Is electromagnetic field momentum due to the flow of 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 (defined as energy density divided by the square of the speed of light). This article investigates the velocity of such a mass flow and finds a conflict between two different definitions of it, both of which originally seem plausible if the flow is to be taken as real. This investigation is careful to respect the transformation rules of special relativity throughout. 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 replacement flow velocity is 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 also allows electromagnetic field momentum density to be the result of relativistic mass flow. Though real, field momentum density cannot be explained as the flow of a relativistic mass density. 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.