ADM, Bondi, Komar, and Quasilocal Charges

Gravitational charges are boundary constructions fixed by a phase space, surface, generator, and reference. ADM, Bondi, Komar, covariant-Hamiltonian, and quasilocal quantities are not interchangeable; equality requires overlapping domains and matching hypotheses. The Asymptotic Symmetries and Gravitational Charges chapter gives the asymptotic setting; Komar, ADM, and Covariant Charges Compared gives the common construction and equality conditions.

Helpful background. Causality and Global Structure supplies infinity; Presymplectic Potentials and Currents supplies variations; Hypersurface Normals and Tangential Projectors supplies surface geometry; Killing Fields and Isometry Groups supplies generators; and The Gibbons–Hawking–York Boundary Term supplies boundary terms.

Data that fix a charge

For every charge, state the allowed fields and boundary conditions, oriented surface, normalized generator, reference zero, and whether the result is integrable, conserved, or governed by a flux law.

Use the site orientation $\epsilon_{0123}=+\sqrt{-g}$. On a spacelike slice with future unit normal $n^a$ and outward spatial normal $s^a$, orient a closed two-surface by

$$\epsilon^{(S)}_{ab}=\epsilon_{abcd}n^cs^d.$$

Reversing this orientation reverses every integral. The signature is $(-,+,+,+)$ and $G$ is explicit.

For a covariant Lagrangian four-form,

$$\delta\boldsymbol L =\boldsymbol E\mathbin{\cdot}\delta\phi +d\boldsymbol\Theta(\phi;\delta\phi),$$

use the GR Formulations ordering

$$\omega(\delta_1,\delta_2) =\delta_2\Theta(\delta_1)-\delta_1\Theta(\delta_2).$$

The asymptotic-symmetry treatment and horizon-Hamiltonian treatment use the opposite ordering $\boldsymbol\omega_{\rm alt}=\delta_1\boldsymbol\Theta(\delta_2) -\delta_2\boldsymbol\Theta(\delta_1)=-\boldsymbol\omega$. Translating a balance law reverses flux or bracket terms linear in $\boldsymbol\omega$, not the surface orientation or $\delta Q_\xi-\xi\mathbin{\lrcorner}\Theta$ Iyer and Wald 1994, §§ 2–4.

ADM and Bondi charges

For an asymptotically Euclidean end with asymptotically Cartesian coordinates, take

$$\gamma_{ij}=\delta_{ij}+O(r^{-1}), \qquad \partial_k\gamma_{ij}=O(r^{-2}), \qquad K_{ij}=O(r^{-2}),$$

with constraints, derivative control, and a declared parity class. With $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$, the time- and space-translation charges are

$$\begin{aligned} E_{\rm ADM} &=\frac{1}{16\pi G}\lim_{r\to\infty} \int_{S_r} (\partial_j\gamma_{ij}-\partial_i\gamma_{jj})s^i\,dS,\\ P_i^{\rm ADM} &=\frac{1}{8\pi G}\lim_{r\to\infty} \int_{S_r}(K_{ij}-K\gamma_{ij})s^j\,dS. \end{aligned}$$

Rotations and boosts require stronger moment and parity control. See Henneaux and Troessaert 2020, §§ 2–5 for falloffs, parity, and the proper-gauge quotient, and Poincaré Symmetry and ADM Charges.

At future null infinity, choose a retarded Bondi frame, a cut $S_u$, and the unit-sphere metric $q_{AB}$. In the site convention,

$$\begin{gathered} g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad q^{AB}C_{AB}=0, \qquad N_{AB}=\partial_uC_{AB},\\ g_{uu}=-1+\frac{2m(u,x^A)}{r}+O(r^{-2}), \qquad T_{uu}=\frac{T_{uu}^{(2)}}{r^2}+O(r^{-3}),\\ M_B(u)=\frac{1}{4\pi G}\int_{S^2}m\,d\Omega. \end{gathered}$$

The asymptotic constraint gives the time-translation balance law

$$\frac{dM_B}{du} =-\frac{1}{32\pi G}\int_{S^2}N_{AB}N^{AB}\,d\Omega -\int_{S^2}T_{uu}^{(2)}\,d\Omega.$$

Positive outward flux lowers $M_B$; translate the metric, news, measure, and coefficient together Compère 2019, “Asymptotically Flat Spacetimes,” pp. 81–102. See Bondi Mass Aspect and the Asymptotic Constraint for the derivation. BMS charges also depend on cut and vacuum frame; Supertranslations and Angular-Momentum Ambiguity states the resulting ambiguity.

Komar, Hamiltonian, and isolated-horizon charges

Let $t^a$ be a stationary Killing field unit-normalized at infinity, and let $\phi^a$ be an axial Killing field with period $2\pi$. With the orientation fixed above,

$$M_{\rm K}=-\frac{1}{8\pi G}\int_S\star dt^\flat, \qquad J_{\rm K}=\frac{1}{16\pi G}\int_S\star d\phi^\flat.$$

The unequal factors are essential. Moving the surface requires the Killing identity and field equations, with matter and gauge-field bulk terms retained ADM, Komar, and Horizon Charges.

For Einstein gravity the Noether two-form is

$$\boldsymbol Q_\xi=-\frac{1}{16\pi G}\star d\xi^\flat,$$

but the Hamiltonian variation is, when integrable,

$$\delta H_\xi =\int_S(\delta\boldsymbol Q_\xi -\xi\mathbin{\cdot}\boldsymbol\Theta).$$

Before integrability, the right side is the one-form $\not\!\delta H_\xi$; finiteness, integrability, and a reference are required. For stationary time translation, $\int Q_t$ is half the normalized Komar mass; the boundary term restores the Hamiltonian mass Iyer and Wald 1994, § VI, Eqs. (83)–(90), arXiv PDF pp. 15–16.

For rationalized Maxwell density $-F_{ab}F^{ab}/4$, orient the surface so that a positive hole has positive outward flux and define

$$Q_S=\int_S\star F.$$

This is the rationalized flux convention. The charged exact-spacetime records use Gaussian variables; the complete Gaussian-to-rationalized Maxwell map preserves the geometric charge parameter and electromagnetic work term.

On an axisymmetric isolated-horizon section, let $D_a\ell^b=\omega_a\ell^b$ and give $\varphi^a$ period $2\pi$. Then

$$J_S[\varphi] =-\frac{1}{8\pi G}\int_S (\varphi\mathbin{\cdot}\omega)\,\epsilon -\int_S(\varphi\mathbin{\cdot}A)\star F.$$

The first term survives $\ell\mapsto f\ell$ on a closed axisymmetric section; the second needs a symmetry-preserving gauge or patch data. Without an axial field $J_S$ is not unique Ashtekar and Krishnan 2004, §§ 2–4. See Quasi-Local Horizon Angular Momentum and Charge.

Comparison by domain

Construction	Defining data	Statement	Failure boundary
ADM	Spatial-end spheres, asymptotic Poincaré generator, origin/frame, Minkowski zero	Constraints and falloff make the limit surface independent	Wrong decay, parity, coordinates, or end
Bondi–BMS	Null-infinity cut, Bondi frame, BMS generator, reference vacuum	Differences equal flux; radiative charges need an integrability prescription	Irregular infinity, unmatched frame, or omitted news/corner terms
Komar	Closed surface and exact Killing field with fixed normalization	Surface-independent only through a source-free region; otherwise two surfaces differ by bulk Ricci or stress-energy terms	Missing symmetry or normalization, or an omitted bulk source term
Covariant Hamiltonian	Boundary cut, covariant phase space, allowed generator, reference	A finite variation becomes a charge only when integrable	Ambiguity, nonintegrability, or inadmissible generator
Isolated horizon	Horizon section, null-normal class, axial field, gauge, reference	Quasi-local conservation or a declared flux law	No axial field, or missing dynamical-horizon foliation/flux data

No row defines a local gravitational-energy density. Invariance is only under transformations allowed by the boundary or horizon data.

A bounded quasi-local mass map

For a closed surface $S$ in a time-symmetric three-slice, let $s^i$ be its outward unit normal, $H=D_i s^i$, and $A$ its area. Use the same outward-normal convention for the Euclidean reference curvature $H_0$. Then

$$\begin{aligned} m_H(S) &=\frac{1}{G}\sqrt{\frac{A}{16\pi}} \left(1-\frac{1}{16\pi}\int_SH^2\,dA\right),\\ m_{\rm BY}(S) &=\frac{1}{8\pi G}\int_S(H_0-H)\,dA, \end{aligned}$$

where $H_0$ comes from a specified Euclidean reference embedding Brown and York 1993, § IV, Eq. (4.8), printed pp. 1412–1414. Bartnik mass is an infimum of ADM masses over a declared admissible-extension class. The horizon-area specialization is

$$M_{\rm irr}=\frac{1}{G}\sqrt{\frac{A}{16\pi}}$$

in four dimensions.

Mass	Extra structure	Main limitation
Hawking	Surface embedding/null expansions, or the displayed time-symmetric data	Positivity and monotonicity are not universal
Brown–York	Fill-in and reference embedding defining $H_0$	Depends on fill-in, embedding, and reference
Bartnik	Admissible extensions, asymptotic end, scalar-curvature and horizon rule	Changing admissibility changes the infimum
Horizon or irreducible	Selected black-hole section, plus spin/charge when needed	Not a mass of an arbitrary finite region

The exact hypotheses come from Huisken and Ilmanen 2001, Main Theorem and §§ 1–4, printed pp. 355–401, Shi and Tam 2002, Theorem 4.2, pp. 117–118, and McCormick 2024, §§ 2–7, printed pp. 458–481. Quasi-Local Mass and Horizon Inequalities supplies the theorem boundaries. There is no universal ordering: these quasilocal definitions are not interchangeable.

Equality and failure checks

Flat reference. ADM, Bondi, Komar, and reference-normalized Hamiltonian charges vanish in Minkowski spacetime; a Euclidean round sphere has $m_H=m_{\rm BY}=0$.

Stationary Schwarzschild. With common time, orientation, $G$, and reference, ADM, Komar, Hamiltonian, zero-news Bondi, and centered Hawking masses equal $M$. Finite-radius Brown–York mass tends to $M$ only at infinity Brown and York 1993, § VI, Eq. (6.14), printed pp. 1415–1417.

Radiation or exterior matter. Bondi mass changes with flux; Komar surfaces differ by bulk terms; horizon and infinity charges can differ. Equality needs the field equations, matching conditions for generator/frame/reference/gauge/ orientation, and every intervening flux.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. DOI.
• Brown, J. David, and James W. York Jr. “Quasilocal Energy and Conserved Charges Derived from the Gravitational Action.” Physical Review D 47 (1993): 1407–1419. DOI.
• Compère, Geoffrey. Advanced Lectures on General Relativity. Lecture Notes in Physics 952. Cham: Springer, 2019. DOI.
• Henneaux, Marc, and Cédric Troessaert. “The Asymptotic Structure of Gravity at Spatial Infinity in Four Spacetime Dimensions.” Proceedings of the Steklov Institute of Mathematics 309 (2020): 127–149. DOI. arXiv:1904.04495.
• Huisken, Gerhard, and Tom Ilmanen. “The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality.” Journal of Differential Geometry 59 (2001): 353–437. DOI.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994): 846–864. DOI. Open PDF.
• McCormick, Stephen. “An Overview of Bartnik’s Quasi-Local Mass.” Beijing Journal of Pure and Applied Mathematics 1 (2024): 455–487. DOI. arXiv:2401.05128.
• Shi, Yuguang, and Luen-Fai Tam. “Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature.” Journal of Differential Geometry 62 (2002): 79–125. DOI.