the positive mass theorem
An isolated gravitating system — a star, a galaxy, a black hole sitting alone in otherwise empty space — has a total mass you can read off from how its gravitational field falls away at great distance. The positive mass theorem says something that sounds obvious but was genuinely hard to prove: that total mass can never be negative, and it is zero only for completely empty, flat spacetime. In other words, gravity, with physically reasonable matter, cannot make the universe weigh less than nothing; you cannot build an isolated system with negative total energy.
Precisely, consider an asymptotically flat Riemannian 3-manifold (M, g) — a model of a time-slice of spacetime that looks like flat R^3 outside a large ball, with the metric approaching the Euclidean metric at a controlled rate. The ADM mass m is a number extracted from the leading correction of g to flatness at infinity, m = lim over large spheres of a specific flux integral of (partial_j g_ij - partial_i g_jj). Assume the dominant/nonnegative energy condition, which on the time-slice becomes nonnegative scalar curvature S >= 0 (in the time-symmetric case). The theorem: m >= 0, with m = 0 if and only if (M, g) is flat Euclidean R^3. There are two famous and completely different proofs. Schoen and Yau (1979) used minimal surfaces: if mass were negative they constructed a stable minimal surface and derived a contradiction from the stability inequality combined with S >= 0. Witten (1981) used spinors and the Dirac operator: the Lichnerowicz formula D^2 = nabla* nabla + S/4 lets one solve for a harmonic spinor whose boundary behaviour computes the mass as a manifestly nonnegative integral — a strikingly short argument when a spin structure is available.
Why it matters: positivity of mass is the statement that flat space is a stable ground state of general relativity, it underlies the proof of the Riemannian Penrose inequality (mass bounds the area of horizons), and it knits together general relativity, minimal-surface theory, and spin geometry. Honest cautions. First, both proofs are conditional on an energy condition (nonnegative scalar curvature on the slice); drop it and mass can be negative — the theorem is about physically reasonable matter, not a tautology. Second, the rigidity case is essential and subtle: m = 0 forces exact flatness, not merely 'small' geometry; this rigidity is what makes the theorem powerful but also what makes both proofs delicate. Third, the two proofs have different reaches: Witten's spinor proof needs a spin structure and generalizes cleanly to higher dimensions only where spinors behave, while Schoen-Yau's minimal-surface proof hit the same codimension-7 singularity wall as geometric measure theory and was only recently extended to all dimensions; do not assume 'the' positive mass theorem holds in every dimension by either method without that caveat.
The Schwarzschild slice of mass m > 0 has metric g = (1 + m/2r)^4 (dx^2 + dy^2 + dz^2) on R^3 minus the origin; it is scalar-flat (S = 0), asymptotically flat, and its ADM mass is exactly m > 0. The theorem says you cannot find any asymptotically flat scalar-nonnegative metric with negative ADM mass — Schwarzschild with m < 0 would be such, and indeed it has a naked singularity and violates the hypotheses.
The Schwarzschild slice realizes positive ADM mass; negative-mass Schwarzschild violates the energy-condition hypothesis.
The theorem requires an energy condition (nonnegative scalar curvature on the slice); without it negative mass is possible, so positive mass is a theorem about physically reasonable matter, and its rigidity case m = 0 forces exact flatness — both facts are routinely dropped in casual statements.