# 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.

## Key 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.