The interaction between gravitational waves and a viscous fluid shell on a Schwarzschild background

Previous work has shown that the interaction between gravitational waves (GWs) and a shell of viscous matter leads to damping of the GWs and heating of the matter, and that these effects may be astrophysically significant. This result was derived using the theory of linear perturbations about a Minkowki background, and in this work the model is extended to be more physically realistic by allowing the background geometry to be Schwarzschild. It is found that the difference between using a Schwarzschild or Minkowski background is minimal when either $r\gg M$ or $\lambda< M$, where $r$ is the radius of the shell, $\lambda$ is the GW wavelength and $M$ is the mass of the system in geometric units (so that $1M_\odot=1.48$km). However, when $r\sim 6M$ and $\lambda\sim 25M$, then the damping and heating effects are about 9 times larger on a Schwarzschild background than on Minkowski, and such situations occur astrophysically.