A solution of linearized Einstein field equations in vacuum used for the detection of the stochastic background of gravitational waves

Abstract

A solution of linearized Einstein field equations in vacuum is given and discussed. First it is shown that, computing from our particular metric the linearized connections, the linearized Riemann tensor and the linearized Ricci tensor, the linearized Ricci tensor results equal to zero. Then the effect on test masses of our solution, which is a gravitational wave, is discussed. In our solution test masses have an apparent motion in the direction of propagation of the wave, while in the transverse direction they appear at rest. In this way it is possible to think that gravitational waves would be longitudinal waves, but, from careful investigation of this solution, it is shown that the tidal forces associated with gravitational waves act along the directions orthogonal to the direction of propagation of waves. The computation is first made in the long wavelengths approximation (wavelength much larger than the linear distances between test masses), then the analysis is generalized to all gravitational waves. In the last sections of this paper it is shown that the frequency dependent angular pattern of interferometers can be obtained from our solution and the total signal seen from an interferometer for the stochastic background of gravitational waves is computed.