# Positivity of Total Gravitational Energy in GR

I read the following statement in the introduction to an article:

Over the last 30 years, one of the greatest achievements in classical general relativity has certainly been the proof of the positivity of the total gravitational energy, both at spatial and null infinity. It is precisely its positivity that makes this notion not only important (because of its theoretical significance), but also a useful tool in the everyday practice of working relativists.

Since I haven't been involved in GR for about 30yrs, I've missed this. Could someone briefly explain how these results are stated and give some references if possible.

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The proofs that I know of are very long and technical. The outline is that if you assume something like the Null or Dominant energy conditions and asymptotic flatness, then you can prove that the Bondi or ADM energies are positive. The earliest one of these was done by Witten using the spinor formulation of relativity sometime in the 80s. I believe they are worked out in Penrose's spinor book, but I don't remember for sure. –  Jerry Schirmer Nov 14 '12 at 17:24
@JerrySchirmer thanks! you're absolutely right, Witten's proof is available here –  twistor59 Nov 14 '12 at 17:47
@JerrySchirmer Schoen and Yau's minimal surface proof of the positive energy theory predates that of Witten's. –  Willie Wong Jan 25 '13 at 12:47
@WillieWong: that's the problem with going from memory. –  Jerry Schirmer Jan 27 '13 at 0:19