From one of the research papers paper on tidal friction in black holes, I came across a statement "The energy which is crossing the horizon is the difference between the energy which is being carried inward and that which is being carried outward at infinity". What I understand by this is - If there is an exterior matter outside of a black hole, then that will produce perturbation. Perturbation means some form of waves, which will be absorbed by the black hole- so that is the energy carried inward. Now gravitational waves from the black hole - means energy carried outward. So up to this, it is fine.
Then the next statement from the paper is - "A stationary perturbation, however, will contain no radiation at infinity either ingoing or outgoing. The energy crossing the horizon is, therefore, zero". Can anyone help me understand what does this means? How is it that a stationary perturbation does not have any outgoing or ingoing radiation? And black holes will always produce outgoing gravitational waves, then how the energy crossing the horizon will be zero? I also want to know how can we write these boundary conditions mathematically? Thank you in advance!