Interaction of weak gravitational waves with a small gyroscope universal-mounted in a macroscopic shell

Abstract

The interaction of a weak plane gravitational wave with a small gyroscope universal-mounted in a macroscopic gimbal is calculated. The microscopic gyroscope can be exemplified by a small hollow sphere with a massive spherical rotor inside. The macroscopic gimbal may be realized by a massless sphere with a rigidly coupled bar-bell of two masses. The gravitational wave causes a torque on the bar-bell, which is transferred to the gyroscope by the gimbal. For the proposed configuration the calculation results in a precession angle of the gyroscope proportional to $$\varGamma \times \omega _r / \omega _G$$ where $$\varGamma $$ is the amplitude of the gravitational wave, $$\omega _r$$ is the angular velocity of the rotor and $$\omega _G$$ is the frequency of the gravitational wave. The precession angle can be measured, in principle, by angular laser interferometry and leads for our configuration to a length difference of $$\varDelta L \approx 10^{-18}\text { m}$$ which is comparable with the formal arm length difference of LIGO for the assumed gravitational wave amplitude $$\varGamma = 10^{-21} \text { with }\omega _G = 1 \text { Hz}$$ .