The Nature of Energy

This posting is a little more advanced than usual. Read on at your own discretion...

I was reading a number of articles on general relativity recently, and I started thinking about exactly what energy is and perhaps more important - how is it actually defined?

The definition I have tended to favour is that energy is the variation in the action due to changes in the g_00 component of the spacetime metric. This definition works very well in almost every situation. It also means that the Einstein equations of general relativity are actually a statement that the total energy (and momentum) - at any point in spacetime- must vanish.

But then I started reading this papers about the energy localization hypothesis, in which it is claimed that gravity does NOT carry energy! And they have several published examples supporting the hypothesis. But that means that my definition cannot be accurate, because it includes gravitational energy through the Einstein tensor.

And suppose that you have two pendulae in space, one swinging and one at rest. The swinging one has energy, but it is also applying a gravitational force on the second one. The second one will start swinging along with the first, and so energy has been transported by the gravity.

So now I am wondering if the energy referred to in these papers in really energy, or is this hypothesis actually stating that gravitational energy cannot curve spacetime any further. Does it actually mean that the energy-momentum tensor in Einstein's equations does not include gravitational energy, but that gravity still has energy?. If anyone knows the answer to this, please post a rely below...