Classical four-vector fields in the relativistic longitudinal gauge

Minkowski four-space uniqueness theorems are used to develop further the author's "relativistic longitudinal gauge" [J. Math. Phys. 40, 4911 (1999)] for four-irrotational classical four-vector fields. A theorem is developed which distinguishes between two and only two "physical" classes of classical four-vector fields. One must satisfy the "relativistic transverse gauge," i.e., the Lorentz condition, while the other must satisfy this new relativistic longitudinal gauge where its four-curl, the Maxwell field tensor itself, is set to zero. The Lagrangian density of the new four-irrotational four-vector field is distinguished from the usual Lorentz constrained Lagrangian density by the incorporation of an additional overall minus sign. Application of the relativistic longitudinal gauge, in the four-irrotational four-vector field case, eliminates the badly behaved terms associated with the spatial degrees of freedom from a most general, fully quadratic, Lagrangian density. The resulting constrained Lagrangian density is bounded from below and therefore a relativistic longitudinal classical four-vector field has the possibility of a physical interpretation.