Is Electromagnetic Field Momentum Due to the Flow of Field Energy?

TL;DR

This paper examines whether electromagnetic field momentum can be explained as the flow of field energy, concluding that such an explanation is incompatible with special relativity and providing a corrected understanding of energy flow velocity.

Contribution

It demonstrates that the standard definition of electromagnetic energy flow velocity conflicts with relativity, and derives a consistent alternative, challenging the common energy flow explanation of field momentum.

Findings

The consensus energy flow velocity is inconsistent with relativity.

A correct relativistic energy flow velocity is derived.

Field momentum cannot be explained as energy flow in a relativistic framework.

Abstract

Conservation laws that are based on the divergence of a second-rank four-tensor are fundamentally different from conservation laws based on the divergence of a four-vector. This article investigates the consequences of this difference for understanding the relation between electromagnetic field momentum and the flow of electromagnetic field energy. Momentum and energy conservation require electromagnetic field momentum and energy to be treated as physically real, even in static fields. This motivates the conjecture that field momentum might be due to the flow of a relativistic mass density (energy density divided by the square of the speed of light). This article investigates the velocity of such flow and finds a conflict between two different definitions of it. This investigation is careful to respect the transformation rules of special relativity. The paper demonstrates that the consensus definition of the flow velocity of electromagnetic energy is inconsistent with the transformation rules of special relativity, and hence is incorrect. A correct flow velocity is then derived which is completely consistent with those transformation rules. The conclusion is that these conflicting definitions of flow velocity cannot be resolved in a way that is consistent with special relativity and that also allows electromagnetic field momentum to be the result of energy flow. Though real, field momentum cannot be explained as the flow of field energy. As a byproduct of the study, it is also shown that there is a comoving system in which the electromagnetic energy-momentum tensor is reduced to a simple diagonal form, with two of its diagonal elements equal to the energy density and the other two diagonal elements equal to plus and minus a single parameter derived from the electromagnetic field values, a result that places constraints on possible fluid models of electromagnetism.