Fluctuation-dissipation relation of test masses in classical stochastic gravitational wave background

Research on pulsar timing arrays has provided preliminary evidence for the existence of a stochastic gravitational background, which, either being primordial or of astrophysical origin, will interact universally with matter distributions in our universe and affect their evolutions. This work, based on general relativity and stochastic dynamics theory, investigates the fluctuation-dissipation relation of isolated celestial bodies within a classical stochastic gravitational wave background. We employ the generalized Langevin model to analyze the fluctuating forces exerted on test masses by random spacetime and how energy dissipation occurs. Through the assumption of equilibrium, we derive the necessary conditions that should be satisfied by the stochastic gravitational wave background in the long wavelength limit, which, as we found, is a result of the back-reactions of the test mass system to the stochastic field. Additionally, as the establishment of the fluctuation-dissipation relation for such a system, certain thermodynamic quantities related to the statistical properties of the stochastic gravitational wave background could be defined and the characteristic of the diffusion process of test masses is obtained.