change of the expression for the energy flux density, if correct, would simply revolutionize classical electromagnetic theory. It cannot be right.

3. In their analysis, PT made use of relativistic arguments, involving the rest mass energy $\delta mc^{2}$. This may be an unfortunate approach, as it tends to obscure the argument. Physically, the present problem involves only weak mechanical forces in a resting medium, and has very little to do with relativity, although it is true that relativity formally turns up in connection with the transformations of energy and momentum that PT made use of. From a general perspective it is known that in order for energy and momentum to transform as the components of a four-vector, the corresponding four-divergence of the energy-momentum tensor has to be zero. This has been shown by Møller and others moller72 . The Minkowski tensor satisfies this criterion, as its energy-momentum tensor (superscript M) obeys the equation $\partial_{\nu}T_{\mu\nu}^{M}=0$ in view of Maxwell’s equations.

4. Remark on Appendix A in the Comment: Equation (A1) is the same as the Minkowski/Abraham conservation equation for energy, when multiplied with the constant $n^{2}$. Equation (A2) is precisely the Minkowski conservation equation for momentum. Evidently, the four-divergence of the energy-momentum tensor $T_{MP}$ in (10) is therefore equal to zero.

In conclusion, I have to uphold my earlier standpoint on the Abraham-Minkowski problem as expressed in Ref. brevik18 . Especially, this concerns Point 2 above.

Iver Brevik

Department of Energy and Process Engineering

Norwegian University of Science and Technology,

7491 Trondheim, Norway