Positive Mass Theorems

Positive-mass theorems are global statements about complete asymptotically flat initial data. The applicable conclusion depends on the data class, spatial dimension, regularity, spin structure, boundary, energy condition, and the charge defined at infinity. The rows below therefore cannot be combined into one dimension-free claim that “mass is positive.”

Detailed treatment. Positive Mass and Geometric Inequalities develops the ADM definitions, geometric and spinorial proof routes, and rigidity arguments used here.

Positive mass by data class

In every row, asymptotic flatness is strong enough to define ADM quantities at each asymptotically Euclidean end. Completeness controls the interior; falloff at infinity alone is insufficient.

Variant	Data and hypotheses	Exact conclusion and boundary
Riemannian Schoen–Yau	An oriented asymptotically flat three-manifold with nonnegative scalar curvature, using the paper’s $C^6$ manifold and $C^5$ metric formulation. The metric is complete; the original statement also permits a compact boundary under its positive-mean-curvature condition. No spin structure is assumed.	Every end has nonnegative ADM mass. With the additional derivative falloff in the rigidity statement, zero mass forces Euclidean three-space. This is the time-symmetric theorem, not a spacetime energy–momentum inequality Schoen and Yau 1979, § 1, Theorems 1–2, pp. 47–48.
Spacetime Schoen–Yau	Three-dimensional asymptotically flat general initial data $(\Sigma,h,K)$ without boundary, satisfying the paper’s $C^3/C^4$ regularity and decay assumptions and the dominant energy condition $\mu\geq\lvert J\rvert_h$. No spin structure is assumed.	The ADM energy of each end is nonnegative. Under the stronger asymptotic hypotheses of the equality theorem, zero energy gives an isometric spacelike embedding in Minkowski spacetime with second fundamental form $K$. This source states endwise $E\geq0$; it should not be quoted as the manifestly Lorentz-covariant inequality Schoen and Yau 1981, § 1, Theorems 1–2, p. 234.
Spinorial Witten	Smooth, complete, asymptotically flat three-dimensional initial data with a global spin structure, an asymptotically constant spinor, solvability of the Witten equation, the dominant energy condition, and no uncontrolled inner-boundary term.	The ADM four-momentum is future causal: $E\geq\lvert P\rvert$. The safe rigidity clause is that $E=0$ gives Minkowski data. The spin and elliptic hypotheses are part of the theorem, not optional proof language Witten 1981, § III, Eqs. (57) and (72), pp. 395–399.
Nonspin spacetime theorem below dimension eight	Complete asymptotically flat initial data without boundary in spatial dimension $3\leq n\lt8$, with $h\in C^2_{\mathrm{loc}}$, $K\in C^1_{\mathrm{loc}}$, the stated weighted Sobolev regularity and decay, and $\mu\geq\lvert J\rvert_h$. No spin assumption is made.	$E\geq\lvert P\rvert$. The cited theorem supplies the inequality only: it does not supply equality rigidity, cover $n\geq8$, or admit an arbitrary inner boundary Eichmair et al. 2016, Definition 3 and Theorem 1, pp. 84 and 88–89.

Reading the hypotheses correctly

The Riemannian row assumes $R(h)\geq0$. For general $K$, the relevant input is the dominant energy condition on the constraint densities; nonnegative scalar curvature alone loses the momentum information. Similarly, “spin proof” and “nonspin theorem” identify different admissible classes. In the oriented three-dimensional setting a spin structure is available, but a higher-dimensional spinorial extension would have to state spin and its own dimension and regularity assumptions explicitly.

Mass is also end-specific. A manifold with several admissible asymptotically flat ends receives a conclusion at each end; the theorem does not define a sum over ends. Removing completeness or replacing an admissible boundary by a singular interior invalidates the global elliptic or minimal-surface step even when an ADM integral still exists.

Charged, horizon-boundary, and AdS exclusions

These four rows concern the ordinary asymptotically flat ADM theorem. A charge-refined inequality requires electromagnetic boundary terms and charge-specific hypotheses; an apparent-horizon boundary needs an admissible boundary argument; and asymptotically anti-de Sitter positivity uses a different asymptotic charge and reference geometry. The charged black-hole and AdS arguments are separate results Gibbons et al. 1983, main argument and Appendix, pp. 295–308. Neither ordinary Maxwell matter satisfying the dominant energy condition nor a change of notation silently supplies those stronger conclusions. Bondi and quasi-local masses are likewise outside this card.

References

• Eichmair, Michael, Lan-Hsuan Huang, Dan A. Lee, and Richard Schoen. “The Spacetime Positive Mass Theorem in Dimensions Less Than Eight.” Journal of the European Mathematical Society 18 (2016): 83–121. DOI.
• Gibbons, Gary W., Stephen W. Hawking, Gary T. Horowitz, and Malcolm J. Perry. “Positive Mass Theorems for Black Holes.” Communications in Mathematical Physics 88 (1983): 295–308. DOI.
• Schoen, Richard, and Shing-Tung Yau. “On the Proof of the Positive Mass Conjecture in General Relativity.” Communications in Mathematical Physics 65 (1979): 45–76. DOI.
• Schoen, Richard, and Shing-Tung Yau. “Proof of the Positive Mass Theorem. II.” Communications in Mathematical Physics 79 (1981): 231–260. DOI.
• Witten, Edward. “A New Proof of the Positive Energy Theorem.” Communications in Mathematical Physics 80 (1981): 381–402. DOI.