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Equality in the Spacetime Positive Mass Theorem
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We affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight. Namely, if an asymptotically flat initial data set satisfies the dominant energy condition and has E=|P|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E=|P|$$\end{document}, then E=|P|=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E=|P|=0$$\end{document}, where (E, P) is the ADM energy-momentum vector. The dimensional restriction can be removed if we assume the positive mass inequality holds. Previously the result was only known for spin manifolds (Beig and Chruściel in J Math Phys37(4):1939–1961, 1996; Chruściel and Maerten in J Math Phys 47(2), 2006).
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TL;DR
The paper proves the rigidity case of the spacetime positive mass theorem: for asymptotically flat initial data sets satisfying the dominant energy condition, \(E=|P|\) forces \(E=|P|=0\), in dimensions \(3\le n\le 7\), and more generally whenever the positive mass inequality holds near the data. The proof is variational, using a modified Regge-Teitelboim Hamiltonian and the modified constraint operator, and does not assume spin.
Key findings
- Theorem 1: for \(3\le n\le 7\), if an \(n\)-dimensional asymptotically flat initial data set satisfies the dominant energy condition and has \(E=|P|\), then \(E=|P|=0\).
- Theorem 3: for \(n\ge 3\), if the positive mass inequality holds near \((g,k)\), then \(E=|P|\) again implies \(E=|P|=0\); the dimensional restriction can be removed under this assumption.
- The result affirms the rigidity conjecture of the spacetime positive mass theorem, previously known for spin manifolds by Beig-Chruściel in three dimensions and Chruściel-Maerten in higher dimensions.
- Combined with prior \(E=0\) characterization results, the theorem implies the equality case embeds into Minkowski spacetime with induced second fundamental form \(k\) (with an additional \(n=3\) decay assumption \(\operatorname{tr}_g k=O(|x|^{-s})\), \(s>2\)).
Methods
- The proof uses a variational approach: a modified Regge-Teitelboim Hamiltonian built from the modified constraint operator \(\Phi_{(g,\pi)}\) of Huang-Corvino, replacing the usual constraint operator.
- Under \(E=|P|\), the Sobolev positive mass inequality is used to show \((g,k)\) locally minimizes this Hamiltonian on its \(\Phi_{(g,\pi)}\) level set; Lagrange multipliers then produce a lapse-shift pair \((f,X)\) in the kernel of the adjoint linearized modified constraint operator, asymptotic to \((E,-2P)\).
- A separate theorem, adapting Beig-Chruściel’s argument, shows that such an asymptotically translational vacuum Killing lapse-shift pair forces \(E=|P|=0\).
Limitations
- The paper states the dimensional restriction \(3\le n\le 7\) comes from possible singularities of marginally outer trapped hypersurfaces in the positive mass inequality; the restriction is removed only if the positive mass inequality is assumed near \((g,k)\).
- The paper notes the nonstandard technical assumption \(q+\alpha>n-2\) in Definition 2.5 is needed only for Theorem 5.4/Lemma A.10, not elsewhere.
- Remark 2.11 states that a weaker regularity assumption in a prior paper was insufficient and that stronger hypotheses are needed for the adjoint-equation unique continuation argument; the present proof relies on the positive mass inequality as an input rather than proving it in all dimensions.
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“Let (M, g, π) be asymptotically flat of type (p, q, q 0 , α) satisfying the dominant energy condition.Assume that the positive mass inequality holds near (g, π).If E = |P |, then there exists a lapse-shift pair (f, X) ∈ C 2,α loc (M ) solving (DΦ (g,π) ) * (f, X) = 0 in M (f, X) = (E, -2P ) + O 2,α (|x| -q ).”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > 5. Main argument
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“Let (M, g, π) be asymptotically flat of type (p, q, q 0 , α) with the dominant energy condition.Suppose the positive mass inequality holds near (g, π).Then there is an open ball U of (g, π) in W 2,p -q × W 1,p -1-q such that if (γ, τ ) ∈ U and Φ (g,π) (γ, τ ) = Φ (g,π) (g, π), we have E(γ, τ ) ≥ |P (γ, τ )|.”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > Theorem 4.1 (Sobolev version of positive mass inequality).
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“We claim that that (g, π) is a local minimizer of H in C (g,π) .”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > 5. Main argument
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In the paper
“Then H is differentiable at (g, π) with derivative given by DH (g,π) (h, w) = - M (h, w) • (DΦ (g,π) ) * (f 0 , X 0 ) dµ g for all (h, w) ∈ W 2,p -q × W 1,p -1-q .”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > 5. Main argument
What the sources say
- Not checked yetcites [4] Bartnik, R.: Phase space for the Einstein equations. Commun. Anal. Geom. 13(5), 845–885 (2005). MR 2216143 source
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“if an asymptotically flat initial data set satisfies the dominant energy condition and has E = |P |, then E = |P | = 0”
- Also stated in: § 1. Introduction · plain
“Let 3 ≤ n ≤ 7. Let (M, g, k) be an n-dimensional asymptotically flat initial data set that satisfies the dominant energy condition and has E = |P |, where (E, P ) is the ADM energymomentum vector.Then E = |P | = 0.”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > Abstract
What the sources say
- Not checked yetcites [11] Eichmair, M., Huang, L.-H., Lee, D., Schoen, R.: The spacetime positive mass theorem in dimensions less than eight. J. Eur. Math. Soc. (JEMS) 18(1), 83–121 (2016). MR 3438380 source
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In the paper
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > Appendix A. Asymptotically Killing lapse-shift pair
“2(n -1)E(nρ 4 |x| -n-2 -ρ 2 |x| -n ) -2E(nρ 4 |x| -n-2 -(n -1)ρ 2 |x| -n )”
§ EQUALITY IN THE SPACETIME POSITIVE MASS THEOREM > Appendix A. Asymptotically Killing lapse-shift pair
“= -2Eρ 4 |x| -n-2 + O 0,α (|x| 2-n-q 1 ).”
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Self-citation
Metadata
- Authors
- Huang, L.-H., & Lee, D. A.
- Affiliations
- University of Connecticut (US)
- Queens College, CUNY (US)
- Corresponding author
- Huang, Lan-Hsuan
- ORCID
- Huang, Lan-Hsuan
- Venue
- Communications in Mathematical Physics
- Published
- 2019-11-18
- Volume
- 376
- Issue
- 3
- Pages
- 2379-2407
- ISSN
- 0010-3616 (ISSN-L)
- 1432-0916
- OpenAlex
- W2986722835
- Semantic Scholar
- da6313edfcb122642e605d584cdcd0ec94e239cb
- Cited by
- 32
- References
- 36
- Citations by year
- 2026: 1 · 2025: 5 · 2024: 5 · 2023: 3 · 2022: 5 · 2021: 8 · 2020: 3 · 2019: 2
- FWCI
- 3.74
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- www.springer.com/tdm (vor)
- Locations
- Communications in Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (DMS~1452477)
- Simons Foundation (Simons Fellowship)
- Institute for Advanced Study (von Neumann Fellowship)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- asymptotically flat initial data, dominant energy condition, spin manifolds
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