The momentum of electromagnetic waves in dielectric materials

Variation of the generally covariant version of Minkowski's Lagrangian of macroscopic Maxwell theory with respect to the metric yields a symmetric tensor T μν which reproduces the Poynting vector i = cT 0i , and therefore would seem to favor Abraham's proposal A i = i /c 2 for the momentum density in a macroscopic electromagnetic wave. However, the T 00 component does not reproduce the standard energy density in the rest frame in the flat limit. The difference between the energy density and T 00 has the same structure as the difference between the Minkowski momentum density M i = n 2 i /c 2 and the Abraham momentum density. The result therefore adds further credibility to Minkowski's original energy‐momentum tensor and his proposal for the total momentum density of electromagnetic waves in dielectric materials. We draw attention to the facts that the covariant formulation of macroscopic Maxwell theory of dielectric materials requires a field strength‐velocity coupling term which could only be inferred in a long wavelength limit from quantum electrodynamics, and that the theory is clearly limited to interactions of sub‐UV photons with matter. Therefore, in spite of the fact that macroscopic Maxwell theory can formally be written in generally covariant terms, the theory for n ≠ 1 should be considered as inherently non‐relativistic, with non‐relativistic corrections of order n ‐ 1. The difference M ‐ c ‐2 between Minkowski's momentum density and the rescaled Poynting vector is an example of an order n ‐ 1 correction.