Quasi-Local Gravitational Energy

Sean A. Hayward

REVIEW

Quasi-Local Gravitational Energy

Sean A. Hayward

General Relativity and Quantum Cosmology · Submitted 26 Mar 1993 · 1 reference

Pith's verdict: The functional is explicit and the flat-space fix is right; the 'dynamically preferred' label leans on two in-press self-citations and the twist term is never exercised by a single worked example.

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arXiv IDgr-qc/9303030 (version 1)

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CitationSean A. Hayward. "Quasi-Local Gravitational Energy." arXiv:gr-qc/9303030 (1993). https://pith.science/paper/PM7DM4WB

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Editor's note

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Hayward's E is solid in one respect and a promissory note in another, and it's worth being clear which is which before you spend time on it. The functional itself is explicit, depends only on the fundamental forms, and the flat-space vanishing in §4 is a two-line consequence of the contracted Gauss equation. The shear term is exactly the correction the Hawking mass needed, and that is a clean fix, not a cosmetic one. The Schwarzschild, Reissner-Nordström and Robertson-Walker evaluations are short and I checked them. The appendix is the part I would actually hand to a student: it works out why the Bondi mass comes out as a limit of E along an area-radius foliation but as a limit of the Hawking mass along an affine one, and it is candid that the answer depends on the choice. That foliation-dependence point is genuinely clarifying and I don't think it had been stated this cleanly. Where I would push back is the framing. 'Dynamically preferred' is the headline claim and it rests on the Hamiltonian 2-form quoted from Hayward 1993a, in press, with one line of justification here. The uniqueness argument in §7 is explicitly informal and concedes the twist coefficient is undetermined, deferred to Hayward 1993c, also in press. And every worked example — Schwarzschild, Reissner-Nordström, Robertson-Walker, plane waves, colliding plane waves — has vanishing twist, so the twist sector is never tested by anything in this paper. That is a bounded problem: the demonstrated properties don't depend on it, but the label does. Two smaller things. The displayed equation in §3, E√A = E1√A1 + E2√A2, is wrong; the additive quantity is E/√A, though the inequality E > E1 + E2 that follows from it is correct. And I would not call the §7 calibration circular: fixing coefficients on flat space and Schwarzschild and then recovering Bondi and ADM is a real check, not a restatement of the inputs. Who should read it: anyone working on quasi-local mass, and anyone who needs the area-radius versus affine distinction straight. The §10 criteria list is opinionated but useful as a map of the field. Worth a referee.

Referee report

Significance

If the stated properties hold, this is a well-defined, parameter-free quasi-local energy with the correct asymptotic limits and the correct values on the standard examples — a real advance over the Hawking mass, which is negative on generic surfaces in flat spacetime, and over definitions that require a symmetry, a preferred 3-surface, or global data. The worked examples are short enough to check by hand, and I verified the Reissner–Nordström, Robertson–Walker and torus normalisations independently; the Appendix's reconciliation of the affine and area-radius foliations, and its explanation of why Penrose's expression for the Bondi mass is a limit of the Hawking mass along a different foliation, is careful and useful work. The weakness is that the headline claim of dynamical preference rests on a Hamiltonian 2-form quoted from an in-press companion paper, and the twist sector is neither derived from the paper's own criteria nor exercised by any example.

Major objections (4)

[§3 and abstract:s0] The paper's title-level claim — that E is 'dynamically preferred' — rests on the Hamiltonian 2-form 8πH = −μ(R + θθ̃ − ½σ_ab σ̃^ab − 2ω_aω^a), which is quoted from Hayward 1993a (Class. Qu. Grav. 10, in press) with the single justification that 'variation of the (full) Hamiltonian yields the Einstein equations in a form adapted to null foliations of spatial 2-surfaces'. Because the cited source is not yet available to readers, and because this is the claim that distinguishes the paper from a further entry in the existing catalogue of quasi-local energies, please add a short appendix sketching the derivation: the 2+2 form of the Einstein–Hilbert Lagrangian, the reduction up to a total divergence, and the variation that yields the field equations. At minimum, state the variational principle precisely enough that a reader can verify it without the companion paper. Relatedly, §3 itself notes that 'when considering the variation of energy between surfaces, the quasi-local coordinate freedom cannot be fixed as in §2 and the full Hamiltonian should be used', but no such variation is computed anywhere in the paper, and §9 concedes that a monotonicity result for E 'seems more difficult to obtain'. As it stands the dynamical justification is programmatic rather than demonstrated.
[§7] The uniqueness argument assumes that E is a boost- and interchange-invariant functional that is second degree in g_ab, and then asserts that 'the only such functions are linear combinations of R, θθ̃, σ_ab σ̃^ab and ω_aω^a'. That classification is not proved, and the section then concedes that the coefficient of ω_aω^a is 'not so clearly determined', deferring its justification to Hayward 1993c (in press). Since every worked example in §8 has ω_a = 0, the twist term is never tested anywhere in the paper, so a reader cannot assess whether the coefficient −2 is correct. Please either prove the classification or present it explicitly as an ansatz, and temper the concluding sentence ('It is certainly remarkable that E automatically satisfies these desiderata') to reflect what has actually been shown.
[§5] The Bondi limit depends on the falloff ω_aω^a = O(r^{-6}), which is precisely what makes the twist term drop out of the limit, and this is asserted as a 'translation from the spin-coefficient expressions (e.g. §9.8 of Penrose and Rindler 1986)' without derivation. The anholonomicity ω_a is not one of the standard Newman–Penrose coefficients, so the translation is not immediate; please supply the argument. It would also help to state explicitly that R + θθ̃ = O(r^{-3}) follows from the vacuum Gauss equation, so that the leading r^{-2} terms in the two expansions cancel by construction rather than by coincidence.
[§8] The paper finds E = 0 for every plane wave, justified only by the remark that 'plane waves propagate linearly'. A pp-wave is an exact vacuum solution with non-zero Weyl curvature, so the vanishing of E on it is a substantive claim about what E measures, not a triviality. Please give a sharper argument — for instance, that a 2-surface transverse to the propagation direction sees no flux — or compare E against a Bel–Robinson-type energy density for these solutions. The same request applies with more force to the colliding plane wave case, where the interpretive claim (E < 0 after the collision, unbounded at the singularity) is left unchecked because 'the exact expression is long and unilluminating'.

Minor objections (7)

[§3] The additivity equation is incorrect as written. Since E = (1/8π)√(A/16π)∫_S μ(...), the additive quantity is E/√A, so the relation should read E/√A = E1/√A1 + E2/√A2. The stated inequality E > E1 + E2 still follows, because √A > √A1 and √A > √A2, so the interpretation as a binding energy survives; only the displayed equation needs correcting.
[§2–§3] It would help to state explicitly that ω_a is invariant under the boost freedom left open in §2: under l → e^λ l, n → e^{-λ} n the extra terms in the commutator are proportional to l and n and are annihilated by h_a^b. This one-line observation is what makes E independent of the boost choice, and the paper currently leaves it implicit.
[§8] For the flat torus with plane-symmetric identifications, ∫_S μR = 8π(1−g) vanishes for g = 1, so the coefficient of R — calibrated on spheres — does not enter the torus result at all. The torus value is therefore a prediction with no independent check, and it would be worth saying so. I have checked the normalisation of the displayed torus expression and it is correct.
[§9] The result 16πE² = A on a static Killing horizon presumably extends to stationary horizons, since the rigidity theorem gives a shear-free horizon and the argument only uses the vanishing of the shear and twist. Please say so, or say why the argument is restricted to the static case.
[§10] The sentence 'To the best of my knowledge, this leaves only the Hawking mass as an unambiguous quasi-local definition' reads as though E does not exist. Please clarify that the survey covers prior definitions and that E is exempt from the disqualification being applied.
[§10] The Newtonian and linearised limits are listed as 'of interest to investigate'. A short check that E reduces to the Newtonian mass in the weak-field limit would strengthen the case considerably and is presumably straightforward given the spherical-symmetry formula E = r(r_ξ r_η + ½).
[References] Hayward 1993a and Hayward 1993c are both listed as 'in press'. Please give the preprint number for 1993c (gr-qc/9303006) so that readers can consult the argument on which the twist coefficient depends.

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The paper

The paper claims

Hayward proposes a single formula for the gravitational energy of any closed spatial two-surface, built from the Hamiltonian of a null-foliation ('2+2') formulation of general relativity, and shows it vanishes in flat spacetime, reproduces$

The reading

General relativity has no agreed local density of gravitational energy: the only broadly accepted totals are the asymptotic masses measured at infinity, Bondi's at null infinity and ADM's at spatial infinity. Hayward asks what a finite region weighs, taking the region to be a compact orientable spatial two-surface, and proposes to read the answer off the Hamiltonian of a '2+2' formulation of Einstein's equations, in which null surfaces are propagated off the two-surface in two independent null directions. The resulting energy depends only on the data a surface actually carries: its induced area element, its two null second fundamental forms (expansion and shear in each null direction), and its twist, the commutator of the null normals. If the construction holds up, one expression does the work of several: it gives the Schwarzschild mass on a sphere of symmetry, reduces to the Hawking mass when shear and twist vanish, tends to the Bondi mass at null infinity and the ADM mass at spatial infinity along a foliation labelled by area radius, and returns the irreducible mass on a static black-hole horizon. The author also states plainly what the argument does not yet settle, namely the coefficient of the twist term, which is fixed by appeal to a separate paper rather than by the criteria used for the other terms.

What carries the argument

The load-bearing object is the Hamiltonian 2-form of the '2+2' (dual-null) formulation, $8pi H = -mu(R + theta tilde{theta} - tfrac{1}{2}sigma_{ab}tilde{sigma}^{ab} - 2omega_aomega^a)$, obtained from the Einstein-Hilbert Lagrangian rewritten in a basis of two null evolution vectors plus two tangent vectors to the surface, up to a total divergence. Two geometric identities do the bookkeeping: the contracted Gauss equation $R + theta tilde{theta} - tfrac{1}{2}sigma_{ab}tilde{sigma}^{ab} = h^a{}_c h^b{}_d R_{abcd}$, which turns the integrand into a curvature contraction and makes the flat-space vanishing immediate, and Gauss-Bonnet, $int_S mu R = 8pi(1-g)$, which separates the topological genus from the dynamical content. The single length scale needed to convert the integrated Hamiltonian into energy units is supplied by the area, which is why $sqrt{A/16pi}$ appears and why the Schwarzschild normalisation comes out right.

Extended reading notes

Core claim

The paper defines the quasi-local energy of a compact orientable spatial two-surface $S$ as $$E = -frac{sqrt{A}}{16pi}int_S H = frac{1}{8pi}sqrt{frac{A}{16pi}}int_S mu left(R + $\theta$ tilde{$\theta$} - tfrac{1}{2}sigma_{ab}tilde{$\sigma$}^{ab} - 2omega_a omega^aright),$$ where $H$ is the Hamiltonian 2-form of the '2+2' formulation, $A$ is the area of $S$, $mu$ its area 2-form, $R$ the Ricci scalar of the induced metric, $theta,tilde{theta}$ the two null expansions, $sigma_{ab},tilde{sigma}_{ab}$ the two shears, and $omega_a$ the twist. The claim is that this is a dynamically preferred quantity rather than an ad hoc guess, because varying the full Hamiltonian returns the Einstein equations adapted to null foliations of two-surfaces. The supporting checks are that $E$ vanishes on every surface in flat spacetime (the shear term is exactly the correction that removes the known negative flat-space values of the Hawking and Geroch masses), that it equals the Hawking mass when shear and twist vanish, that it evaluates to $m$ for Schwarzschild, $m - tfrac{1}{2}e^2 r^{-1}$ for Reissner-Nordström, $tfrac{4}{3}pi r^3 rho$ for Robertson-Walker, zero for plane waves and negative for the Khan-Penrose colliding plane wave, and that on a static horizon it satisfies $16pi E^2 = A$, making it the irreducible mass.

Load-bearing premise

The coefficient of the twist term, $-2omega_a omega^a$, is not fixed by any of the paper's stated criteria: section 7 concedes that flat-space vanishing plus the Schwarzschild and Bondi/ADM limits determine the coefficients of $R$, $theta tilde{theta}$ and $sigma_{ab}tilde{sigma}^{ab}$ but leave the twist coefficient open, justified only by reference to a separate in-press paper on black-hole dynamics.

Editorial extensions

If this is right

A single functional unifies the null-infinity and spatial-infinity limits: $M_B = \lim_{r\to\infty} E$ along an area-radius foliation and $M_{\mathrm{ADM}} = \lim_{\rho\to\infty} E$, so the Bondi and ADM masses become two limits of one quasi-local quantity rather than unrelated definitions.
The long-standing failure of the Hawking mass to vanish on generic surfaces in flat spacetime is repaired by a specific shear term dictated by the Gauss equation, with no free parameter introduced for that purpose.
On a Killing horizon the twist and internal shear vanish, so $16\pi E^2 = A$ holds exactly, giving the irreducible mass of a static black hole directly from the quasi-local definition.
Quasi-local energy is not monotonic and need not be positive: it is zero for plane waves, unbounded and negative at the singularity of a colliding plane wave, and negative inside the inner horizon of a charged black hole, so the definition tracks non-linear gravitational interaction rather than a conserved charge.
For a disconnected surface $S = S_1 \cup S_2$ the energy satisfies $E > E_1 + E_2$, giving a positive binding energy and making coalescence energetically favourable.

Reading between the lines

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

The definition supplies a ready-made diagnostic for numerical relativity: tracking $E$ across a foliation of surfaces during a black-hole merger would test whether the binding-energy inequality is realised dynamically, since the paper only establishes it algebraically for disjoint surfaces.
Because $E$ is built from the same data that characterise trapped surfaces, it sits naturally alongside horizon-finding algorithms: the marginal-trapping condition $\theta\tilde{\theta}=0$ turns the energy formula into a relation between $E$, the shear and the twist on the apparent horizon, which is a checkable identity in any numerical simulation.
The twist coefficient is the one place where the construction could be repaired or broken without touching the rest, so a natural extension is to test boost invariance of $E$ on a surface carrying both shear and twist; that is a purely algebraic computation in the '2+2' variables.
The paper's remark that $E$ controls the signature and trapping of singularities in spherical symmetry, and its two-dimensional dilaton analogue, hints at a broader use of the same Hamiltonian as a censorship diagnostic rather than only as a mass.

Original abstract

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.34 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.

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	A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity.	contribution 0.98	0.91	0.63	asserted 1.00
s1	The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only.	result 0.57	0.80	0.13	asserted 1.00
s2	The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist.	result 0.99	0.76	0.17	asserted 1.00
s3	For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius.	result 0.98	0.81	0.18	asserted 1.00
s4	The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves.	result 0.48	0.90	0.07	asserted 1.00
s5	Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon.	result 0.89	0.80	0.11	asserted 0.96
s6	Criteria for an adequate definition of quasi-local energy are discussed.	contribution 0.48	0.63	0.06	unclear 0.76

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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References

Forward citations

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How the literature uses it

32 arXiv papers cite this in 45 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 · 33 sentences without a confident label

Uses its method or tool 3 papers

In this space-time, the physically relevant mass is the Hawking-Hayward quasi-local mass [ref] [equation]

Dark Energy Accretion onto a Black Hole in an Expanding Universe · the Solution with Imperfect Fluid and Radial Mass Flow · 0.84
To probe horizon formation, we compute the quasi‐local Hawking mass [ref] M(T,x) ≡r̅^2- (∇r̅)^2, r̅ =e^- T r (T,x).

Quantum Critical Collapse Abhors a Naked Singularity · Introduction · 0.99
With the help of the Gauss-Codazzi equation for codimension-$2$ spatial surfaces [ref] , [equation] we get

Dynamics and Observer-Dependence of Holographic Screens · Cross-focusing Equations · 0.98

Compares with it 1 paper

Now focusing on the first term on the RHS of ([ref]), we identify it as the variation of the energy associated with the null surface $S_t$ under the virtual displacement $δλ_(k)$, i.e [equation] Performing an indefinite integration over $λ_(k)$ allows us to have an expression for the energy associated with the $2$-surface $S_t$, [equation] Before proceeding ahead, we note that the expression of the energy as obtained in ([ref]) is reminiscent of the Hawking-Hayward energy definition [ref] .

Covariant approach to the thermodynamic structure of a generic null surface · $R_abl^ak^b$, favourable candidate for thermodynamic interpretation: a covariant approach · 0.97

Cites it as background 8 papers

The Hawking-Hayward quasilocal mass [ref] reduces to the Misner-Sharp-Hernandez mass in spherical symmetry and is the Noether charge associated with the conservation of the Kodama current [ref] .

The curious case of the Buchdahl-Land-Sultana-Wyman-Iba\~nez-Sanz spacetime · Quasilocal mass · 0.80
There is a host of quasi-local definitions, see e.g. Ref. [ref] and the works discussed there.

Conserved charges in (Lovelock) gravity in first order formalism · Quasi-local generators and conservation · 0.99
The irreducible mass is closely related to another quasilocal measure of mass: the Hawking mass [ref] , M_H := M_irr(1+1/8π∫_H_v ϑ_+ ϑ_- √(γ)d^2θ).

Deformed Schwarzschild horizons in second-order perturbation theory: mass, geometry, and teleology · Surface area and mass · 0.86

Discussion

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[1] [1]

Arnowitt R, Deser S & Misner C W 1962 in Gravitation, an Introduction to Current Research ed Witten (Wiley) Ashtekar A 1980 in General Relativity and Gravitation ed Held (Plenum) Ashtekar A & Hansen R O 1978 J. Math. Phys. 19 1542 Ashtekar A & Horowitz G T 1982 Phys. Lett. 89A 181 Ashtekar A & Magnon-Ashtekar A 1979 Phys. Rev. Lett. 43 181 Bartnik R 1989 ...

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