A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity

Andreas Zoupas; Robert M. Wald

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A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity

Robert M. Wald, Andreas Zoupas

General Relativity and Quantum Cosmology · Submitted 24 Nov 1999 · 25 references

Pith's verdict: The paper that turned conserved quantities at null infinity into a Hamiltonian prescription: the modified Hamiltonian equation (eq. 26) and the flux formula (eq. 32) are the parts that lasted, and the Dray–Streubel derivation is the payoff.

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CitationAndreas Zoupas, Robert M. Wald. "A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity." arXiv:gr-qc/9911095 (1999). https://pith.science/paper/UUSHOOE6

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Editor's note

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This is the Wald–Zoupas paper, and it earns its place. The core move is simple to state and hard to get right: when no Hamiltonian exists because symplectic current escapes through the cross-section, replace δH = ∫ω with eq. 26, adding a boundary symplectic potential Θ that must vanish on stationary solutions. That yields a definition for every BMS generator and every cross-section, plus the flux formula F_ξ = Θ(g, L_ξ g) in eq. 32, which vanishes whenever the solution is stationary or ξ is an exact symmetry. Section 3's existence criterion (eq. 18) is clean and is the part I would hand a student first. The second contribution is deriving Dray–Streubel from Hamiltonian considerations. Section 5 does the vacuum GR work honestly: ω is shown to extend continuously and nonvanishingly to I, Θ is identified as the Bondi news contracted with the perturbation (eq. 73), and both cases are checked — ξ tangent to the cross-section (eq. 92, the Geroch–Winicour linkage form) and ξ a supertranslation (eq. 98, Geroch's expression). Agreement with the Ashtekar–Streubel flux is an independent cross-check, so circularity is not a worry here. Where it is soft: the uniqueness of Θ in §5 is a scaling-dimension count, not a proof. The authors write “presumably” and then “it is easily verified” — those are exactly the two steps that would need filling in. If an allowed W with δW vanishing on stationary solutions exists, Θ shifts and H_ξ shifts by a locally defined, cross-section-independent term, which the paper itself notes would be a genuinely conserved quantity. That is a real gap, but a bounded one: existence of Θ and the Dray–Streubel agreement do not depend on it. The claim that only Minkowski satisfies the consistency condition (36) is likewise asserted rather than shown. The step from eq. 97 to eq. 98 leans on unpublished Iyer notes [20], though it is cross-checked against Dray–Streubel, so it is not load-bearing. Worth noting: the abstract does flag that general-context existence and uniqueness are not investigated, so the classifier's “no limitations found” is wrong. Footnote 24 corrects an error in Geroch–Winicour; that kind of care is worth crediting. Who it is for: anyone working on asymptotic symmetries, Bondi mass, or the Hamiltonian formulation of gravity with boundaries. It is a standard citation and deserves to be. Send it out.

Referee report

Significance

The paper supplies a Hamiltonian-foundation derivation of the Dray–Streubel conserved quantities at null infinity, which previously rested on twistor-motivated considerations, and it does so within a framework general enough to cover arbitrary diffeomorphism-covariant Lagrangian theories. The construction has no free parameters: Θ is fixed by the requirements of local construction, analyticity, background independence, and vanishing on stationary solutions, and the additive constant in H_ξ is fixed by a reference solution. The result is cross-checked against two independent published prescriptions — the Ashtekar–Streubel flux formula (Eq. 82) and the Dray–Streubel quantities (Eqs. 92 and 98) — which is a genuine external validation rather than an internal consistency check. The paper is also candid about scope: the abstract states that existence and uniqueness in the most general context are not investigated, and §5 flags where its own arguments are heuristic. The careful separation of the two ambiguities, that of θ from the Lagrangian (Eq. 3) and that of Θ from ω (Eq. 27), is a conceptual contribution in its own right.

Major objections (1)

[§5, uniqueness of Θ (Eq. 73 and the paragraph following it)] The uniqueness claim for Θ rests on a scaling-dimension count that is sketched rather than carried out, and the paper's own language marks the gap: 'Presumably, this will imply that we can write w as a convergent sum of terms...' followed by 'it is then easily verified that there does not exist an allowed w such that δw is nonzero in general... but δw vanishes whenever the physical metric is stationary.' Two steps are unproved: (i) that any admissible scalar w of scaling dimension −3 admits such a convergent expansion in the unphysical metric, curvature, n^a and their derivatives; (ii) that the resulting finite list contains no term with the stated property. Since the consequence is substantive — if such a W exists, Θ is ambiguous up to δW and H_ξ shifts by ∫_∂Σ ξ·δW, which the paper itself observes (remark following Eq. 82) would be a locally defined, cross-section-independent quantity — the enumeration should be exhibited explicitly, at least as a table of the surviving candidate terms and the reason each is excluded. I want to be clear that this does not threaten the paper's headline result: the existence of Θ (Eq. 73) and the agreement with Dray–Streubel (Eqs. 92, 98) are established for the exhibited Θ and are independent of uniqueness. The concern is confined to the word 'unique' in the §5 conclusion and to the parallel claim in §4 that the prescription 'uniquely determines H_ξ'.

Minor objections (7)

[§4, definition of equivalence of representatives] The equivalence relation on representatives is defined partly through the asymptotic integral I of Eq. (19), which depends on φ and δφ; this reintroduces at the level of the symmetry algebra exactly the field dependence that §2 says it wants to avoid by working on F rather than on phase space. Footnote 17 notes that in general relativity agreement of ξ^a and ξ'^a on I already implies equality of I, so the extra clause is idle there. It would help to say explicitly whether the equivalence is intended to be field-independent in general, and if not, how H_ξ is to be viewed as a function on F̄ for a fixed ξ^a.
[§5, Eq. (97) to Eq. (98)] The passage from Eq. (97) to Geroch's expression Eq. (98) is attributed to unpublished notes [20]. The endpoint is a published formula (Geroch [3]), so the result itself is supported, but the derivation route is not available to the reader. Either cite [3] directly for the closed form and note that Eq. (97) reproduces it, or include enough of the calculation to make the step checkable.
[§5, reference solution] The sentence 'it should be possible to show that no other stationary solution can satisfy eq. (36)' is asserted without argument. The prescription only requires one stationary φ0 satisfying (36), and Minkowski is verified explicitly, so the uniqueness of φ0 is not load-bearing; either supply the argument or soften the sentence to a remark.
[Abstract] The generality claim would read more accurately if the central hypothesis of §4 were stated alongside it: the construction requires the symplectic current (n−1)-form ω to extend continuously to the boundary B. This is the condition that decides whether the framework applies at all, and it is the first thing a reader needs in order to judge the scope.
[§4, notation] The distinction between a true Hamiltonian and the quantity defined by the modified equation (26) is carried by an overbar or underline on H_ξ that is easy to lose in typesetting and in reading. A distinct symbol, or an explicit reminder at each use, would prevent confusion in Eqs. (26), (28) and (31).
[§3, Eq. (16) and Eq. (18)] The contraction ξ·ω and the orientation conventions entering the boundary integrals are fixed only implicitly, through footnotes 2, 3 and 8, and the sign differs between future and past null infinity. One explicit sentence at Eq. (16) stating the orientation convention used for ∂Σ would make the signs in Eqs. (23), (24) and (28) easier to check.
[Footnote 24] The correction to Geroch–Winicour [17] regarding the intrinsic-divergence claim is a useful erratum that readers of that paper would benefit from seeing; consider whether it belongs in the body of §5 rather than in a footnote.

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The paper

The paper claims

Wald and Zoupas propose a replacement for the Hamiltonian equation that still defines mass, momentum and angular momentum at null infinity, where radiation destroys the ordinary Hamiltonian and the quantities stop being conserved.

The reading

In general relativity, mass, momentum and angular momentum at spatial infinity are read off from the Hamiltonian that generates an asymptotic symmetry. At null infinity that route breaks down: symplectic current can flow out through the cross-section, so for a general BMS generator no Hamiltonian exists — which is the Hamiltonian way of saying the quantity is not conserved. The paper proposes a modified equation, keeping the usual surface term $[δQ - ξ·θ]$ and adding an integral of $ξ·Θ$ over the cross-section, where $Θ$ is a symplectic potential for the pullback of the symplectic current to the boundary, required to vanish on stationary solutions. The payoff is a definition that applies to every infinitesimal asymptotic symmetry and every cross-section, comes with a local flux formula $F_ξ = Θ(g_{ab}, L_ξ g_{ab})$, and in vacuum general relativity at null infinity gives a unique answer that matches what Dray and Streubel had proposed from entirely different, twistor-motivated reasoning.

What carries the argument

The load-bearing object is the boundary symplectic potential $Θ$, defined on the boundary $B$ by $ω̄(φ,δ_1φ,δ_2φ) = δ_1Θ(φ,δ_2φ) - δ_2Θ(φ,δ_1φ)$ for the pullback of the symplectic current. It is the extra term that repairs the failed consistency check $∫_{∂Σ} ξ·ω = 0$ for the existence of a Hamiltonian. Three conditions pin it down: it must be locally constructed from the dynamical fields and the fixed background structure, it must be independent of arbitrary choices in that background structure (in the null-infinity case, conformally invariant), and it must vanish on stationary solutions. In vacuum general relativity at null infinity those conditions select $Θ = -N_{ab}τ^{ab}$ times the boundary volume element, which is what makes the flux equal to the Bondi news contracted with the perturbation. The Noether charge two-form $Q_{ab}[ξ] = -(1/16π)ε_{abcd}∇^cξ^d$ supplies the other half of the definition, and the Geroch–Winicour gauge condition $∇_aξ^a = 0$ is what makes $∫_{∂Σ}Q$ well defined as a representative-independent charge.

Extended reading notes

Core claim

The central claim is that a 'conserved quantity' can be defined even where no Hamiltonian exists, by weakening the Hamiltonian equation rather than abandoning it. The proposed definition is $δH_ξ = ∫_{∂Σ}[δQ - ξ·θ] + ∫_{∂Σ} ξ·Θ$, where $Q$ is the Noether charge two-form, $θ$ the presymplectic potential, and $Θ$ a symplectic potential on the boundary whose variation reproduces the pullback of the symplectic current there. Requiring $Θ$ to vanish on stationary solutions makes the associated flux vanish when there is no radiation and when $ξ^a$ is an exact symmetry, and gives the simple flux formula $F_ξ = Θ(φ, L_ξφ)$. For vacuum general relativity at null infinity the authors argue that $Θ = -N_{ab}τ^{ab}$ (times the boundary volume element, with $N_{ab}$ the Bondi news) is the unique admissible choice, so that the flux is exactly the Ashtekar–Streubel news-tensor flux. The resulting charges are then shown, case by case, to agree with the Dray–Streubel prescription: for a generator tangent to the cross-section the answer reduces to the Geroch–Winicour linkage expression, and for a supertranslation it reduces to Geroch's formula built from the news and the Weyl tensor.

Load-bearing premise

The claim of a unique definition rests on the assertion that no other admissible boundary term $W$ exists whose variation vanishes on stationary solutions but not in general, and the paper supports that by a scaling-dimension counting argument introduced with the hedge 'presumably' rather than by a complete enumeration.

Editorial extensions

If this is right

Every infinitesimal BMS generator and every cross-section of null infinity gets a definite 'conserved quantity', even though these quantities are not conserved when radiation is present.
The flux of each such quantity is fixed by a single local formula, $F_ξ = Θ(g_{ab}, L_ξ g_{ab})$, which vanishes for stationary spacetimes and for exact symmetries regardless of radiation.
In vacuum general relativity the resulting charges coincide with the Dray–Streubel quantities, which supplies a Hamiltonian-formulation derivation of a definition previously obtained from twistor-inspired considerations.
The construction is stated for any diffeomorphism-covariant theory derived from a Lagrangian, so the same recipe is expected to extend to matter couplings, higher-derivative gravity, two-dimensional dilaton models, and asymptotically anti-de Sitter settings.
The method gives a general diagnostic for when a genuine Hamiltonian does exist: precisely when the pullback of the symplectic current to the boundary vanishes, the 'case I' that covers spatial infinity and anti-de Sitter.

Reading between the lines

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

The same recipe, applied to a horizon rather than an asymptotic boundary, would define a flux through the horizon and might constrain how entropy of a non-stationary black hole is defined; the authors raise this only as a possibility, and they doubt a unique $Θ$ exists in that setting.
Because the whole scheme turns on whether an admissible $Θ$ exists and is unique, the method's reach into modified gravity depends on a case-by-case scaling-dimension count like the one done here for the news tensor; a theory where two independent boundary tensors share the same scaling weight would be the natural place to test the uniqueness argument.
The agreement with Dray–Streubel is derived by splitting a BMS generator into a cross-section-tangent part and a supertranslation; that split suggests the charge is naturally a sum of a 'Komar-like' piece and a 'news-and-Weyl' piece, which is a structural statement that could be checked independently of the derivation.
If an admissible $W$ with $δW$ vanishing on stationary solutions but nonzero otherwise were found on null infinity, the definition would become a one-parameter family rather than a single prescription — a concrete target for anyone wanting to test the uniqueness claim.

Original abstract

In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.96 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry.	background 1.00	0.09	0.05	asserted 1.00
s1	However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry.	background 0.59	0.12	0.04	asserted 0.99
s2	(This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian.	contribution 0.94	0.78	0.51	asserted 1.00
s3	Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts.	limitation 0.51	0.73	0.17	asserted 0.96
s4	In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.	result 0.96	0.67	0.12	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.83 · evidence from proof or derivation 0.93.

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arXiv listing abstract v1

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (25)

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hep-th 2008-05 conditional novelty 7.0 of 10

Counter-term contributions to the bulk symplectic structure make fluctuations of the AdS boundary metric normalizable, so Neumann boundary conditions define a consistent bulk dual of a CFT with a dynamical metric.
Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions
gr-qc 2006-10 accept novelty 7.0 of 10

The charge algebra of asymptotically flat three-dimensional gravity carries a nontrivial central extension: the J-P commutator picks up a term proportional to (1/4G) m(m^2-1) delta_{m+n,0}.
Dynamical Horizons and their Properties
gr-qc 2003-08 conditional novelty 7.0 of 10

The change in a dynamical horizon's area equals a local, positive flux of matter plus gravitational energy, yielding a finite first law of black hole mechanics in fully non-linear general relativity.
Covariant theory of asymptotic symmetries, conservation laws and central charges
hep-th 2001-11 accept novelty 7.0 of 10

Equivalence classes of asymptotic reducibility parameters are shown to correspond one-to-one with asymptotically conserved (n-2)-forms, yielding a universal covariant charge formula and a covariant central-charge 2-cocycle.
Thermodynamics of Swirling Black Holes and Kerr-Taub-NUT
hep-th 2026-10 conditional novelty 6.0 of 10

Using a surface-charge method with string-defect corrections, the authors build first laws for swirling and Kerr-Taub-NUT black holes and find entropy that departs from horizon area in the defected cases.
On the geometric origin of central charges in gravity
hep-th 2026-10 accept novelty 6.0 of 10

Central charges in gravity are derived from the Noetherian split of the covariant phase space, with explicit formulas for vanishing and integrable flux, and applied to AdS3.
Can a Wormhole Induce a Monopole?
gr-qc 2026-10 accept novelty 6.0 of 10

Periodic mass monopole induction across a wormhole throat is incompatible with conservation laws and the vacuum Einstein equations.
Towards a Complete Kerr Soft-Hair Theory: Integrability, Counterterm Equivalence,and Quantum State Counting
physics.gen-ph 2026-10 accept novelty 6.0 of 10

The HHPS central charge calculation for Kerr black holes is correct but insufficient to determine the quantum state count; additional gravitational data are required.
Thermodynamics of Kerr-Newman-Bertotti-Robinson black holes
gr-qc 2026-09 accept novelty 6.0 of 10

Using covariant phase-space methods, the authors derive the mass, angular momentum, first law, and Smarr formula for the full Kerr-Newman-Bertotti-Robinson black hole family.
Model dependent analytic spin torsion corrections to Blandford Znajek energy extraction in Einstein Cartan gravity
physics.gen-ph 2026-08 conditional novelty 6.0 of 10

A leading-order analytic formula for the Blandford Znajek power correction from an Einstein Cartan spin source is derived, conditional on a prescribed conserved completion and imposed optimal load.
Inclusive Radiation and Backreaction from the Phase-Space S-Matrix
hep-th 2026-08 conditional novelty 6.0 of 10

A partially Weyl-transformed S-matrix shows that classical waveform, impulse, position shift, and angular momentum are projections of a single symbol, with radiation reaction encoded in a Berry connection.
Gravitational Memory Beyond Null Infinity through Finite-Distance Carrollian Screens
hep-th 2026-07 conditional novelty 6.0 of 10

Finite-distance null screens carry a Carrollian memory whose leading tracefree large-radius part reproduces the standard Bondi displacement memory in Robinson–Trautman spacetimes.
On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity
hep-th 2026-07 conditional novelty 6.0 of 10

In the diagonal (Cartan) sector of AdS3 gravity, the radial flow of the quasi-local stress tensor satisfies an exact T Tbar-like equation, while the boundary time evolution forms an integrable bi-Hamiltonian hierarchy.
Large-$N$ Carrollian Thermodynamics from AdS Black-Hole Phase-Space Contractions
hep-th 2026-06 unverdicted novelty 6.0 of 10

Finite Carrollian limit of extended AdS first law is reinterpreted as double-scaled large-N low-temperature holographic ensemble with finite products, new boundary stress tensor, and celestial correlator representations.
The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory
gr-qc 2026-06 unverdicted novelty 6.0 of 10

Introduces a gauge transformation framework for BMS frames in multiscale black hole perturbation theory on Kerr that incorporates memory effects and avoids infrared divergences.
An algebra of proper observables at null infinity: Dirac brackets, Memory and Goldstone probes
hep-th 2026-05 unverdicted novelty 6.0 of 10

The algebra of proper observables at null infinity admits Goldstone probes that measure the memory mode, but none can be built from shear or news alone, and the Dirac brackets acquire non-local corrections.
Toward claiming a detection of gravitational memory
gr-qc 2026-01 unverdicted novelty 6.0 of 10

A framework using scale separation in the Isaacson description defines observable gravitational memory rise for compact binary coalescences, providing a basis for hypothesis testing in LISA data.
Sound as a gauge theory and its infrared triangle
hep-th 2025-12 conditional novelty 6.0 of 10

Linear acoustic perturbations admit a memory effect that, in a dual Kalb-Ramond formulation, is encoded by large gauge transformations plus a monopole piece.
Modular Witten Diagrams and Quantum Extremality
hep-th 2025-12 unverdicted novelty 6.0 of 10

Modular Witten diagrams reproduce the O(λ² G_N) correction to holographic entanglement entropy, matching the canonical energy term in the quantum Ryu-Takayanagi formula with wedge shape deformation.
Twisting asymptotically-flat spacetimes
gr-qc 2025-11 conditional novelty 6.0 of 10

Twisting asymptotically-flat spacetimes are brought into a generalized Bondi gauge with finite radial expansion, producing new flux-balance laws, Carroll-boost symmetries, and finite supertranslated Kerr–Taub–NUT metrics.
Localization and anomalous reference frames in gravity
hep-th 2025-10 unverdicted novelty 6.0 of 10

Constructs a phase space for gravitational degrees of freedom on null ray segments with commuting localized observables via edge modes and dressing time, then introduces an effective classical theory with Virasoro def...
From the Corner Proposal to the Area Law
hep-th 2025-10 reject novelty 6.0 of 10

For spherically symmetric gravity, the entanglement entropy of 'classical' quantum-corner coherent states scales with horizon area — but the Bekenstein–Hawking coefficient 1/4 is fixed by hand-picked central charges, ...
Conformal Blocks in 2d Carrollian/Galilean CFTs and Excited State Entanglement Entropy
hep-th 2025-10 unverdicted novelty 6.0 of 10

Derives heavy-light conformal blocks in 2d C/G CFTs and computes excited-state entanglement entropy via replica trick, finding thermal form that reproduces holographic EE and establishes dictionary between boundary we...
Covariant phase space and the semi-classical Einstein equation
hep-th 2025-10 unverdicted novelty 6.0 of 10

A semi-classical symplectic two-form is defined as the sum of the gravitational symplectic form and the Berry curvature of the quantum matter state; it is shown to be independent of the Cauchy slice and to satisfy a q...
Surg-InvNeRF: Invertible NeRF for 3D tracking and reconstruction in surgical vision
cs.CV 2025-08 unverdicted novelty 6.0 of 10

An invertible-NeRF test-time optimization approach tracks points in 2D and 3D across surgical videos, reporting roughly 50% higher average precision than prior optimization-based 2D trackers.
Foundational Structure of Local Amplitudes in Quantum Gravity
gr-qc 2025-08 conditional novelty 6.0 of 10

Local amplitudes for causal diamonds can be constructed from null-slab boundary states, projectors, and vacuum intertwiners, with Ward identities and charge conservation as consequences.
Radiation in Fluid/Gravity and the Flat Limit
hep-th 2025-08 unverdicted novelty 6.0 of 10

Establishes a holographic link between bulk gravitational radiation and dissipative corrections plus entropy production in boundary fluids, then constructs Carrollian analogues and celestial observables in the flat limit.
Gauge Symmetries, Exact Symmetries and Conserved Charges in Minimal Massive Gravity
hep-th 2025-07 conditional novelty 6.0 of 10

Minimal Massive Gravity has three gauge symmetries, and a newly constructed transformation generates exact symmetries and a conserved charge in a restricted parameter limit.
Entanglement Entropy of Quantum Corners
hep-th 2025-07 conditional novelty 6.0 of 10

For a two-dimensional corner symmetry algebra, coherent corner states give an entanglement entropy that scales with the area when mapped to near-extremal Reissner-Nordström black holes.
Field-dependent diffeomorphisms and the transformation of surface charges between gauges
hep-th 2024-12 conditional novelty 6.0 of 10

The Weyl charge in (A)dS3 gravity is kinematical: it can be toggled on or off by a field-dependent diffeomorphism between Bondi-Sachs and Fefferman-Graham gauges.
Linearization (in)stabilities and crossed products
hep-th 2024-11 conditional novelty 6.0 of 10

Second-order Taub charge constraints are justified as linearization stability conditions in spatially closed spacetimes, while in boundary or no-isometry cases the justification is weaker and depends on interest in se...
Loop-corrected subleading soft theorem and the celestial stress-tensor
hep-th 2022-05 conditional novelty 6.0 of 10

The one-loop exact subleading soft graviton theorem follows from BMS charge conservation once the soft superrotation flux includes specific quadratic terms, yielding a celestial stress tensor with no divergent counterterms.
The partial Bondi gauge: Further enlarging the asymptotic structure of gravity
hep-th 2022-05 conditional novelty 6.0 of 10

A partial Bondi gauge unifies the Bondi-Sachs and Newman-Unti gauges and reveals an extra abelian radial translation symmetry in four dimensions, extending the Λ-BMSW group.
Carrollian Perspective on Celestial Holography
hep-th 2022-02 unverdicted novelty 6.0 of 10

A 3d sourced conformal Carrollian field theory is proposed to holographically capture 4d flat gravity kinematics, with Ward identities matching 2d celestial CFT after relating operators.
Embeddings and Integrable Charges for Extended Corner Symmetry
hep-th 2021-11 conditional novelty 6.0 of 10

Treating boundary embeddings as dynamical variables in the covariant phase space makes all extended corner symmetry charges integrable and gives a Poisson bracket representation without central extension.

See all 136 Pith citations

How later papers used it

357 later papers cite it

Evidence: substantial. 352 of the citing papers share no author with it, and 137 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 916 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

12 later papers by other authors are listed below: 9 used its method or tool, 2 compared their own results with it and 1 cited it as support.

Uses its method or tool 9 of 113 papers with this label

Calculated conserved charges and fluxes with the Wald-Zoupas formalism of this paper for BMS algebra.

Conserved charges in Chern-Simons modified theory and memory effects (2021) · 1.00 · paraphrased: the licence does not allow quoting
Used the Wald-Zoupas prescription of this paper, along with the covariant phase space formalism, to compute boundary charges and their fluxes.

Symmetries, charges and conservation laws at causal diamonds in general relativity (2019) · 1.00 · paraphrased: the licence does not allow quoting
Used the integral of the Noether charge 2-form given by Eq. (44) of this paper, the Komar formula, for the contribution to the charge.

Conserved charges of the extended Bondi-Metzner-Sachs algebra (2015) · 1.00 · paraphrased: the licence does not allow quoting
Used the covariant phase space formalism of this paper to show that the charge is associated to divergent large gauge transformations.

Asymptotic Symmetries in $(d+2)$-Dimensional Gauge Theories (2019) · 1.00 · paraphrased: the licence does not allow quoting
We list a set of criteria that these should satisfy to be physically reasonable and, using the prescription provided by Wald and Zoupas in [ref] , provide an explicit computation of the general expressions for these quantities.

The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity (2021) · Introduction · 0.99
Used covariant phase space methods, including those of this paper, to analyze the Hamiltonian phase space near the background metric.

Schwarzschild/CFT from soft black hole hair? (2018) · 0.99 · paraphrased: the licence does not allow quoting
Followed this paper to read off the (pre) symplectic potential current density and the symplectic form.

Lectures on the Infrared Structure of Gravity and Gauge Theory (2017) · 0.99 · paraphrased: the licence does not allow quoting
Since the background radiative electromagnetic field vanishes, the electromagnetic flux has no first-order contribution, $n̂^an̂^bδT̂^EM_abℐ^+=0$, the contribution of Eq. [ref] on the $ℐ^+$ gives [equation] Following the scheme in [ref] , after canceling the boundary terms in the expression above, the modified canonical energy can be defined as [equation] Consequently, the change in $ℰ'[Σ(t)]$ between $Σ(t_1)$ and $Σ(t_2)$ is [equation] The first and third terms are the gravitational contributions associated with horizon shear and Bondi news, respectively.

The entropy of dynamical black holes deviating from electrovacuum in second order (2026) · Modified Canonical Energy · 0.83
Identified the non-integrable piece as in this paper and treated it in the current algebra as in another work.

Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity (2013) · 0.96 · paraphrased: the licence does not allow quoting

Compares with it 2 of 31 papers with this label

Showed that the asymptotic limit of its own quasi-local BMS charges agrees with the BMS charges constructed by this paper at null infinity.

Quasi-local conserved charges in General Relativity (2019) · 0.99 · paraphrased: the licence does not allow quoting
Checked that its own result matches the one given by this paper, here specialized to Bondi coordinates and for a shear associated with the u-foliation.

Center-less BMS charge algebra (2024) · 0.99 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 5 papers with this label

Justified its viewpoint that the boundary location has no physical meaning by the fact that a transversal vector field of compact support generates a field variation lying in the kernel of the pre-symplectic two-form, as in this paper.

Generating functional for gravitational null initial data (2019) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

333 arXiv papers cite this in 866 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 535 sentences without a confident label

Contests it 2 papers

Otherwise, (94) of [ref] is missing an additional finite term coming from the $O(^2)$ terms of $ξ$.

Wald-Zoupas prescription with (soft) anomalies · Charges' archeology · contest 0.74
The fact that one can assign a unique symplectic form to the pair $(L,ℓ)$ is an important point which was overlooked in the early references, such as [ref] , where this type of changes in Lagrangian and symplectic potential are often considered to be ambiguities of the covariant phase space.

Extended corner symmetry, charge bracket and Einstein's equations · Boundary Lagrangian shift · contest 0.72

Builds on it 6 papers

Following with the discussion in Sec. [ref], [ref] and [ref], we may define [equation] The $ℰ_0$ is the divergent part from linear dilaton background, and $ℰ$ is the finite part that contributes to the first law of thermodynamics hence we may rewrite this as [equation] For more general case, we may generalise the Eq. ([ref]) and Eq. (20) in [ref] with [equation] Here the $ℰ_0$ and $𝒥_0$ describe the quantity defined by what we choose as background.

Weak cosmic censorship conjecture is not violated for a rotating linear dilaton black hole · Review of Wald's geometrical formulation · 0.83
In developing the field theoretical description of the boundary spinors, we will also develop further the covariant Hamiltonian formalism [ref] for complex Ashtekar variables in the presence of inner null boundaries, thereby generalising the isolated horizon framework [ref] to generic lightlike boundaries.

New boundary variables for classical and quantum gravity on a null surface · Introduction: Why spinors? · 0.96
We are proposing a new perspective on these questions which builds on the Wald–Zoupas approach [ref] and on the recent work of Chandrasekaran and Speranza [ref] .

Extended corner symmetry, charge bracket and Einstein's equations · Introduction · 0.81

Uses its method or tool 102 papers

The prescription of Wald and Zoupas [ref] provides a method of determining charges and fluxes at null infinity, and can be applied to any local and covariant theory.

The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity · Asymptotic charges and fluxes: The Wald-Zoupas prescription · 0.90
Its value on the desired solution will be our definition of the mass of the metric [We are specifying here to a particular symmetry generator since we are interested in the mass, but the same procedure con be applied to the most general asymptotic symmetry generator [ref] .] $ℋ_ξ$ is defined via its variation with respect to a generic fluctuation $δϕ$, obeying the linearised equations of motion in a given background obeying the full equations of motion [ref] : [equation] where $Σ$ is a 9 dimensional submanifold of the spacetime without boundary, a “slice” corresponding to the vector field $ξ$.

Half BPS states in AdS_5 x Y^{p,q} · Definition of charges in asymptotically $AdS_5 ×X^5$ · 0.99
In this paper, we apply the prescription set out in Ref. [ref] , which uses the covariant phase space formalism [ref] to propose a systematic method for determining, in principle, all possible gravitational charges, to give a Hamiltonian derivation of a recently discovered tower of dual BMS charges [ref] .

Hamiltonian derivation of dual gravitational charges · Introduction · 0.98

Compares with it 28 papers

These covariant expressions are equal to those given by [ref] , and are therefore independent of foliation; however, picking a choice of foliation allows us to write the charge in terms of quantities defined entirely at null infinity.

The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity · Introduction · 0.92
For vacuum GR, the charge expression <ref> coincides with the charges defined by Wald and Zoupas [ref] .

Angular momentum at null infinity in Einstein-Maxwell theory · Computation of the Wald-Zoupas charges and fluxes at null infinity for Einstein-Maxwell theory · 0.95
Here we show that the asymptotic limit of the quasi-local BMS charges [ref] agrees with the BMS charges constructed by Wald and Zoupas [ref] at null infinity.

Quasi-local conserved charges in General Relativity · Asymptotic behaviour · 0.99

Cites it as support 3 papers

In general relativity it can be shown that the $3$-form $θ(δĝ)$ is finite at  [ref] and thus $(Ω^2 W) ·θ(δĝ) 0$.

The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity · Asymptotic charges and fluxes: The Wald-Zoupas prescription · 0.97
This viewpoint is motivated by background invariance, which implies that the location of the boundary with respect to a given coordinate system has no physical meaning, and is justified by the fact that a transversal vector field $ξ^a∈Tℳ:ξ^a|_𝒩∉T𝒩$ of compact support on the null boundary generates a field-variation $δ_ξ[·]=ℒ_ξ(·)$ on the covariant phase space that lies in the kernel of the pre-symplectic two-form [ref] .

Generating functional for gravitational null initial data · Field strength, constraints, evolution equations · 1.00
This interpretation is supported by explicit examples such as the Kerr-Newman and Vaidya metric, the Landau-Lifschitz approach to defining balance laws [ref] and put on a firm ground by an analysis of the covariant phase space of asymptotically flat spacetimes [ref] .

BMS-like symmetries in cosmology · Conformal Bondi-Sachs coordinates in the unphysical spacetime · 0.98

Cites it as background 104 papers

The seminal Wald-Zoupas (WZ) paper [ref] provides a prescription for the gravitational charges in the two cases of conservative boundary conditions and of leaky boundary conditions, with the latter that make the system non-conservative and thus the infinitesimal Hamiltonian generators non-integrable.

Wald-Zoupas prescription with (soft) anomalies · Introduction · 0.97
It is well known that $Θ(𝒩)$ is subject to the so-called $W$ and $Y$ freedoms (or ambiguities) [ref] .

Dynamical Entropy Is a Noether Charge · Introduction · 0.80
They were first derived in [ref] using the Barnich–Brandt formalism [ref] and later in [ref] from a split following the prescription laid out by Wald–Zoupas [ref] .

The Weyl BMS group and Einstein's equations · BMSW group and Einstein's equations · 0.85

Discussion

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