Massless spin 2 particle: cylindrical symmetry, projective operators, gauge degrees of freedom

In the present paper, we have developed the theory of a massless spin 2 particle. We apply the matrix equation in Minkowski space-time, specifying it in cylindrical coordinates t, r, φ, z and tetrad. By diagonalizing energy operators, the third projection of total angular momentum, and the third projection of linear momentum, we derive the system of 39 differential equations in a polar coordinate r. In order to resolve this system, we apply the Fedorov–Gronskiy method based on the projective operator method. In accordance with this method, the dependence of all 39 functions is determined only by five different functions of the polar variable r that in the considered case are expressed in terms of Bessel functions. We find the explicit form of six independent solutions of the basic matrix equation. In order to eliminate gauge degrees of freedom, we use the general structure of gauge solutions according to the Pauli-Fierz approach, when the gauge solutions for the spin 2 field are constructed on the basis of the exact solution for a massless spin 1 field (in Bessel functions as well). In this way, we find the explicit form of two independent gauge solutions for the spin 2 field. In the end, we derive the explicit form of two gauge-free solutions for the massless spin 2 field, as should be expected by physical reason.