Energy flows of a gravitational wave in a liquid sphere

We consider the energy transfer problem in a static space–time background created by a liquid spherical body. In the accompanying reference frame, we derive the Lagrangian of gravity and the liquid that is of the fourth order in perturbations. We impose gauge conditions and integrate over the angular coordinates at the level of the action, which makes the problem two-dimensional. We derive the density and flow of the perturbation energy. Different gauge choices are considered. The energy conservation law is ensured by the static property of the metric and by the vanishing of the Lagrange variations.