I have read some papers on positive mass theorem, mainly those by Schoen and Yau. I am awed by their minimal surface technique. Yet, I found little information about the motivation of this conjecture. Is it simply because we suppose mass in a naive sense must not be negative? Why scalar curvature makes sense to determine positivity of mass?

Thank you!


1 Answer 1


The motivation of course comes from physics: under suitable simplifying assumptions, a Lorentzian spacetime is locally $(M \times \mathbb R,g - dt^2)$ for $(M,g)$ a Riemannian 3-manifold, and the local mass density $T_{tt}$ of the stress-energy tensor is equal to the scalar curvature of $g$. Thus the assumption of non-negative scalar curvature of $(M,g)$ in the the positive mass theorem corresponds to the weak energy condition for the spacetime. In colloquial terms, the total energy must be non-negative if the local energy density is non-negative everywhere.

You should be able to find detailed discussion of this matter in many texts on general relativity - I won't try to go in to detail because my physics is very rusty, and there are a lot of subtleties when trying to define local energy. (The root of this is the lack of any well-defined notion of "gravitational energy density" in GR.) My first recommendation would be Chapter 11 of Wald's book.

Maybe something shorter to start: I wrote a little about this in the first chapter of my undergraduate thesis a few years ago. I wouldn't vouch for its accuracy or depth, but hopefully it gives a decent picture of the intuition for the ADM mass and the difficulties in defining local energy.


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