Energy flows of gravitational waves

The problem of the energy transfer produced by weak gravitational waves on a static background is solved by using the Killing vectors without introducing the momentum–energy pseudotensor. The Lagrangian, which is of the second order in metric excitations and which determines the flow and density of the energy of gravitational waves on a static spherically symmetrical space–time background, is derived. Integration over angular variables in the action integral thus reduces the problem to a two-dimensional effective problem. Energy flows on the Schwarzschild metric background are calculated.