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Quasi-local conserved charges in General Relativity

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Pith's one-line read A new prescription for quasi-local conserved charges in general relativity produces a BMS zero-mode charge that vanishes in Minkowski spacetime, becomes the Bondi mass at null infinity, and equals known masses in symmetric and horizon…

desk verdict Serious, clearly written Wald-Zoupas extension for quasi-local BMS charges, but the general prescription outruns the proof: reference-term existence is only shown in a special case and the Newman-Unti preference is deferred to a companion paper. read the letter →

arxiv 1908.07504 v1 pith:WAXP647C submitted 2019-08-20 gr-qc hep-th

Henk Bart This is my paper · ORCID
classification gr-qchep-th MSC 83C4083C57
keywords quasi-localenergyBMSchargesNewman-UntigaugeconservedEinstein-HilbertactionMisner-SharpcovariantphasespaceReissner-Nordströmblackhole
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a general method for assigning conserved charges to closed spacelike two-surfaces in general relativity, the type of quantity called quasi-local energy. It adapts the standard null-infinity correction-term construction so that it works in the bulk of a spacetime, then applies the method to BMS symmetry generators in Newman-Unti gauge. The resulting zero-mode charge, if the construction holds, gives a well-defined quasi-local energy: it vanishes in Minkowski spacetime, tends to the Bondi mass at null infinity, equals the Misner-Sharp energy on round spheres, and equals the irreducible mass at the outer horizon of a Reissner-Nordström black hole. These are the pragmatic criteria that make the candidate worth taking seriously, especially because several existing quasi-local energies fail at least one of them, most notably by not vanishing in flat spacetime.

What carries the argument

The load-bearing object is the boundary reference term $K_0$, constrained by four conditions: consistency (independence of the auxiliary hypersurface), linearity in the symmetry generator, orthogonality (the three-form $V$ has the ingoing null direction $n$ in its kernel), and a zero point (vanish on a reference solution). The reference term is built by a four-step embedding: choose a Newman-Unti foliation, the coordinate system in which $v$ labels a null foliation and $r$ is an affine parameter of the null generators, whose asymptotic shear matches the physical one at past null infinity; isometrically embed $B$ into the reference spacetime; construct the reference hypersurface by matching $C\theta(n)$ and $C^{-1}\mathcal{L}_h C$; and pull back the reference extrinsic curvature. This yields the identity $K-K_0 = -(\kappa-\hat\kappa+\theta(l)-\hat\theta(l))V$, which converts the Noether charge integral into the explicit charge formula (38), and for BMS generators in Newman-Unti gauge into (42). The machinery turns the problem of defining quasi-local energy into the problem of finding meaningful boundary conditions and a preferred bulk extension of the symmetry generators.

What would settle it

Evaluate the zero-mode BMS charge on the outer horizon of a Kerr black hole using the known Newman-Unti form of the Kerr metric: if the result is not the irreducible mass, the proposed quasi-local energy fails for rotating black holes.

Watch

Extended reading notes

Core claim

The central claim is that conserved charges can be defined at any closed spacelike two-surface $B$ by choosing a correction term in the covariant phase space Hamiltonian equation whose defining condition is not stationarity but vanishing of its integral over $B$ for variations preserving chosen boundary conditions on an auxiliary hypersurface. With canonical boundary conditions and a reference term constructed by isometrically embedding $B$ into a reference spacetime along a matched Newman-Unti foliation, the Einstein-Hilbert charge takes the explicit form (38); for BMS generators in Newman-Unti gauge it simplifies to (42). The zero mode $f=1$ of that charge is put forward as a quasi-local energy because it vanishes in Minkowski spacetime, asymptotes to the Bondi mass at null infinity, equals the Misner-Sharp energy on round spheres in spherically symmetric spacetimes, and equals the irreducible mass at the outer Reissner-Nordström horizon. The paper is explicit that the BMS charge is gauge-dependent in the bulk and that the physical preference for Newman-Unti gauge is argued in a companion paper.

Load-bearing premise

The construction is only physically meaningful if BMS generators in Newman-Unti gauge are the preferred bulk extension of BMS symmetries; the paper states that the justification is provided in a separate companion paper and that the charges remain gauge-dependent in the bulk.

Editorial extensions

If this is right

  • The zero-mode BMS charge gives a candidate quasi-local energy for any closed spacelike two-surface covered by Newman-Unti coordinates, including regions inside the exterior of sufficiently weak matter around a black hole.
  • At null infinity the charge reduces to the standard BMS charge, so the construction recovers the Bondi mass as the quasi-local energy of a cut.
  • In spherically symmetric spacetimes the charge equals the Misner-Sharp energy on round spheres, tying the new definition to an established surface mass.
  • On the outer Reissner-Nordström horizon the gravitational part equals the irreducible mass, so black hole entropy bounds based on energy would see the expected horizon value.
  • Because the prescription is formulated for diffeomorphism-covariant theories, the same correction-term logic can define quasi-local charges for other gravitational theories and other asymptotic fall-off conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the irreducible-mass result extends to Kerr, as the paper suggests but does not check, the same zero-mode charge would provide a rotating black hole energy that the angular momentum BMS charges complement; evaluating it in the known Newman-Unti form of Kerr would settle the test.
  • The paper's reliance on Newman-Unti gauge suggests a physical principle: bulk gravitational memory selects the preferred BMS extension, so charges built from any other gauge extension would be physically disfavored rather than merely different.
  • A natural stress test is to compute the charge on a surface that crosses a gravitational wave region and compare its rate of change with the flux of the correction term; the construction predicts the flux is exactly the change in quasi-local energy.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a general covariant-phase-space prescription for constructing quasi-local conserved charges on an arbitrary closed spacelike two-surface B. The construction modifies Wald and Zoupas by replacing the stationarity condition on the symplectic potential with an integral condition that the potential integrates to zero over B for variations respecting boundary conditions on an auxiliary hypersurface (3)B, and it fixes the remaining reference-term ambiguity through consistency, linearity, orthogonality, and zero-point conditions. For Einstein-Hilbert gravity with canonical boundary conditions, the resulting charge is given by Eq. (22); with the reference term (35) it evaluates to Eq. (38). Applied to BMS generators in Newman-Unti gauge, the charge becomes Eq. (42). The paper then argues that the zero mode of this BMS charge is a promising quasi-local energy: it vanishes in Minkowski spacetime, reproduces the Wald-Zoupas BMS charge at null infinity, equals the Misner-Sharp energy on round spheres in the spherically symmetric case, and equals the irreducible mass at the outer horizon of a Reissner-Nordstrom black hole (for the gravitational part of the charge).

Significance. If the construction is made fully well-defined, it would be a useful contribution to the quasi-local energy literature: the derivation from the covariant phase space to Eqs. (38) and (42) is coherent, the asymptotic matching with the Wald-Zoupas charges is explicit, and the spherical and Reissner-Nordstrom checks are concrete. The paper is also commendably honest about its limitations: it states in Section 3.5 that existence and uniqueness of the reference term have not been investigated in general, it acknowledges that the BMS charge is gauge dependent in the bulk, and it flags the gauge dependence of the electromagnetic contribution in the Reissner-Nordstrom check. The consistency and linearity conditions introduced in Section 3 are an interesting structural improvement over previous bulk extensions of the Wald-Zoupas framework. The central claims, however, currently rest on two load-bearing points that are not fully established in the manuscript: the existence of the reference term for generic surfaces, and the physical preference for Newman-Unti gauge, which is deferred to an unpublished companion paper.

major comments (3)
  1. [§4.5 and §3.5] The reference term K0, which is needed to define B_X in Eq. (12) and hence the charge at a generic surface B, is proven to exist only in the special case where the reference spacetime is Minkowski and the slice has vanishing asymptotic shear C_AB|v=0=0. The conditions (31)-(33) that determine the reference hypersurface for a generic B form a nontrivial system, and the manuscript explicitly disclaims general existence and uniqueness in Section 3.5. Since the spherical and Reissner-Nordstrom checks in Section 5.5 all lie in the regime C_AB=0, the advertised quasi-local charge is not yet shown to be defined for the generic surfaces that motivate the paper.
  2. [§5.1 and §1] The BMS charge is gauge dependent in the bulk, as the paper acknowledges: 'We do not solve the issue of gauge dependence of the BMS charge.' The preferred status of the Newman-Unti gauge extension is asserted to follow from gravitational memory only in the companion paper [31], which is listed as 'to appear' and is not available for inspection. Since the zero-mode charge is presented as a quasi-local energy, the physical interpretation depends on this unpublished premise; as the manuscript stands, the charge is one of many gauge-dependent quantities rather than a uniquely defined quasi-local BMS charge.
  3. [§5.5 and §6(ii)] The Reissner-Nordstrom check is performed only for the gravitational part of the charge; the gauge-field contribution is discarded because it is gauge dependent. The full covariant phase space charge for the Einstein-Maxwell system is not gauge invariant, so the statement that the zero-mode charge equals the irreducible mass at the outer horizon is a statement about a truncated gravitational charge, not about the charge defined by the proposed general framework. The horizon property (iii) is therefore not yet a test of the full construction.
minor comments (5)
  1. [Throughout] There are several typographical errors, including 'assocoiated', 'arbitary', and 'Minkowksi', which should be corrected.
  2. [§5.3] The statement that every BMS vector field in Minkowski space is a linear combination of the supertranslation vectors ζ and isometries k is asserted rather than derived; a short coordinate argument would make the vanishing check self-contained.
  3. [§5.4] The assertion that the terms containing I^A ω_A + L_l f vanish asymptotically is stated without an explicit expansion; providing the expansion would help the reader verify the match with Eq. (45).
  4. [Reference [31]] Reference [31] is cited as 'to appear' with no preprint identifier; since the gauge-preference argument is load-bearing, a public version or a summary of its argument in an appendix would be needed for the present claims to be fully verifiable.
  5. [§4.3] The construction in Step 1 sets up the foliation near past null infinity and also requires B to lie in Σ0; the interaction between these requirements and the possible breakdown of Newman-Unti coordinates at caustics in the bulk deserves a more explicit discussion, building on the domain-of-applicability remarks in Section 5.1.

Circularity Check

1 steps flagged · score 4.0 of 10

The charge construction and its spherical/horizon evaluations are self-contained, but the physical preference for Newman-Unti gauge rests solely on a to-appear companion paper by the same author, making that load-bearing premise a self-citation.

  1. self citation load bearing [Section 1, p. 5; repeated in Section 5 introduction.]
    "We do not solve the issue of gauge dependence of the BMS charge. However, we provide in a separate paper [31] a justification for why the BMS generators in Newman-Unti gauge are physically preferred. Namely, that BMS generators in Newman-Unti gauge are connected to the gravitational memory effect."

    The bulk BMS charge is explicitly gauge dependent, so the unique physical interpretation of the zero-mode charge as quasi-local energy depends on why Newman-Unti extensions are preferred over Bondi or other extensions. That preference is not demonstrated here; the only cited support is [31], an unpublished companion paper by the same author. The physical claim therefore reduces to an unverified self-citation, even though the explicit charge formula, the Vaidya evaluation, and the Reissner-Nordstrom horizon computation remain independent mathematical content.

full rationale

The central derivation is not circular: the charge formula (38)/(42) follows from the covariant phase-space procedure with explicitly stated conditions, and the spherical and horizon checks are direct evaluations rather than fitted outputs. The Minkowski vanishing is partly imposed through the zero-point condition (18), but the paper additionally verifies that the Noether charge of BMS generators vanishes in Minkowski, so the property is not a pure tautology. The asymptotic match to Wald-Zoupas is designed through the reference-shear condition (31), yet the final formula (45) still requires the nontrivial theta(l) expansion, so it is a checked target rather than an identity. The paper itself disclaims general existence and uniqueness of the reference term in Section 3.5 and proves existence only for a Minkowski reference with vanishing asymptotic shear in Section 4.5; that is a real gap for the advertised generic quasi-local energy, but it is an omitted proof rather than circularity. The one load-bearing circular element is the justification for privileging Newman-Unti gauge, which is deferred to the author's own to-appear companion paper [31]; hence the moderate score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The construction rests on the existence and regularity of auxiliary structures, namely the Newman-Unti foliation, the reference hypersurface in Minkowski spacetime, and the isometric embedding, and on a gauge-preference claim deferred to a companion paper. No numerical free parameters are fitted to data; the choices are structural. No new physical entities are introduced.

assumptions (6)
  • domain assumption The closed surface B lies inside a Newman-Unti null hypersurface Sigma_0 generated by a congruence that does not focus before reaching B.
    The BMS generators (41) and the matching of the reference foliation are defined only where Newman-Unti coordinates remain regular; caustics are excluded by the footnote in section 5.1 and the paragraph 'Domain of applicability'.
  • domain assumption There exists a reference hypersurface (3)B-hat in Minkowski spacetime satisfying the matching conditions (32) and (33).
    Existence and uniqueness are demonstrated only for the special case of a surface in a slice with vanishing asymptotic shear using the uniformization theorem (section 4.5); the general case is left to future work.
  • ad hoc to paper The BMS generators in Newman-Unti gauge are the physically preferred bulk extension of BMS symmetries.
    The paper states this justification is given in a separate companion paper [31] (to appear); the charge is gauge dependent in the bulk and this choice is load-bearing for the quasi-local energy interpretation (section 5 introductory paragraph).
  • ad hoc to paper The orthogonality condition (17) picks the ingoing null normal n to define the reference three-form V.
    The choice xi_perp = n is one of two natural choices at B, the outgoing null direction l being the other, and is made by hand in section 3.4.
  • domain assumption Canonical boundary conditions fix the induced metric on the auxiliary hypersurface (3)B.
    The charge is defined with respect to canonical boundary conditions (section 3.2); other boundary conditions give different charges with different interpretations.
  • standard math Uniformization theorem for the two-sphere.
    Used in section 4.5 to establish existence and uniqueness of the isometric embedding of B into Minkowski spacetime for the special case of vanishing asymptotic shear.

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Pith. "Pith review of Quasi-local conserved charges in General Relativity." pith.science (2026). https://pith.science/paper/WAXP647C

@misc{pith2026190807504,
  author       = {Pith},
  title        = {Pith review of: Quasi-local conserved charges in General Relativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WAXP647C}},
  note         = {Machine review of arXiv:1908.07504}
}
read the original abstract

A general prescription for constructing quasi-local conserved quantities in General Relativity is proposed. The construction is applied to BMS symmetry generators in Newman-Unti gauge, so as to define quasi-local BMS charges. It is argued that the zero mode of this BMS charge is a promising definition of quasi-local energy.

Figures

Figures reproduced from arXiv: 1908.07504 by the authors.

Figure 1
Figure 1. The Bondi mass at null infinity may be thought of as a quasi- [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A spacetime containing a stationary shell of matter betwe [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. for a situation where this condition should apply. B ξ (3)B′ (3)B [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The linearity condition is a constraint on the relation betwee [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: A pictorial representation of the vectors [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Foliations by Newman-Unti (or Bondi) coordinates [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: A planet in the presence of a black hole. The trajectory of [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

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