The four-dimensional group velocity

The advantages of using the 4-dimensional group velocity are demonstrated. After first giving its purely kinematic definition for an arbitrary wave field, the discussion turns to packets of electromagnetic waves in electrically and magnetically anisotropic media which possess space-time dispersion and are smoothly nonuniform in space and slowly varying in time. In particular, a simple expression is obtained for the energy-momentum 4-tensor in terms of the group-velocity 4-vector. Finally, it is shown how the 4-dimensional notation simplifies the derivation of the conditions of orthogonality and conservation of the adiabatic invariant. A note by M. L. Levin which follows this paper contains a brief account of the basis results of W. R. Hamilton's investigations relating to the velocity of wave motion.