Solutions with Cylindrical Symmetry for Graviton Field in Linear Approximation: The Gauge Degrees of Freedom

Abstract

The theory of the graviton field in linear approximation is studied. We apply the matrix equation in Minkowski space-time, specifying it the cylindrical coordinates t, r, φ, z and tetrad. By diagonalizing the operators of the energy, the third projection of the total angular momentum, and the third projection of the linear momentum, we derive the system of 39 differential equations in polar coordinate r. It is resolved with the use of the method by Fedorov–Gronskiy. In accordance with this, dependance of all 39 functions is determined through only five different functions of the variable r, in the case under consideration they are expressed in terms of Bessel functions. We have constructed six linearly independent solutions of the basic equation. In order to eliminate the gauge degrees of freedom, we use the general definition for gauge solutions according to the Pauli–Fierz approach, now adjusted to tetrad formalism. These gauge solutions are constructed with the use the exact solutions for massless spin 1 field. In this way, we find explicit form of four independent gauge solutions for spin 2 field. In the end, we find a explicit form of two gauge-free solutions for spin 2 field, as it should be expected by physical reason.