Do gravitational waves carry energy?

It seems like a simple question, with an obvious answer, and yet this question is still debated by some of the leading experts in the theory of gravity. There is no doubt that electromagnetic waves carry energy, or that pressure waves or sound waves or any other form of wave carries energy, and yet gravity is somehow quite different from all other known phenomena.

The reason for this is the general theory of relativity, first proposed in 1915 by Albert Einstein (and possibly David Hilbert, but that is an argument for another day). One of the fundamental concepts in general relativity is that the force of gravity is an illusion, and that objects are actually travelling through a curved spacetime that appears to us to be a flat spacetime that contains an extra force. If there is no force of gravity, then there can be no energy contained in it. 

Adding to the confusion are the two facts about relativistic theories in general - that the energy depends on the observer, so that every detector records a different value for the energy, and the fact that gravitational fields are nonlinear, meaning that the gravitational field will create its own gravitational field which will alter our calculations.

The experts on the general theory of relativity are divided on whether gravitational waves carry energy. As yet, there is no clear method for calculating the energy of a gravitational field, and there are numerous competing proposals. Several well regarded textbooks on general relativity have given definitive proofs, only to be shown false by later research and calculations. 

There is however one famous thought experiment that is often quoted, which demonstrates that gravitational waves must definitely carry energy. It is called the sticky bead argument, and it goes as follows.

Suppose that somewhere deep in space, there is some energetic astrophysical process happening. For the sake of argument, and because of recent discoveries, suppose that it is a pair of black holes orbiting each other.

Because the two masses are moving, the gravitational field that they generate will also be changing over time. The result is a series of gravitational waves being produced that then propagate out into space. The kinetic and potential energy of the black holes will be decreasing as they lose energy to these waves, and it is argued that the waves will carry energy off into space.

A some later point in time, one of these gravitational waves will hit a very simple detector. This detector is in the form of a cylindrical rod, on which there is a bead that can move up and down the rod. The gravitational wave will cause the bead to move, absorbing energy from the wave. However the bead and the rod are made of materials that have a very high friction coefficient (hence the name sticky bead), so the bead quickly loses its energy to friction.

The rod and the bead heat up, proving definitively that the binary black hole system has transferred some of its energy to the sticky bead through a gravitational wave. This simple thought experiment proves that gravitational waves, and thus gravitational fields, must contain energy.

This argument is often quotes in textbooks, websites, and academic research papers. And it is wrong.