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Initial data sets with vanishing mass are contained in pp-wave spacetimes
Open accessAbstract
In 1981, Schoen–Yau and Witten showed that in general relativity both the total energy E and the total mass m of an initial data set modeling an isolated gravitational system are non-negative. Moreover, if E=0 , the initial data set must be contained in Minkowski space. In this paper, we show that if m=0 , i.e. if E equals the total momentum |P| , the initial data set must be contained in a pp-wave spacetime. Our proof combines spinorial methods with spacetime harmonic functions and works in all dimensions. Additionally, we find the decay rate threshold where the embedding has to be within Minkowski space and construct non-vacuum initial data sets with m=\nobreak0 in the borderline case. As a consequence, this completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.
Overview
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TL;DR
The paper proves that a spin asymptotically flat initial data set with vanishing ADM mass, i.e. \(E=|P|\neq 0\), must isometrically embed into a pp-wave spacetime, and identifies the decay-rate threshold where the pp-wave must reduce to Minkowski space. It also constructs non-vacuum examples with \(E=|P|\neq 0\) in the borderline decay case, settling the rigidity of the spacetime positive mass theorem for spin manifolds.
Key findings
- If \((M^n,g,k)\), \(n\ge 3\), is a \(C^{2,\alpha}\)-asymptotically flat spin initial data set with decay rate \(q\in(\frac{n-2}{2},n-2]\), satisfying the dominant energy condition and \(E=|P|\neq 0\), then \((M^n,g)\) isometrically embeds into a pp-wave spacetime with second fundamental form \(k\) (Theorem 1.1).
- If \(E=|P|\) and \(q>n-1-s-\alpha\), or if \(E=0\) and \(q>\frac{n-2}{2}\), then the initial data set embeds into Minkowski spacetime; the paper states these decay conditions are optimal (Theorem 1.3).
- For \(n\ge 5\), the authors construct \(C^{2,\alpha}\)-asymptotically flat initial data sets with decay rate \(q=n-3-\alpha\), satisfying the dominant energy condition and \(E=|P|\neq 0\), which embed into a non-trivial pp-wave spacetime; no such examples exist for \(n=3,4\) (Theorem 1.4).
- The result “completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.”
Methods
- The proof combines spinorial methods with spacetime harmonic functions: it constructs a spacetime harmonic function \(u\) whose level sets become the “planes” of the pp-wave, and solves the spacetime Dirac equation \(\not D\psi=\frac12 \operatorname{tr} k\, e_0\psi\) to obtain a spinor asymptotic to a constant spinor.
- Witten’s mass formula is used to derive \(X=\nabla u\) and \(\nabla^\Sigma Z=0\), showing the level sets are flat; a Killing development then constructs the pp-wave spacetime.
- The decay-threshold and borderline-example proofs use elliptic estimates, Gauss–Codazzi identities, asymptotic analysis of the PDE for the wave profile \(F\), and Reeb stability for the topology.
Limitations
- The paper assumes spin, asymptotic flatness, the dominant energy condition, and specific decay-rate ranges; Theorem 1.1 also assumes \(E=|P|\neq 0\).
- The non-vacuum borderline examples require \(n\ge 5\); the paper states there are no such examples for \(n=3\) and \(n=4\).
- The paper notes that the proofs of Theorems 1.3 and 1.4 are “very technical” and depend on precise asymptotics and the underlying PDE for \(F\). It also remarks that interior boundaries are not allowed in the main setup, though the results extend to MOTS/MITS boundaries with adjustments.
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In the paper
“Therefore, C ∼ = B × Υ is topologically trivial.”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > □
plain- Asserted: passes — The authors put it forward as true.
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- Rests on 3 assumptions not shown here.
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In the paper
“(5) The second fundamental form h of the level sets Σ satisfies h = -k| T Σ⊗T Σ .”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 4. Construction of a spacetime harmonic function
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In the paper
“We have R annb = A nab”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > □
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“(6.12)k ab = |∇u|∇ Σ ab ℓ = O s-1,β (ρ -1-q ) for any β ∈ (0, α).”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 6.2. Applications of the Codazzi equations.
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“From the estimates above, g is only a C 2,α 4+α-n -asymptotically flat metric because of the cross term 2ℓ ρ dudρ in the metric.”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 8. Non-trivial pp-wave spacetimes and the proof of Theorem 1.4
plainstates a cause- Asserted: passes — The authors put it forward as true.
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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated major: it would touch a result — a value a result rests on, reported and recomputed p-values on opposite sides of the significance level, two passages that conflict with no sign which is right, or a cited work retracted before this paper was published.
In the paper
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 3.2.
“Moreover, for any initial data set (M n , g, k), we have µ = - 1 2 F -1 ∆ R n-1 F.”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 6.3. Defining the spacetime metric.
“µ = - 1 2 |∇u| 2 ∆ Σ F.”
Rated major: it would touch a result — a value a result rests on, reported and recomputed p-values on opposite sides of the significance level, two passages that conflict with no sign which is right, or a cited work retracted before this paper was published.
In the paper
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 7. Analyzing the PDE and proof of Theorem 1.3
“(7.7) R(u, ρ) = s-1 i=1 (u -u i ) (s -2)! ∂ s-2 u F (Φ(u, ρ), ρ) with Φ(u, ρ) ∈ [u 0 , u s ].”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 7. Analyzing the PDE and proof of Theorem 1.3
“(7.6) F (u, ρ) -1 = s-1 i=1 L i (u)[F (u i , ρ) -1] + R(u, ρ)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 4.4. Adjustments for non-empty interior boundary.
“Then Witten's mass formula with boundary, cf.[23], [32,Theorem 8.29] and [8, Theorem 11.4], yields again that ψ satisfies ∇ i ψ = 1 2 k ij e j e 0 ψ.Now all parts of the proof of Theorem 4.1 go through verbatim”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > Let ψ ∞
“∇ i ψ = - 1 2 k ij e j e 0 ψ,”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 3.1.
“κ : R n-1 → R is a fixed positive superharmonic function such that κ = O(|y| -(n-3) ) near ∞”
§ INITIAL DATA SETS WITH VANISHING MASS ARE CONTAINED IN PP-WAVE SPACETIMES > 3.1.
“Here (κ -1) = O(|y| -(n-3) ) denotes that (κ -1) decays like |y| -(n-3) as |y| → ∞.”
Citations
Cites
Multiple Integrals in the Calculus of Variations
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A new proof of the positive energy theorem
Witten, E. (1981) · Communications in Mathematical Physics · doi:10.1007/bf01208277 · Cited by 1,595
On the proof of the positive mass conjecture in general relativity
Schoen, R., & Yau, S.-T. (1979) · Communications in Mathematical Physics · doi:10.1007/bf01940959 · Cited by 870
Proof of the positive mass theorem. II
Schoen, R., & Yau, S.-T. (1981) · Communications in Mathematical Physics · doi:10.1007/bf01942062 · Cited by 533
Einstein spaces which are mapped conformally on each other
Brinkmann, H. W. (1925) · Mathematische Annalen · doi:10.1007/bf01208647 · Cited by 507
Observation of Gravitational Waves from a Binary Black Hole Merger
Abbott, B. P. (2017) · Centennial of General Relativity · doi:10.1142/9789814699662_0011 · Cited by 323
Any Space-Time has a Plane Wave as a Limit
Penrose, R. (1976) · Differential Geometry and Relativity · doi:10.1007/978-94-010-1508-0_23 · Cited by 254
Positive mass theorems for black holes
Gibbons, G. W., Hawking, S. W., Horowitz, G. T., & Perry, M. J. (1983) · Communications in Mathematical Physics · doi:10.1007/bf01213209 · Cited by 229
Positive scalar curvature and minimal hypersurface singularities
Schoen, R., & Yau, S. (2019) · Surveys in Differential Geometry · doi:10.4310/sdg.2019.v24.n1.a10 · Cited by 197
Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensorPreprint
Bryant, R. L. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0004073 · Cited by 102
Research works printed but not linked: 2
- C. Camacho and A. Neto, Geometric theory of foliations, Springer Science & Business Media, 2013.
- H.B. Lawson, and M.-L. Michelsohn, Spin Geometry, (PMS-38), Volume 38, Princeton University Press, 2016.
Cited by
Hawking Mass Monotonicity for Initial Data SetsPreprint
Hirsch, S. (2022) · arXiv (Cornell University) · doi:10.48550/arxiv.2210.12237 · Cited by 4
Self-citationMonotonicity of Causal Killing Vectors and Geometry of ADM Mass MinimizersPreprint
Hirsch, S., & Huang, L.-H. (2025) · arXiv (Cornell University) · doi:10.48550/arxiv.2510.10306 · Cited by 4
Self-citationInitial data set rigidity results for polyhedra
Chai, X., & Wan, X. (2026) · Transactions of the American Mathematical Society · doi:10.1090/tran/9810 · Cited by 2
Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass
Hirsch, S., Jang, H. C., & Zhang, Y. (2025) · Communications in Mathematical Physics · doi:10.1007/s00220-025-05485-1 · Cited by 2
Self-citationThe Positive Mass Theorem for Creased Initial DataPreprint
Kazaras, D., Khuri, M., & Lin, M. (2025) · arXiv (Cornell University) · doi:10.48550/arxiv.2508.17585 · Cited by 2
Metadata
- Authors
- Hirsch, S., & Zhang, Y.
- Affiliations
- Columbia University (US)
- BIMSA, Bejing, P. R. China
- Venue
- Journal of the European Mathematical Society
- Published
- 2025-11-07
- ISSN
- 1435-9855 (ISSN-L)
- 1435-9863
- OpenAlex
- W4415988127
- Cited by
- 9
- References
- 3
- Citations by year
- 2026: 3 · 2025: 5
- FWCI
- 1.61
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Licence
- cc-by
- Locations
- Journal of the European Mathematical Society (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1926686)
- National Natural Science Foundation of China
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- positive mass theorem, pp-wave spacetimes, initial data sets, total momentum, spin manifolds, rigidity
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