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

The theory of the graviton field in linear approximation is studied. We apply the matrixequation in Minkowski space-time, specifying it the cylindrical coordinates <em>t</em>, <em>r</em>, <em>&phi;</em>, <em>z</em> andtetrad. By diagonalizing the operators of the energy, the third projection of the total angularmomentum, and the third projection of the linear momentum, we derive the system of 39differential equations in polar coordinate <em>r</em>. It is resolved with the use of the method byFedorov&ndash;Gronskiy. In accordance with this, dependance of all 39 functions is determinedthrough only five different functions of the variable <em>r</em>, in the case under consideration theyare expressed in terms of Bessel functions. We have constructed six linearly independentsolutions of the basic equation. In order to eliminate the gauge degrees of freedom, we usethe general definition for gauge solutions according to the Pauli&ndash;Fierz approach, now adjustedto tetrad formalism. These gauge solutions are constructed with the use the exact solutionsfor massless spin 1 field. In this way, we find explicit form of four independent gauge solutionsfor 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.