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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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    $\begingroup$ 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. $\endgroup$
    – Jerry Schirmer
    Nov 14, 2012 at 17:24
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    $\begingroup$ @JerrySchirmer thanks! you're absolutely right, Witten's proof is available here $\endgroup$
    – twistor59
    Nov 14, 2012 at 17:47
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    $\begingroup$ @JerrySchirmer Schoen and Yau's minimal surface proof of the positive energy theory predates that of Witten's. $\endgroup$
    – Willie Wong
    Jan 25, 2013 at 12:47
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    $\begingroup$ en.wikipedia.org/wiki/Positive_energy_theorem $\endgroup$
    – Willie Wong
    Jan 25, 2013 at 12:48
  • $\begingroup$ @WillieWong: that's the problem with going from memory. $\endgroup$
    – Jerry Schirmer
    Jan 27, 2013 at 0:19

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The key idea for the positive mass theorem is that asymptotically flat spacetime always has non-negative energy. Furthermore, of all spacetime which are asymptotically flat, empty Minkowski space is the only one which has zero energy.

This is an important result because it tells us that spacetimes such as Minkowski are inherently stable.

Now, the proof of this was lacking. How would one define 'mass' in spacetime? The ADM formulation of GR allowed you to do this. (You can look up: http://homepage.univie.ac.at/piotr.chrusciel/teaching/Energy/Energy.pdf and references therein) Soon after, there was the proof by Robert Schoen and Shing Tung Yau which is a nice exercise in differential geometry. Then came the famous Witten proof in which he constructed initial data on a Cauchy hypersurface for asymptotically flat spacetimes and showed that the energy density would always be non-negative. It should be mentioned that people know that supergravity theories had this positve mass spacetimes because the mass was defined in terms of squares of real numbers/ real, positive valued functions. Witten's paper is very readable and well articulated. (https://projecteuclid.org/euclid.cmp/1103919981 - Witten).

Following the work of Witten, there was an extension of the theorem to higher dimensional asymptotically flat spacetimes by Parker and Taubes (http://users.math.msu.edu/users/parker/Witten.pdf)

Theis inspired a host of papers, most famously the paper by Gibbons, Horowitz, Perry and Hawking for black holes (http://projecteuclid.org/euclid.cmp/1103922377)

There is still a huge amount of interest in the community regarding the stability of spacetime. For example, stability of vacua in string theory is a highly researched topic and so is the stability in dS spacetimes.

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