The Positive Mass Theorem

The positive mass theorem turns a local energy inequality into a global rigidity statement. For a complete, asymptotically flat three-dimensional initial data set satisfying the dominant energy condition, the ADM energy–momentum is future causal. Equality first forces the whole ADM vector to vanish. The stronger conclusion that the data embed as a spacelike hypersurface in Minkowski spacetime uses the rigidity theorem’s additional asymptotic decay; in the time-symmetric case, zero mass forces Euclidean geometry. Completeness, asymptotics, regularity, and the energy condition are all theorem hypotheses, not background decoration.

Required background. Analysis for Mathematical Relativity supplies elliptic estimates, Hypersurfaces and Foliations supplies initial-data geometry, Curvature supplies scalar and spacetime curvature, and Einstein Equations supplies the constraint identities.

Helpful background. Causality and Global Structure clarifies future-causal energy–momentum, while Partial Differential Equations and Green Functions supports the elliptic boundary problems used in both proof routes.

The three-dimensional theorem

Let $(\Sigma,h,K)$ be a smooth, complete, asymptotically flat initial data set with one chosen asymptotic end and no inner boundary. Use

$$K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu.$$

Suppose the constraint densities obey the dominant energy condition $\rho\ge |j|_h$. For the equality theorem used below, take asymptotic flatness in the type-$(p,q,q_0,\alpha)$ class of Huang and Lee 2020, Definition 2.5, printed pp. 4–5: in spatial dimension three, $p>3$, $1/2<q<1$, $\alpha\in(0,1)$, $q+\alpha>1$, and $q_0>0$, with the weighted Hölder, Sobolev, and constraint-density decay specified there. These assumptions are stronger than merely asking the ADM surface integrals to converge. Then

$$E_{\mathrm{ADM}}\ge |P|, \qquad |P|=\sqrt{\delta^{ij}P_iP_j}.$$

The null-equality theorem states that $E=|P|$ already forces $E=|P|=0$. To pass from this charge statement to Minkowskian initial data in three dimensions, impose the extra asymptotic condition $\operatorname{tr}_hK=O(r^{-s})$ for some $s>2$. Then the data arise from an isometric embedding of $\Sigma$ as a spacelike hypersurface in Minkowski spacetime with second fundamental form $K$; see Huang and Lee 2020, Theorem 1 and Corollary 4, printed pp. 1–3. When $K=0$, the constraints reduce to $R(h)=16\pi G\rho\ge0$, and $E_{\mathrm{ADM}}=0$ forces $(\Sigma,h)$ to be Euclidean space.

Compact apparent-horizon or minimal-surface boundaries require an additional boundary formulation or a doubling argument and are not silently included in this basic statement. Higher-dimensional variants have separate dimension, spin, regularity, and singular-set hypotheses. The time-symmetric theorem is proved in Schoen and Yau 1979, theorem and pp. 45–76. For general $K$, the nonspin spacetime inequality in dimensions $3\le n<8$ is Eichmair, Huang, Lee, and Schoen 2016, Theorem 1, pp. 83–85. Its null-equality case is the separate Huang–Lee result above. The distinction matters: their Theorem 1 proves that a null ADM vector must vanish, whereas the Minkowski-embedding conclusion is Corollary 4 and includes the stronger trace decay in spatial dimension three.

Why zero mass is rigid

Positivity alone would allow a nonflat zero-energy geometry. Rigidity excludes it by identifying where every nonnegative term in the proof can vanish. In a spinorial proof with the regularity and asymptotic decay needed for its equality analysis, choose a spinor approaching a constant spinor at infinity and solve the Sen–Witten equation. The integrated Weitzenböck identity has the schematic form

$$E-P_iv^i =c_G\int_\Sigma \left(|\nabla^{\mathrm{SW}}\psi|^2 +4\pi G\,\mathcal Q(\rho,j;\psi)\right)d\mu_h,$$

where $v$ is the asymptotic future-null direction selected by the constant spinor, $c_G>0$, and the dominant energy condition makes $\mathcal Q\ge0$. Varying $v$ yields $E\ge|P|$. If $E=0$, positivity also gives $P=0$, so the identity vanishes for every choice of constant asymptotic spinor. Solving for a basis of such values gives a full family of parallel spinors. Their integrability conditions, together with the constraint and global equality analysis, give the Minkowski embedding. A single parallel Lorentzian spinor would not suffice to conclude flatness: special nonflat geometries can preserve one spinor. The exact identity and equality analysis are given in Witten 1981, §§2–4, pp. 384–397.

The geometric route reaches the same sign conclusion differently. In the time-symmetric case, a hypothetical negative mass is converted through a stable minimal surface into a contradiction with its stability inequality and Gauss–Bonnet in three dimensions. General $K$ requires additional machinery: the classical Schoen–Yau route uses the Jang equation Schoen and Yau 1981, Introduction, pp. 232–233, and Eq. (2.26) (PDF), while the modern dimension-$<8$ proof uses marginally outer trapped hypersurfaces and a density theorem. These are not consequences of the 1979 Riemannian theorem.

Proof architecture and independent checks

Both proofs have four essential stages: establish a global elliptic solution; integrate an identity whose boundary term is ADM energy–momentum; use the dominant energy condition to make the bulk nonnegative; analyze simultaneous equality of every bulk term. The analytic stage depends on completeness and asymptotic control. The geometric stage depends on dimension or spin structure. The rigidity stage is not obtained merely by replacing a strict inequality with equality.

If the initial manifold has several asymptotically flat ends, the construction is performed relative to a chosen end while the other ends enter the global elliptic problem. The conclusion is positivity at each admissible end, not a single mass obtained by adding end charges. This is another reason the theorem cannot be reduced to a local curvature calculation: the boundary term is tied to one infinity, whereas the rigidity argument constrains the entire complete manifold and its other ends.

Time-symmetric Schwarzschild data provide the normalization check. In isotropic coordinates,

$$h=\left(1+\frac{GM}{2r}\right)^4\delta, \qquad K=0,$$

and the ADM energy is $M>0$. The limit $M\to0$ is Euclidean. By contrast, the formal negative-mass Schwarzschild metric has a singular incomplete end. It is therefore a counterexample to the claim that the sign conclusion survives after completeness and interior regularity are removed, not a counterexample to the theorem.

Scope and limitations

For general $K$, the dominant energy condition is not equivalent to $R(h)\ge0$, and it need not imply that inequality. Indeed, $R(h)=16\pi G\rho-K^2+|K|_h^2$ has an indefinite extrinsic-curvature contribution. Replacing the dominant energy condition by $R(h)\ge0$ loses both this distinction and the momentum information. Spinorial and minimal-surface proofs are also not interchangeable in arbitrary dimension. In three dimensions orientability removes much of the spin obstruction, but higher-dimensional statements must name it explicitly.

The theorem is asymptotically flat and concerns ADM energy. It does not directly establish positivity of Bondi, anti-de Sitter, or quasi-local masses. Nor does $E>0$ quantify horizon area or prove a Penrose inequality. Those stronger-looking conclusions require additional global geometry. This limitation prevents the word “mass” from erasing its boundary and asymptotic definition.

Check the missing hypothesis

Why does the negative-mass Schwarzschild expression not contradict the positive mass theorem, even though its asymptotic expansion gives a negative ADM coefficient?

Solution

The negative-mass metric fails the theorem’s completeness and regularity hypotheses: its interior contains a naked singularity at finite distance rather than a complete smooth end or an admissible horizon boundary. The ADM integral can still read the negative asymptotic coefficient, but the positivity proof cannot solve its global elliptic problem across the singular interior. This shows exactly why asymptotic decay plus $R=0$ near infinity is insufficient; the theorem is global.

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.
• Huang, Lan-Hsuan, and Dan A. Lee. “Equality in the Spacetime Positive Mass Theorem.” Communications in Mathematical Physics 376 (2020): 2379–2407. 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. Open PDF.
• Witten, Edward. “A New Proof of the Positive Energy Theorem.” Communications in Mathematical Physics 80 (1981): 381–402. DOI.