The structure of the plane waves for spin $2$ field, massive and massless cases

A general theory for a field with spin based on the $30$-component system of first-order Fedorov – Regge equations is presented. As a result of the elimination of the additional vector and the third-rank tensor in these equations, the Pauli – Fierz second-order equations for the scalar and symmetric tensor are derived. Transition to the massless limit is analyzed in detail; the gauge symmetry available according to the Pauli – Fierz analysis is investigated. There are explicitly constructed solutions in the form of plane waves for a massive particle, which correspond to five linearly independent states. In the case of a massless field, 6 independent solutions are found, and it is shown that four of them are gauge ones and, therefore, can be excluded as nonphysical. Two independent solutions that do not contain gauge degrees of freedom are found explicitly.