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arxiv: 2604.07168 · v1 · submitted 2026-04-08 · 🧮 math.AP · gr-qc· math.DG

Geometrically defined asymptotic coordinates in General Relativity

Carla Cederbaum , Jan Metzger This is my paper

Pith reviewed 2026-05-10 17:26 UTC · model grok-4.3

classification 🧮 math.AP gr-qcmath.DG
keywords asymptotic flatnessgeneral relativityinitial data setsCMC foliationSTCMC foliationmassangular momentumcenter of mass
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The pith

Asymptotic flatness and its invariants in general relativity can be defined geometrically using CMC and STCMC foliations instead of coordinates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews and announces results that recast asymptotic flatness for relativistic initial data sets as a property of certain geometric foliations. These foliations, including constant mean curvature surfaces and constant scalar mean curvature surfaces, encode the usual asymptotic invariants such as mass, energy, linear momentum, angular momentum, and center of mass. A reader would care because this replaces coordinate-dependent definitions with purely geometric ones, making the invariants intrinsic to the spacetime geometry at large distances. The approach ties physical quantities directly to the shape and behavior of these foliations as one moves outward. It thereby offers a coordinate-free framework for studying the ends of initial data sets.

Core claim

The authors establish that asymptotic flatness and the associated geometric invariants in asymptotically Euclidean relativistic initial data sets admit a geometrization through their relations to asymptotic foliations, in particular the CMC-foliations and STCMC-foliations, which capture mass, energy, linear momentum, angular momentum, and center of mass without reference to asymptotic coordinates.

What carries the argument

The CMC- and STCMC-foliations, which serve as geometric asymptotic foliations that encode the invariants through their mean curvature and scalar mean curvature properties.

If this is right

  • Mass and energy become expressible as limits of geometric quantities on the foliations.
  • Angular momentum and linear momentum acquire direct relations to the twisting or shear of the foliation leaves.
  • The center of mass is located by the asymptotic centering of the foliation surfaces.
  • Relations among the invariants follow from the geometry of the foliations rather than coordinate expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same foliation approach may extend to numerical constructions of initial data where coordinate choices are inconvenient.
  • Connections could appear between these foliations and those used in black-hole uniqueness theorems or stability problems.
  • The framework might allow re-examination of the positive mass theorem in a purely geometric setting.

Load-bearing premise

That asymptotically Euclidean relativistic initial data sets admit CMC- and STCMC-foliations with the required asymptotic properties.

What would settle it

An explicit asymptotically Euclidean initial data set on which no CMC foliation exists whose asymptotic expansion recovers the standard mass and momentum values.

read the original abstract

We review and announce recent results on the asymptotic behavior of asymptotically Euclidean relativistic initial data sets and asymptotic foliations thereof. In particular, we discuss the geometrization of asymptotic flatness and of asymptotic geometric (in-)\-variants such as mass, energy, linear momentum, angular momentum and center of mass as well as their relations to certain geometric asymptotic foliations such as the CMC- and STCMC-foliations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript reviews and announces recent results on the asymptotic behavior of asymptotically Euclidean relativistic initial data sets in general relativity. It focuses on the geometrization of asymptotic flatness and of asymptotic geometric invariants such as mass, energy, linear momentum, angular momentum, and center of mass, together with their relations to geometric asymptotic foliations including the CMC- and STCMC-foliations.

Significance. If the announced results hold, the work offers a coordinate-independent geometric framework for asymptotic flatness and conserved quantities in GR. Linking these invariants to specific foliations could provide new tools for analyzing isolated systems and gravitational radiation, with potential impact on both the mathematical theory and applications in mathematical physics.

major comments (1)
  1. [Introduction] The central claims on geometrization rest on the existence of CMC- and STCMC-foliations with controlled asymptotics for the given initial data sets. While the manuscript refers these existence and uniqueness statements to prior literature, the precise hypotheses (decay rates, regularity, etc.) under which the foliations are guaranteed should be stated explicitly, for example in a dedicated subsection of the introduction, to make the presentation self-contained and to clarify the scope of the geometrization.
minor comments (1)
  1. [Abstract] The abstract contains the typographical error 'in-)-variants'; this should be corrected to 'invariants'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment. The suggestion to clarify the hypotheses for the foliations will help make the presentation more self-contained.

read point-by-point responses

  1. Referee: [Introduction] The central claims on geometrization rest on the existence of CMC- and STCMC-foliations with controlled asymptotics for the given initial data sets. While the manuscript refers these existence and uniqueness statements to prior literature, the precise hypotheses (decay rates, regularity, etc.) under which the foliations are guaranteed should be stated explicitly, for example in a dedicated subsection of the introduction, to make the presentation self-contained and to clarify the scope of the geometrization.

    Authors: We agree that explicitly summarizing the relevant hypotheses would improve accessibility. In the revised version we will add a short subsection in the introduction that collects the precise decay rates, regularity assumptions, and other conditions (drawn from the cited literature) under which the CMC- and STCMC-foliations exist with the required asymptotic control. This addition will delineate the scope of the geometrization results without reproducing the proofs from the referenced works. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript is a review announcing results on the geometrization of asymptotic flatness and invariants (mass, energy, momentum, angular momentum, center of mass) via relations to CMC- and STCMC-foliations. All load-bearing existence and uniqueness statements for the foliations are explicitly referred to prior literature rather than derived or fitted within the present text. No equations, definitions, or self-citations reduce any claimed prediction or uniqueness result to the paper's own inputs by construction. The derivation chain remains self-contained and conditional on external results, with no self-definitional, fitted-input, or ansatz-smuggling patterns exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions from general relativity and differential geometry for asymptotically flat manifolds; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Initial data sets are asymptotically Euclidean.
    Stated directly in the abstract as the setting for the results.
  • domain assumption Existence of CMC and STCMC foliations with controlled asymptotics.
    Invoked as the geometric tools for defining invariants.

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    On the (non)-existence of asymptotically S chwarzschildian coordinates for boosted initial data sets

    Rodrigo Avalos and Carla Cederbaum. On the (non)-existence of asymptotically S chwarzschildian coordinates for boosted initial data sets. in preparation

    work page

  2. [2]

    Regularity of conformal structures on closed 3-manifolds, 2025

    Rodrigo Avalos, Albachiara Cogo, and Andoni Royo Abrego . Regularity of conformal structures on closed 3-manifolds, 2025. arXiv:2511.00178 https://arxiv.org/abs/2511.00178

    work page arXiv 2025

  3. [3]

    Richard Arnowitt, Stanley Deser, and Charles W. Misner. Dynamical structure and definition of energy in general relativity. Phys. Rev. (2) , 116:1322--1330, 1959

    work page 1959

  4. [4]

    Sobolev regularity of compactified 3-manifolds and the A D M C enter of M ass, 2024

    Rodrigo Avalos. Sobolev regularity of compactified 3-manifolds and the A D M C enter of M ass, 2024. arXiv:2403.04034 https://arxiv.org/abs/2403.04034

    work page arXiv 2024

  5. [5]

    The mass of an asymptotically flat manifold

    Robert Bartnik. The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. , 39(5):661--693, 1986

    work page 1986

  6. [6]

    On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth

    Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Inventiones Mathematicae , 97(2):313--349, 1989

    work page 1989

  7. [7]

    The P oincar\' e group as the symmetry group of canonical general relativity

    Robert Beig and Niall \' O Murchadha. The P oincar\' e group as the symmetry group of canonical general relativity. Ann. Physics , 174(2):463--498, 1987

    work page 1987

  8. [8]

    On the relevance of super-translations for the definition of center of mass and angular momentum for isolated relativistic systems

    Carla Cederbaum. On the relevance of super-translations for the definition of center of mass and angular momentum for isolated relativistic systems. Work in Progress, talk given at BIRS Granda, May 2025

    work page 2025

  9. [9]

    Coordinates are messy---not only in general relativity

    Carla Cederbaum and Melanie Graf. Coordinates are messy---not only in general relativity. In Betti Hartmann and Jutta Kunz, editors, Gravity, Cosmology, and Astrophysics: A Journey of Exploration and Discovery with Female Pioneers , pages 273--288. Springer Nature Switzerland, Cham, 2023

    work page 2023

  10. [10]

    Initial data sets that do not satisfy the R egge-- T eitelboim conditions

    Carla Cederbaum, Melanie Graf, and Jan Metzger. Initial data sets that do not satisfy the R egge-- T eitelboim conditions. Work in Progress, talk given at Simons' Center, September 2023 https://scgp.stonybrook.edu/video_portal/video.php?id=5340

    work page 2023

  11. [11]

    Chru\'sciel

    Piotr T. Chru\'sciel. On angular momentum at spatial infinity. Classical Quantum Gravity , 4:L205--L210, 1987

    work page 1987

  12. [12]

    Chru\'sciel

    Piotr T. Chru\'sciel. On the I nvariant M ass C onjecture in G eneral R elativity. Commun. Math. Phys. , 120:233--248, 1988

    work page 1988

  13. [13]

    On the validity of the definition of angular momentum in general relativity

    Po-Ning Chen, Lan-Hsuan Huang, Mu-Tao Wang, and Shing-Tung Yau. On the validity of the definition of angular momentum in general relativity. Annales Henri Poincar \'e , 17(2):253--270, Feb 2016

    work page 2016

  14. [14]

    Asymptotically euclidean vacuum initial data sets of prescribed decay

    Carla Cederbaum and David Maxwell. Asymptotically euclidean vacuum initial data sets of prescribed decay. Work in progress

    work page

  15. [15]

    Explicit R iemannian manifolds with unexpectedly behaving center of mass

    Carla Cederbaum and Christopher Nerz. Explicit R iemannian manifolds with unexpectedly behaving center of mass. Ann. Henri Poincar\' e , 16(7):1609--1631, 2015

    work page 2015

  16. [16]

    On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in G eneral R elativity

    Carla Cederbaum and Anna Sakovich. On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in G eneral R elativity. Calc. Var. , 60(214), 2021

    work page 2021

  17. [17]

    On the center of mass of isolated systems

    Justin Corvino and Haotian Wu. On the center of mass of isolated systems. Classical and Quantum Gravity , 25, 2008

    work page 2008

  18. [18]

    Deturck and Jerry L

    Dennis M. Deturck and Jerry L. Kazdan. Some regularity theorems in riemannian geometry. Annales scientifiques de l'\'E.N.S. 4 s\'erie , 14:249--260, 1981

    work page 1981

  19. [19]

    Optimal rigidity estimates for nearly umbilical surfaces

    Camillo De Lellis and Stefan M \"u ller. Optimal rigidity estimates for nearly umbilical surfaces. J. Differential Geom. , 69(1):75--110, 2005

    work page 2005

  20. [20]

    Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres

    Michael Eichmair and Thomas K\"orber. Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres. J. Differential Geom. , 128(3):1037--1083, 2024

    work page 2024

  21. [21]

    Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

    Michael Eichmair and Jan Metzger. Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions. Invent. Math. , 194(3):591--630, 2013

    work page 2013

  22. [22]

    Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics

    Lan-Hsuan Huang. Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics. Comm. Math. Phys. , 300(2):331--373, 2010

    work page 2010

  23. [23]

    Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature

    Gerhard Huisken and Shing-Tung Yau. Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. , 124(1-3):281--311, 1996

    work page 1996

  24. [24]

    Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean , 2024

    Klaus Kr\"oncke and Markus Wolff. Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean , 2024. doi:10.48550/arXiv.2412.17563

    work page doi:10.48550/arxiv.2412.17563 2024

  25. [25]

    Uniqueness of the foliation of constant mean curvature spheres in asymptotically flat 3-manifolds

    Shiguang Ma. Uniqueness of the foliation of constant mean curvature spheres in asymptotically flat 3-manifolds. Pacific J. Math. , 252:145--179, 2011. MR 2862146

    work page 2011

  26. [26]

    Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature

    Jan Metzger. Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. J. Differential Geom. , 77(2):201--236, 2007

    work page 2007

  27. [27]

    Geometric invariance of mass-like asymptotic invariants

    Benoit Michel. Geometric invariance of mass-like asymptotic invariants. J. Math. Phys. , 52:052504, 2011. https://doi.org/10.1063/1.3579137

    work page doi:10.1063/1.3579137 2011

  28. [28]

    Local space time constant mean curvature and constant expansion foliations

    Jan Metzger and Alejandro Pe \ n uela Diaz. Local space time constant mean curvature and constant expansion foliations. J. Geom. Phys. , 188:Paper No. 104823, 23, 2023

    work page 2023

  29. [29]

    Time evolution of A D M and C M C center of mass in general relativity, 2013

    Christopher Nerz. Time evolution of A D M and C M C center of mass in general relativity, 2013. arXiv:1312.6274

    work page arXiv 2013

  30. [30]

    Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry

    Christopher Nerz. Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry. Calc. Var. Partial Differential Equations , 54(2):1911--1946, 2015

    work page 1911

  31. [31]

    Geometric characterizations of asymptotic flatness and linear momentum in general relativity

    Christopher Nerz. Geometric characterizations of asymptotic flatness and linear momentum in general relativity. J. Funct. Anal. , 269(12):3812--3867, 2015

    work page 2015

  32. [32]

    Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry

    Christopher Nerz. Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry . J. Differential Geom. , 109(2):257--289, 2018

    work page 2018

  33. [33]

    A geometric choice of asymptotically euclidean coordinates via stcmc-foliations

    Annachiara Piubello and Olivia Vi c \' a nek Mart\' i nez . A geometric choice of asymptotically euclidean coordinates via stcmc-foliations. In preparation

    work page

  34. [34]

    On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

    Jie Qing and Gang Tian. On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds. J. Amer. Math. Soc. , 20:1091--1110, 2007. MR 2328717

    work page 2007

  35. [35]

    Role of surface integrals in the H amiltonian formulation of general relativity

    Tullio Regge and Claudio Teitelboim. Role of surface integrals in the H amiltonian formulation of general relativity. Ann. Physics , 88:286--318, 1974

    work page 1974

  36. [36]

    Infinitesimal B oosts of the S T C M C - C enter of M ass

    Saradha Senthil Velu. Infinitesimal B oosts of the S T C M C - C enter of M ass. work in progress

    work page

  37. [37]

    Time-evolution of the C M C - and S T C M C - C enter of M ass

    Saradha Senthil Velu. Time-evolution of the C M C - and S T C M C - C enter of M ass. work in progress

    work page

  38. [38]

    The P oincar\'e S tructure and the C entre-of- M ass of A symptotically F lat S pacetimes

    L\'aszl\'o Szabados. The P oincar\'e S tructure and the C entre-of- M ass of A symptotically F lat S pacetimes. In Analytical and Numerical Approaches to Mathematical Relativity , volume 692 of Lect. Notes Phys. , pages 157--184. Springer, 2006

    work page 2006

  39. [39]

    Volume preserving spacetime mean curvature flow and foliations of initial data sets

    Jacopo Tenan. Volume preserving spacetime mean curvature flow and foliations of initial data sets. Journal of Functional Analysis , 290(6):111313, 2026

    work page 2026

  40. [40]

    Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

    Jacopo Tenan and Carlo Sinestrari. Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces. Ann. Henri Poincar\'e , 2026

    work page 2026

  41. [41]

    Geometrizing physical properties of spacetime via spectral theory

    Olivia Vi c \' a nek Mart\' i nez. Geometrizing physical properties of spacetime via spectral theory. dissertation in preparation

    work page

  42. [42]

    Foliation by constant mean curvature spheres

    Rugang Ye. Foliation by constant mean curvature spheres. Pacific J. Math. , 147(2):381--396, 1991

    work page 1991

  43. [43]

    Foliation by constant mean curvature spheres on asymptotically flat manifolds

    Rugang Ye. Foliation by constant mean curvature spheres on asymptotically flat manifolds. In J\"urgen Jost, editor, Geometric analysis and the calculus of variations , pages 369--383. Int. Press, Cambridge, MA, 1996. MR 1449417

    work page 1996