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

Iver Brevik

arxiv: 1907.07927 · v1 · pith:WDTRFJPDnew · submitted 2019-07-18 · ⚛️ physics.class-ph

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

Iver Brevik This is my paper

Pith reviewed 2026-05-24 19:36 UTC · model grok-4.3

classification ⚛️ physics.class-ph

keywords Abraham-Minkowski problemenergy-momentum tensorMinkowski tensorAbraham tensorradiation pressureelectromagnetic momentumrelativistic field theorydielectric media

0 comments

The pith

Only the Minkowski energy-momentum tensor is divergence-free, so only it lets a radiation pulse's total energy and momentum form a four-vector.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defends an earlier analysis against a comment by showing that a radiation-pressure experiment on a dielectric slab measures boundary forces and interior Lorentz forces but supplies no information on the Abraham-Minkowski controversy. It further shows that an electromagnetic pulse in matter carries mechanical momentum yet produces no mechanical energy density comparable to the field energy itself. The central relativistic claim is that the electromagnetic four-momentum of the pulse can be treated as a true four-vector only when the electromagnetic energy-momentum tensor is divergence-free, a property satisfied by the Minkowski tensor but not by the Abraham tensor.

Core claim

When electromagnetic theory in matter is cast in relativistic form, the total electromagnetic energy and momentum of a radiation pulse constitute a four-vector if and only if the electromagnetic energy-momentum tensor is divergence-free. This condition holds for the Minkowski tensor but fails for the Abraham tensor.

What carries the argument

The divergence-free condition on the electromagnetic energy-momentum tensor, required for the integrated energy and momentum to transform as a four-vector.

If this is right

The Abraham tensor cannot be used when four-vector properties of the field momentum are required.
The Kundu experiment measures ordinary electromagnetic forces and does not resolve the momentum controversy.
An electromagnetic pulse in matter transports mechanical momentum without a matching mechanical energy density of the same magnitude.
Relativistic consistency selects the Minkowski tensor over the Abraham tensor on fundamental grounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Momentum definitions that violate the divergence-free condition may produce inconsistent results when the medium moves or when boosts are applied.
The mechanical-momentum-without-energy result suggests that any stored energy in the medium must be treated separately from the propagating pulse.

Load-bearing premise

The Kundu radiation-pressure experiment can be shown by direct force calculation to involve only boundary and Lorentz forces and therefore has no bearing on the Abraham-Minkowski momentum question.

What would settle it

A calculation or measurement demonstrating that the Abraham tensor remains divergence-free inside a dielectric for a propagating pulse, or that the integrated Abraham four-momentum transforms correctly under Lorentz boosts.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Brevik's reply defends his earlier take on the Kundu experiment and mechanical energy but the relativistic four-vector distinction rests on a claim that does not match standard field theory.

read the letter

The main takeaway is that this reply restates Brevik's 2018 positions without new derivations or data. It handles the Kundu experiment by showing the measured forces come from boundaries and interior Lorentz terms, not the Abraham-Minkowski momentum issue. The numerical check that an electromagnetic pulse carries mechanical momentum but no comparable mechanical energy density is straightforward and addresses one specific objection cleanly. Those sections are clear and proportionate to the claims being defended. The third point on relativistic formulation is the soft spot. The abstract asserts that only the Minkowski tensor is divergence-free, so only its integrated energy-momentum forms a four-vector, while Abraham fails this. Standard relativistic electromagnetism in linear media gives a non-zero four-divergence for both tensors, equal to the negative of the four-force density on the material. Neither vanishes inside the dielectric, which removes the claimed distinction. The reply presents the divergence-free property as established without re-deriving it or engaging the force-density coupling. This is a reply paper, so it is mainly useful to readers already tracking the specific exchange around the 2017 Kundu work and the 2018 Brevik comment. It shows consistent reasoning on the experimental reinterpretation but does not supply independent evidence or resolve the broader controversy. I would not bring it to reading group or cite it. It does not rise to the level that would justify sending it out for peer review.

Referee Report

1 major / 2 minor

Summary. This manuscript is a reply to a comment by Partanen and Tulkki on the author's 2018 paper criticizing their 2017 analysis of the Abraham-Minkowski momentum problem. It addresses three points: (1) reinterpreting the Kundu et al. radiation pressure experiment as demonstrating only boundary and Lorentz forces with no connection to the momentum problem; (2) arguing via numerical analysis that an electromagnetic pulse carries mechanical momentum but not mechanical energy of comparable magnitude; (3) asserting that in a relativistic formulation, the electromagnetic energy and momentum of a radiation pulse form a four-vector only if the energy-momentum tensor is divergence-free, a condition satisfied by the Minkowski tensor but not the Abraham tensor.

Significance. If the relativistic claim in point (3) holds, the paper would strengthen the case for the Minkowski tensor on fundamental field-theoretic grounds and help clarify experimental interpretations in the Abraham-Minkowski controversy. The emphasis on divergence-free conditions and the separation of mechanical momentum from energy could provide useful constraints for relativistic treatments of EM in media.

major comments (1)

[Abstract, point (3)] Abstract, point (3): The central claim that the Minkowski tensor is divergence-free (while the Abraham tensor is not), allowing only the former's integrated EM energy-momentum to form a four-vector, is asserted without equations or derivation. Standard relativistic EM in linear media gives ∂_μ T^{μν} = -f^ν (nonzero inside the dielectric) for both tensors, with f the 4-force density on matter; this removes the claimed distinction and undermines the preference for Minkowski on this basis.

minor comments (2)

The abstract references a 'simple numerical analysis' for point (2) and a 'mathematical analysis' for point (1), but provides no equations, tables, or figures; the full manuscript should include these explicitly for verifiability.
The reply relies on prior self-citations (2018 paper and 2017 experiment) without re-deriving the divergence property here; adding a brief self-contained derivation or reference to the specific equation establishing divergence-freeness would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading of our reply manuscript and for highlighting the need for greater clarity on point (3). We respond to the major comment below.

read point-by-point responses

Referee: [Abstract, point (3)] Abstract, point (3): The central claim that the Minkowski tensor is divergence-free (while the Abraham tensor is not), allowing only the former's integrated EM energy-momentum to form a four-vector, is asserted without equations or derivation. Standard relativistic EM in linear media gives ∂_μ T^{μν} = -f^ν (nonzero inside the dielectric) for both tensors, with f the 4-force density on matter; this removes the claimed distinction and undermines the preference for Minkowski on this basis.

Authors: We agree that the abstract presents the claim concisely without derivation or equations. In the revised manuscript we will add a short derivation (or explicit reference to the standard relativistic treatment) showing why the Minkowski tensor satisfies the divergence-free condition required for the electromagnetic energy-momentum of an isolated radiation pulse to form a four-vector, while the Abraham tensor does not. Although both tensors obey ∂_μ T^{μν} = -f^ν inside the dielectric, the explicit forms of the tensors differ: the Minkowski tensor is the canonical one whose four-divergence vanishes in the relevant sense for the pulse's total 4-momentum, whereas the Abraham tensor contains additional terms that violate this property. We will expand the text to make this distinction explicit and to address the referee's concern directly. revision: yes

Circularity Check

1 steps flagged

Central 4-vector claim asserted without re-derivation, relying on self-citation to 2018 paper

specific steps

self citation load bearing [Abstract, point (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."

The load-bearing distinction (Minkowski tensor divergence-free hence 4-vector; Abraham not) is presented as given without derivation or external verification in this document. It is justified only by reference to the author's own 2018 paper (Phys. Rev. A 98, 043847), making the relativistic preference for Minkowski dependent on that self-citation chain rather than independent content here.

full rationale

The paper states the divergence-free property for Minkowski (but not Abraham) as a fact enabling the 4-vector property, without deriving it in this reply. This traces directly to the author's prior 2018 work cited in the abstract. However, points (1) and (2) involve independent mathematical analysis of the experiment and energy densities, so the central claim has partial independent content and is not fully reduced by construction. No other patterns (self-definitional, fitted predictions, etc.) appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper invokes standard properties of the Minkowski and Abraham tensors from classical field theory without introducing new fitted parameters or postulated entities.

axioms (1)

domain assumption The Minkowski energy-momentum tensor is divergence-free while the Abraham tensor is not, allowing only the former to yield a four-vector for pulse energy and momentum.
Stated directly in abstract point (3) as a fundamental condition from field theory.

pith-pipeline@v0.9.0 · 5867 in / 1382 out tokens · 27003 ms · 2026-05-24T19:36:19.113659+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

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.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

The Minkowski tensor satisfies this criterion, as its energy-momentum tensor (superscript M) obeys the equation ∂νT^M_μν = 0 in view of Maxwell’s equations.

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