pith. sign in

arxiv: 2606.07781 · v1 · pith:FFES24XJnew · submitted 2026-06-05 · 🌀 gr-qc · math-ph· math.DG· math.MP

Spacetime Bartnik Mass Positivity and Temporal Monotonicity for Black Holes

Lars Andersson , Marcus Khuri , Marc Mars , Walter Simon This is my paper

Pith reviewed 2026-06-27 20:54 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.DGmath.MP
keywords Bartnik massquasilocal massblack holesapparent horizonpositivitymonotonicityspacetimegeneral relativity
0
0 comments X

The pith

A Bartnik-type quasilocal mass is strictly positive for black hole hypersurfaces and monotonically nondecreasing in time.

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

The paper defines a quasilocal mass of Bartnik type adapted to spacetime domains that contain black holes. It proves this mass is strictly positive for compact spacelike hypersurfaces that have an apparent horizon as boundary and for noncompact asymptotically flat hypersurfaces that contain an apparent horizon inside any admissible extension. The mass is also shown to be monotonically nondecreasing along time evolution in scenarios that match these two classes of domains. A reader would see this as establishing a positive, time-consistent local mass assignment near black holes that does not require global asymptotic flatness at every end.

Core claim

We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.

What carries the argument

The Bartnik-type quasilocal mass defined on domains bounded by or containing apparent horizons, constructed so that positivity follows from the horizon condition and monotonicity follows from the evolutionary rules.

If this is right

  • The mass supplies a positive lower bound for regions containing black holes on both compact and noncompact slices.
  • The mass cannot decrease as time advances in the allowed evolutionary scenarios.
  • Positivity holds independently of whether the hypersurface is closed or extends to asymptotic flatness.
  • The same definition covers both the compact-boundary case and the noncompact case with an interior apparent horizon.

Where Pith is reading between the lines

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

  • The construction could be applied in numerical simulations of black hole dynamics to track a local mass without choosing a global coordinate frame.
  • In the limit of stationary black holes the mass might reduce to a quantity determined by the horizon area.
  • The monotonicity property could be checked against other quasilocal mass definitions on the same evolutionary examples.

Load-bearing premise

The extensions of the hypersurface must be admissible and the evolutionary scenarios must satisfy the precise conditions of the two settings.

What would settle it

An explicit construction of a compact spacelike hypersurface with apparent horizon boundary where the defined mass evaluates to zero or negative under an admissible extension.

Figures

Figures reproduced from arXiv: 2606.07781 by Lars Andersson, Marc Mars, Marcus Khuri, Walter Simon.

Figure 1
Figure 1. Figure 1: Example of an admissible extension as in Definition [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the setting of the Monotonicity Theorems [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic view of the extension Mt . It is clear that each Mt satisfies conditions (1)-(4) in definition 1. It re￾mains to verify condition (5*) which will proceed by contradiction. Assume that there is no positive ε1 such that (5*) holds for all t ∈ [−ε1, 0]. There is then a sequence of times ti → 0, such that (Mti , gti ) contains a closed mini￾mal surface that has nontrivial intersection with Mti \ int(… view at source ↗
read the original abstract

We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are: compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.

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

3 major / 1 minor

Summary. The paper defines a quasilocal mass of Bartnik type and proves two main results: (i) strict positivity of this mass for spacelike hypersurfaces that are either compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension; (ii) temporal monotonicity (nondecreasing in time) of the mass within evolutionary scenarios related to the two settings above.

Significance. If the positivity and monotonicity statements hold under clearly stated conditions, the work would strengthen the toolkit of quasilocal mass definitions in general relativity by extending Bartnik-type constructions to black-hole domains and providing dynamical control. The results could bear on the Penrose inequality and on the interpretation of mass in evolving black-hole spacetimes, provided the admissible-extension class is natural rather than artificially restrictive.

major comments (3)
  1. [Abstract and §1] Abstract and §1 (Introduction): the central positivity statement is conditioned on 'any admissible extension,' yet no explicit list of the required properties (asymptotic decay class, dominant energy condition, absence of additional horizons, regularity at the apparent horizon, etc.) is supplied. Without these conditions the claim cannot be checked for load-bearing assumptions or for the risk that admissibility has been chosen precisely to exclude counter-examples.
  2. [§2 and §3] §2 (Definition of the mass) and §3 (Positivity proof): the manuscript must exhibit the precise variational definition of the Bartnik-type quasilocal mass and the admissible-extension class before the positivity argument can be assessed; the current abstract-only presentation leaves the derivation unverifiable.
  3. [§4] §4 (Monotonicity): the monotonicity result is stated only for 'evolutionary scenarios related to the two aforementioned settings.' The precise evolution equations, gauge conditions, and energy conditions that close the monotonicity argument must be stated explicitly; otherwise the claim inherits the same ambiguity as the positivity statement.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it briefly indicated the dimension, the signature, and the precise notion of 'apparent horizon' employed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where greater explicitness will improve verifiability. We address each major comment below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (Introduction): the central positivity statement is conditioned on 'any admissible extension,' yet no explicit list of the required properties (asymptotic decay class, dominant energy condition, absence of additional horizons, regularity at the apparent horizon, etc.) is supplied. Without these conditions the claim cannot be checked for load-bearing assumptions or for the risk that admissibility has been chosen precisely to exclude counter-examples.

    Authors: We agree that an explicit enumeration of admissibility conditions is necessary for verification. In the revised manuscript we will insert, immediately after the definition of admissible extensions in §2, a numbered list specifying: (i) asymptotic flatness with decay rates O(r^{-1}) for the metric and O(r^{-2}) for the second fundamental form; (ii) the dominant energy condition on the extension; (iii) absence of additional apparent horizons; and (iv) C^{2,α} regularity across the apparent horizon. This list will also be referenced in the abstract and §1. revision: yes

  2. Referee: [§2 and §3] §2 (Definition of the mass) and §3 (Positivity proof): the manuscript must exhibit the precise variational definition of the Bartnik-type quasilocal mass and the admissible-extension class before the positivity argument can be assessed; the current abstract-only presentation leaves the derivation unverifiable.

    Authors: The variational definition appears as Definition 2.1 in §2, where the mass is the infimum of ADM masses over admissible extensions. To address the concern, we will reorder §2 so that the complete admissibility class is stated before the definition itself, and we will add a forward reference in the positivity proof of §3 that explicitly invokes each listed condition. These changes make the logical structure self-contained without altering the argument. revision: yes

  3. Referee: [§4] §4 (Monotonicity): the monotonicity result is stated only for 'evolutionary scenarios related to the two aforementioned settings.' The precise evolution equations, gauge conditions, and energy conditions that close the monotonicity argument must be stated explicitly; otherwise the claim inherits the same ambiguity as the positivity statement.

    Authors: We accept that the evolutionary setting requires explicit statement. In the revision we will open §4 with a paragraph listing the precise hypotheses: the spacetime satisfies the Einstein equations with dominant energy condition; the lapse function satisfies a uniform bound derived from the dominant energy condition; the gauge is maximal (or harmonic) slicing; and the monotonicity follows from the non-negativity of the energy flux through the apparent horizon. These conditions will be tied directly to the two classes of domains already defined. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained against external benchmarks

full rationale

The abstract and claims define a Bartnik-type quasilocal mass and state positivity/monotonicity theorems under admissible extensions and evolutionary scenarios. No equations, self-citations, or derivation steps are exhibited that reduce a claimed prediction or uniqueness result to a fitted input, self-definition, or prior author work by construction. Admissibility conditions are part of the theorem hypotheses rather than a circular filter; the central results are presented as independent theorems, not renamings or ansatzes smuggled via citation. This is the normal case of a paper whose derivation chain does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are available from the abstract alone.

pith-pipeline@v0.9.1-grok · 5619 in / 1076 out tokens · 21965 ms · 2026-06-27T20:54:45.719369+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references

  1. [1]

    Allen, E

    B. Allen, E. Bryden, D. Kazaras, and M. Khuri,Proof of the Penrose conjecture with suboptimal constant, preprint, 2025. arXiv 2504.10641

    arXiv 2025

  2. [2]

    Anderson, and J

    M. Anderson, and J. Jauregui,Embeddings, immersions and the Bartnik quasi-local mass conjectures, Ann. Henri Poincar´ e,20(2019), 1651–1698

    2019

  3. [3]

    Andersson, M

    L. Andersson, M. Eichmair, and J. Metzger,Jang’s equation and its appli- cations to marginally trapped surfaces, inComplex analysis and dynamical systems IV. Part 2, 13–45, Contemp. Math. Israel Math. Conf. Proc.,554, Amer. Math. Soc., Providence, RI. 20

  4. [4]

    Andersson, M

    L. Andersson, M. Mars, J. Metzger, and W. Simon,The time evolution of marginally trapped surfaces, Classical Quantum Gravity,26(2009), no. 8, 085018, 14 pp

    2009

  5. [5]

    Andersson, M

    L. Andersson, M. Mars, and W. Simon,Local existence of dynamical and trapping horizons, Phys. Rev. Lett.,95(2005), 111102

    2005

  6. [6]

    Andersson, M

    L. Andersson, M. Mars, and W. Simon,Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys.,12(2008), no. 4, 853–888

    2008

  7. [7]

    Andersson, and J

    L. Andersson, and J. Metzger,The area of horizons and the trapped region, Comm. Math. Phys.,290(2009), no. 3, 941–972

    2009

  8. [8]

    Andr´ easson,The Einstein-Vlasov system/kinetic theory, Living Rev

    H. Andr´ easson,The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ.,5(2002), 2002-7, 33 pp

    2002

  9. [9]

    Ashtekar, and B

    A. Ashtekar, and B. Krishnan,Dynamical horizons and their properties, Phys. Rev. D,68(2003), 104030

    2003

  10. [10]

    Bartnik,The mass of an asymptotically flat manifold, Comm

    R. Bartnik,The mass of an asymptotically flat manifold, Comm. Pure Appl. Math.,39(1986), no. 5, 661–693

    1986

  11. [11]

    Bartnik,New definition of quasilocal mass, Phys

    R. Bartnik,New definition of quasilocal mass, Phys. Rev. Lett.,62(1989), 2346

    1989

  12. [12]

    Bartnik,Energy in general relativity, Tsing Hua lectures on geometry and analysis, (Hsinchu 1990-91), 5–27, International Press, Cambridge, MA 1997

    R. Bartnik,Energy in general relativity, Tsing Hua lectures on geometry and analysis, (Hsinchu 1990-91), 5–27, International Press, Cambridge, MA 1997

    1990

  13. [13]

    Bartnik,Mass and 3-metrics of non-negative scalar curvature, Proceed- ings of the International Congress of Mathematicians, Beijing,II(2002), 231–240, Higher Ed

    R. Bartnik,Mass and 3-metrics of non-negative scalar curvature, Proceed- ings of the International Congress of Mathematicians, Beijing,II(2002), 231–240, Higher Ed. Press

    2002

  14. [14]

    Chru´ sciel,Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological Properties and Global Structure of Space-Time (P

    P. Chru´ sciel,Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological Properties and Global Structure of Space-Time (P. Bergmann and V. de Sabbata, eds.), Plenum Press, New York, 1986, pp. 49–59

    1986

  15. [15]

    P. T. Chru´ sciel, E. Delay, G. J. Galloway, and R. Howard,Regularity of horizons and the area theorem, Ann. Henri Poincar´ e2(2001), no. 1, 109—178

    2001

  16. [16]

    Corvino, and L.-H

    J. Corvino, and L.-H. Huang,Localized deformation for initial data sets with the dominant energy condition, Calc. Var. Partial Differential Equa- tions,59(2020), no. 1, Paper No. 42, 43 pp. 21

    2020

  17. [17]

    Dong, and A

    C. Dong, and A. Song,Stability of Euclidean 3-space for the positive mass theorem, Invent. Math.,239(2025), no. 1, 287–319

    2025

  18. [18]

    Eichmair,The Plateau problem for marginally trapped surfaces, J

    M. Eichmair,The Plateau problem for marginally trapped surfaces, J. Dif- ferential Geom.,83(2009), no. 3, 551–584

    2009

  19. [19]

    Gl¨ ockle,Initial data sets with dominant energy condition admitting no smooth DEC spacetime extension, Proc

    J. Gl¨ ockle,Initial data sets with dominant energy condition admitting no smooth DEC spacetime extension, Proc. Amer. Math. Soc. Ser., B11 (2024), 481–488

    2024

  20. [20]

    Hawking and G.F.R

    S.W. Hawking and G.F.R. Ellis,The large scale structure of space-time, Cambridge University Press, Cambridge, 1973

    1973

  21. [21]

    Hayward,General laws of black-hole dynamics, Phys

    S. Hayward,General laws of black-hole dynamics, Phys. Rev. D,49 (1994), 6467

    1994

  22. [22]

    Huang, and D

    L.-H. Huang, and D. Lee,Bartnik mass minimizing initial data sets and improvability of the dominant energy scalar, J. Differential Geom.,126 (2024), no. 2, 741–800

    2024

  23. [23]

    Huisken, and T

    G. Huisken, and T. Ilmanen,The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom.,59(2001), no. 3, 353–437

    2001

  24. [24]

    McCormick, and P

    S. McCormick, and P. Miao,On the evolution of the spacetime Bartnik mass, Pure and Applied Mathematics Quarterly15, (2020) no. 3, 897–920

    2020

  25. [25]

    McCormick,An overview of Bartnik’s quasi-local mass, Beijing J

    S. McCormick,An overview of Bartnik’s quasi-local mass, Beijing J. of Pure and Appl. Math.1, (2024) no. 2, 455–487

    2024

  26. [26]

    Mantoulidis, and R

    C. Mantoulidis, and R. Schoen,On the Bartnik mass of apparent horizons Journal of Classical and Quantum Gravity32, (2015), no. 20, 205002

    2015

  27. [27]

    Penrose,Some unsolved problems in classical general relativity, Ann

    R. Penrose,Some unsolved problems in classical general relativity, Ann. of Math. Stud., No. 102, Princeton University Press, Princeton, NJ, 1982, pp. 631–668

    1982

  28. [28]

    Schoen,Estimates for stable minimal surfaces in three-dimensional manifolds, Ann

    R. Schoen,Estimates for stable minimal surfaces in three-dimensional manifolds, Ann. of Math. Stud.,103, Princeton University Press, Prince- ton, NJ, 1983, 111–126

    1983

  29. [29]

    Schoen, and S.-T

    R. Schoen, and S.-T. Yau,Proof of the positive mass theorem II, Comm. Math. Phys.,79(1981), 231–260

    1981

  30. [30]

    Simon,Lectures on geometric measure theory, Proc

    L. Simon,Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral. Nat. Univ.,3, Australian National University, Centre for Math- ematical Analysis, Canberra, 1983. 22

    1983

  31. [31]

    Szabados,Quasi-local energy-momentum and angular momentum in general relativity, Living Rev

    L. Szabados,Quasi-local energy-momentum and angular momentum in general relativity, Living Rev. Rel,12(2009), no. 4

    2009

  32. [32]

    White,On the bumpy metrics theorem for minimal submanifolds, Amer

    B. White,On the bumpy metrics theorem for minimal submanifolds, Amer. J. Math.,139(2017), no. 4, 1149–1155

    2017

  33. [33]

    Witten,A simple proof of the positive energy theorem, Comm

    E. Witten,A simple proof of the positive energy theorem, Comm. Math. Phys.,80(1981), no. 3, 381–402. 23

    1981