Stochastic oscillations of general relativistic disks

We analyze the general relativistic oscillations of thin accretion disks around compact astrophysical objects interacting with the surrounding medium through non-gravitational forces. The interaction with the external medium (a thermal bath) is modeled via a friction force, and a random force, respectively. The general equations describing the stochastically perturbed disks are derived by considering the perturbations of trajectories of the test particles in equatorial orbits, assumed to move along the geodesic lines. By taking into account the presence of a viscous dissipation and of a stochastic force we show that the dynamics of the stochastically perturbed disks can be formulated in terms of a general relativistic Langevin equation. The stochastic energy transport equation is also obtained. The vertical oscillations of the disks in the Schwarzschild and Kerr geometries are considered in detail, and they are analyzed by numerically integrating the corresponding Langevin equations. The vertical displacements, velocities and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases.

💡 Research Summary

This paper investigates the stochastic dynamics of thin accretion disks orbiting compact objects within a fully general‑relativistic framework. The authors model the interaction between the disk and an ambient thermal bath by introducing two non‑gravitational forces: a viscous friction term proportional to the particle’s velocity and a random force representing thermal fluctuations. Starting from the geodesic motion of test particles in the equatorial plane, they perturb the trajectories by a small displacement vector ξμ and derive the first‑order perturbation equation. By adding the friction coefficient γ and a stochastic term ξμ(t) to the covariant geodesic deviation equation, they obtain a general‑relativistic Langevin equation

 m D²ξμ/Dτ² + m γ Dξμ/Dτ = –Rμ ναβ uν ξα uβ + ξμ(t),

where Rμ ναβ is the Riemann tensor of the background spacetime (Schwarzschild or Kerr) and uν is the four‑velocity of the unperturbed circular orbit. The stochastic force is taken to be Gaussian white noise with zero mean and correlation

 ⟨ξμ(t) ξν(t′)⟩ = 2 m γ kB T gμν δ(t–t′),

ensuring that the fluctuation‑dissipation theorem holds in the relativistic setting. From this Langevin equation they also derive an energy transport equation

 dE/dt = –γ v² + v·ξ(t),

which balances viscous dissipation against stochastic energy injection.

The focus of the analysis is on vertical (z‑direction) oscillations of the disk. In the Schwarzschild geometry the effective restoring frequency is

 Ωz² = (M/r³)(1 – 6M/r),

while in the Kerr geometry the expression becomes more intricate due to frame‑dragging,

 Ωz² = (M/r³)