Effective mass of a spatially structured mode of the electromagnetic field: a single invariant for the group-velocity defect and the radiation-pressure defect

For a monochromatic, spatially structured mode of the free electromagnetic field we consider the quantity m_eff = sqrt(E² − (p‖c)²)/c², where E is the photon energy and p‖ is the projection of the total momentum onto the propagation axis.We show that three independently measured characteristics of such a mode — the group-velocity defect, the radiation-pressure defect, and the form of the dispersion relation — are all expressible through this single quantity and are identically related to one another. In particular, the mode obeys the relativistic relation p‖ = E·v_g/c², i.e. momentum, energy and group velocity are connected exactly as for a massive particle of mass m_eff. It follows that the radiation-pressure defect and the group-velocity defect are the same quantity: ΔF/(2P/c) = −δ_g = −½(m_eff c²/E)². The effective mass and all quantities derived from it are conserved under free propagation and are fully determined by preparation.The work is unifying rather than predictive: it introduces no new effect, but reduces results previously obtained by different groups and by different methods — the subluminal group velocity of structured light (Giovannini et al., Science 347, 857, 2015), the reduction of radiation pressure of structured beams (Trowbridge, Wright & Dholakia, 2025), and the formal analogy with massive dispersion (Saari, Phys. Rev. A 97, 063824, 2018) — to one invariant and one identity.A strict distinction is drawn between m_eff and the photon rest mass m₀, which differ numerically by 11–23 orders of magnitude for laboratory modes; experimental bounds on m₀ do not apply to m_eff. A direct experimental test is proposed: a joint measurement of radiation force and arrival-time delay on the same mode must yield the same number.AI use disclosure: Claude (Anthropic) was used as an adversarial interlocutor, for symbolic and numerical verification, for locating and verifying the primary literature, and for drafting the English text. All references were independently checked against primary sources and all numerical results recomputed. The author takes full responsibility for the entire contents.