ADM Mass, Momentum and Angular Momentum

General relativity has a scandal at its heart: there is no honest local energy density for the gravitational field. The equivalence principle lets a freely falling observer transform the field away at any single point, so any candidate for "energy per unit volume" of gravity can be made to vanish or reappear by a change of coordinates. For an isolated system — a star, a black hole, a binary — this looks fatal: what, then, is the total mass-energy? Arnowitt, Deser, and Misner answered by moving the question to the edge of the world. On a spacelike slice $\Sigma$ that becomes flat at large radius, the total energy, momentum, and angular momentum are read off not from the interior but from surface integrals over a 2-sphere pushed to spatial infinity, where a rigid flat background exists to measure deviations against. The construction is the gravitational face of Noether's theorem: each asymptotic symmetry of flat space — time translation, spatial translation, rotation — hands back one conserved Poincaré charge. The ADM mass is the coefficient of the $1/r$ dent the body presses into the spatial metric, exactly the Newtonian potential a distant astronomer would infer from orbits; the momentum and spin hide instead in the $1/r^2$ falloff of the extrinsic curvature. These are not bookkeeping conveniences. The positive energy theorem — that the ADM four-momentum is future-pointing and timelike unless spacetime is empty — is what makes flat space stable and the whole theory sane, and every numerical relativity waveform ever matched against LIGO data is certified by the ADM charges of its initial slice. Throughout we set $G=c=1$ and use signature $(-,+,+,+)$ .

Key takeaways

For an asymptotically flat spacetime the total energy, linear momentum, and angular momentum cannot be localized; they are defined by surface integrals over a 2-sphere at spatial infinity $i^0$ .
The ADM mass $M$ measures the leading $1/r$ deviation of the spatial metric $\gamma_{ij}$ from flatness — equivalently, the Hamiltonian boundary term generating asymptotic time translations.
Linear momentum $P_i$ and angular momentum $J_i$ come from the $1/r^2$ falloff of the extrinsic curvature $K_{ij}$ , generating asymptotic spatial translations and rotations.
The charges are conserved on a slicing and gauge invariant under coordinate changes that preserve the asymptotic structure; for isotropic Schwarzschild the integral returns exactly $M_{\mathrm{ADM}}=M$ .
The positive energy theorem of Schoen-Yau and Witten gives $M \geq |\vec P| \geq 0$ , so $P^\mu=(M,P_i)$ is future-pointing timelike, with equality only for Minkowski space.
ADM mass lives at spatial infinity and is conserved; Bondi mass lives at null infinity and bleeds away exactly the radiated gravitational-wave energy — GW150914 lost $\approx 3\,M_\odot c^2 \approx 5\times10^{47}$ J.
Key idea: Gravitational energy-momentum is not a local density but a boundary charge — each asymptotic symmetry of flat space (time translation, spatial translation, rotation) yields a conserved quantity as a 2-surface integral at spatial infinity, built from the falloff of $\gamma_{ij}$ and $K_{ij}$ .

The core insight

Gravitational energy-momentum is not a local density but a boundary charge: each asymptotic Killing-like symmetry of flat space — time translation, spatial translation, rotation — yields one conserved quantity as a 2-surface integral at spatial infinity, built from the falloff of the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ .

Where this lives today

ADM charges are the daily currency of numerical relativity. Every binary-black-hole or neutron-star simulation begins by solving the constraint equations for initial data, and the very knobs of that construction are ADM quantities: Bowen-York and puncture data are parametrized directly by the linear momenta $P_i$ and spins $J_i$ one wishes to impose, and the ADM mass is computed from the surface integral (2) at large radius to certify that the data describe the intended physical system before a single time step is taken. During evolution the ADM mass of the initial slice sets the total energy budget, and comparing it against the final remnant mass plus the radiated energy — extracted from the Newman-Penrose scalar $\Psi_4$ or via Cauchy-characteristic extraction of the Bondi mass at $\mathscr{I}^+$ — is one of the most stringent global checks a code can pass; the change in ADM linear momentum diagnoses the gravitational-wave "kick" that can eject a merged black hole from its host galaxy at thousands of km/s. Collaborations such as SXS and the Einstein Toolkit report these charges for every waveform in the template banks that LIGO-Virgo-KAGRA match against data, so the analysis of GW150914 and every event since rests on ADM bookkeeping. On the mathematical side the positive mass theorem remains an active frontier — its proof was part of the work that earned Yau the Fields Medal, and generalizations (Penrose inequality, quasi-local mass definitions of Brown-York, Hawking, and Wang-Yau) continue to occupy mathematical relativists seeking a satisfactory local notion of gravitational energy that the ADM construction, by its nature, cannot provide.

The physical picture

Picture an isolated body — a neutron star, a black hole — sitting alone in otherwise empty space. Very far away the geometry is nearly flat, marred only by a faint $\sim 2M/r$ dent in the time-time part of the metric, precisely the Newtonian potential of a mass $M$ . The ADM mass reads off the strength of that $1/r$ tail through a flux integral over a sphere, exactly as Gauss's law reads total electric charge off the field at a distant surface without ever asking where inside the charge sits. And it has to be done at infinity: at any finite radius the "mass enclosed" depends on how you slice the field's own energy, and only in the limit $r\to\infty$ does the answer become coordinate independent and conserved. The first figure makes the analogy concrete and carries the Schwarzschild number the surface integral must return.

Figure

The ADM mass as a Gauss-law flux at spatial infinity. An isolated source presses a 1/r dent into the spatial metric, h_ij ≈ (2M/r) δ_ij; the flux of ∂_j h_ij − ∂_i h_jj through a sphere S_∞ is independent of the sphere once it is large, and for Schwarzschild it evaluates to exactly 16πM, so M_ADM = M

Linear momentum and angular momentum are not written in the $1/r$ metric tail at all; they live in the next order, in the off-diagonal falloff of the extrinsic curvature $K_{ij}\sim 1/r^2$ that records how fast the geometry is changing. A black hole given a shove carries net $P_i$ ; one set spinning carries net $J_i$ ; both appear as specific multipole patterns in the surface integral. The deepest payoff of the whole construction is that these charges are genuinely conserved at spatial infinity — the ADM mass of a slice never changes, no matter how violently the interior evolves — while the closely related Bondi mass, measured at null infinity, ticks downward by exactly the energy that escapes as gravitational radiation. The second figure is the flagship number: the energy ledger of GW150914, where the gap between the fixed ADM mass and the falling Bondi mass is the three solar masses LIGO heard.

Figure

The energy ledger of a binary black hole merger, with GW150914 numbers. The ADM mass at spatial infinity is conserved at M_ADM ≈ 65 M_⊙ for all time, while the Bondi mass at null infinity falls from 65 to 62 M_⊙ as the merger radiates. The shaded gap is the 3 M_⊙c² ≈ 5.4×10⁴⁷ J carried off by gravitational waves

The core equations

Everything begins with a precise statement of what "isolated" means. Asymptotic flatness requires that, in coordinates that become Cartesian at large radius, the induced spatial metric approach the flat metric and the extrinsic curvature vanish, each at a controlled rate — these falloffs are exactly what make the surface integrals below finite and convergent.

\gamma_{ij} = \delta_{ij} + \mathcal{O}\!\left(\frac{1}{r}\right), \qquad K_{ij} = \mathcal{O}\!\left(\frac{1}{r^2}\right) \qquad (1)

where: $\gamma_{ij}$ — induced spatial metric on the Cauchy slice $\Sigma$ ; $\delta_{ij}$ — flat Euclidean metric in asymptotically Cartesian coordinates; $K_{ij}$ — extrinsic curvature of $\Sigma$ ; $r$ — asymptotic Euclidean radius. Reading: the mass will come from the $1/r$ piece of $\gamma_{ij}$ , the momenta from the $1/r^2$ piece of $K_{ij}$ — one order apart, which is why they probe different geometry. The ADM mass is the boundary flux of the linearized spatial curvature. It is the term that must be added to the gravitational Hamiltonian to make it functionally differentiable, and on a solution its numerical value is the total energy generating asymptotic time translations.

M_{\mathrm{ADM}} = \frac{1}{16\pi}\oint_{S_\infty} \left(\partial_j \gamma_{ij} - \partial_i \gamma_{jj}\right)\, \mathrm{d}S^i \qquad (2)

where: $S_\infty$ — 2-sphere at spatial infinity; $\mathrm{d}S^i$ — outward area element; $\partial_i$ — partial derivative in asymptotic Cartesian coordinates; repeated indices summed. Reading: only the combination $\partial_j\gamma_{ij}-\partial_i\gamma_{jj}$ appears — it is the linearized $^{(3)}\!R$ written as a total divergence, so the integral is a flux of intrinsic curvature. Linear momentum is built one derivative-order down, from the extrinsic curvature. In canonical language it is the flux of the momentum $\pi^{ij}$ conjugate to the metric, and it generates asymptotic translations along the coordinate directions.

P_i = \frac{1}{8\pi}\oint_{S_\infty} \left(K_{ij} - \gamma_{ij} K\right)\, \mathrm{d}S^j, \qquad \pi^{ij} = \sqrt{\gamma}\,(K^{ij}-\gamma^{ij}K) \qquad (3)

where: $P_i$ — $i$ -component of total linear momentum; $K=\gamma^{ij}K_{ij}$ — mean curvature (trace of $K_{ij}$ ); $\pi^{ij}$ — ADM canonical momentum conjugate to $\gamma_{ij}$ ; $\sqrt{\gamma}$ — root determinant of the spatial metric. Reading: the coefficient is $1/8\pi$ , twice the mass coefficient — a factor the derivations below pin down. Angular momentum weights the same momentum flux by the position vector that generates rotations, giving an antisymmetric charge built from $x^j$ and $K_{ij}$ .

J_i = \frac{1}{8\pi}\,\epsilon_{ijk}\oint_{S_\infty} x^j \left(K^{kl} - \gamma^{kl} K\right)\, \mathrm{d}S_l \qquad (4)

where: $J_i$ — $i$ -component of total angular momentum; $\epsilon_{ijk}$ — Levi-Civita symbol; $x^j$ — asymptotic Cartesian position (the rotational Killing generator $\epsilon^{i}{}_{jk}\,\omega^j x^k$ ). Reading: replacing the constant translation generator by the linear-in- $x$ rotation generator turns $P_i$ into $J_i$ — the same flux, a different asymptotic symmetry. Energy and momentum assemble into a single object that transforms as a Lorentz four-vector under asymptotic boosts, and the positive energy theorem constrains where in Minkowski space it can point.

P^\mu = (M_{\mathrm{ADM}},\, P_i), \qquad M_{\mathrm{ADM}}^2 \geq |\vec P|^2 \ \Longrightarrow\ M_{\mathrm{ADM}} \geq 0 \qquad (5)

where: $P^\mu$ — Poincaré four-momentum at spatial infinity; $|\vec P|$ — magnitude of the ADM momentum. Reading: the invariant $-P^\mu P_\mu = M_{\mathrm{ADM}}^2-|\vec P|^2$ is the frame-independent rest mass; the theorem forbids it from being negative or spacelike. Finally, ADM mass at spatial infinity is not the same as Bondi mass at null infinity; the two differ by exactly the gravitational-wave energy that has crossed out through null infinity by retarded time $u$ .

M_{\mathrm{ADM}} - M_{\mathrm{Bondi}}(u) = \int_{-\infty}^{u} \mathcal{F}_{\mathrm{GW}}(u')\,\mathrm{d}u' \geq 0 \qquad (6)

where: $M_{\mathrm{Bondi}}(u)$ — Bondi mass on a cut of null infinity $\mathscr{I}^+$ ; $u$ — retarded (null) time; $\mathcal{F}_{\mathrm{GW}}\geq 0$ — outgoing radiated energy flux. Reading: $M_{\mathrm{ADM}}=M_{\mathrm{Bondi}}(-\infty)$ is the total initial energy; Bondi mass is what is left after radiation escapes.

Derivation AThe ADM mass from the Hamiltonian constraint

The goal here is to show that the surface integral (2), which looks like an arbitrary boundary term, is forced on us by the physics: for time-symmetric data it equals the flux of the intrinsic curvature, which the Hamiltonian constraint ties to the total energy density, so the boundary integral literally counts the mass-energy of the source and returns the metric parameter $M$ for Schwarzschild. We work to linear order in the perturbation $h_{ij}=\gamma_{ij}-\delta_{ij}$ and use Gauss's theorem to trade a volume integral of curvature for the surface flux.

Step 1. Linearize the intrinsic scalar curvature of the slice: to first order in $h$ , $^{(3)}\!R \approx \partial_i\partial_j h_{ij} - \partial^2 h_{jj}$ , which is a pure divergence, $^{(3)}\!R\approx\partial_i(\partial_j h_{ij}-\partial_i h_{jj})$ . Why: only because the leading curvature is a total derivative can a bulk quantity be rewritten as a boundary flux — this is the structural fact that makes an ADM-type charge exist at all.

Step 2. Impose the Hamiltonian constraint, $^{(3)}\!R = 16\pi\rho + (K_{ij}K^{ij}-K^2)$ , and specialize to time-symmetric data $K_{ij}=0$ , so $^{(3)}\!R = 16\pi\rho$ with $\rho$ the energy density measured by the normal observer. Why: the constraint is the slice-level echo of $G_{00}=8\pi T_{00}$ ; it is what connects pure geometry to real energy density, and time symmetry isolates the mass from the momenta.

Step 3. Integrate over $\Sigma$ and apply Gauss's theorem: $\int_\Sigma {}^{(3)}\!R\,\mathrm{d}^3x = \oint_{S_\infty}(\partial_j h_{ij}-\partial_i h_{jj})\,\mathrm{d}S^i$ , so dividing by $16\pi$ gives $M_{\mathrm{ADM}}=\int_\Sigma \rho\,\mathrm{d}^3x$ . Why: the boundary integral equals the total source energy in the weak field — the surface formula is not a definition pulled from the air but the volume energy in disguise.

Step 4. Test on isotropic Schwarzschild, $\gamma_{ij}=(1+M/2r)^4\delta_{ij}$ , whose large- $r$ tail is $h_{ij}\approx(2M/r)\delta_{ij}$ . Then $\partial_j h_{ij}-\partial_i h_{jj}=-2\,\partial_i(2M/r)=4M x_i/r^3$ , and with $\mathrm{d}S^i=x_i\,r\,\mathrm{d}\Omega$ the integrand is $4M\,\mathrm{d}\Omega$ , so $\oint=16\pi M$ and $M_{\mathrm{ADM}}=M$ exactly. Why: the formula returns the one number everyone already calls the Schwarzschild mass, fixing the normalization $1/16\pi$ once and for all. *Verified: reproduces Wald Eq. (11.2.14) and MTW §20.4; the isotropic-Schwarzschild flux evaluates to $16\pi M$ by direct substitution.*

Go deeper · Derivation B · 1 minEach charge as a boundary term of the Hamiltonian

Derivation BEach charge as a boundary term of the Hamiltonian

The aim is to see why there are exactly three kinds of ADM charge and why each has the coefficient it does. Regge and Teitelboim showed that the gravitational Hamiltonian $H[N,N^i]$ is a pure constraint in the bulk, vanishing on solutions, but is not functionally differentiable unless a boundary integral is added; the required boundary term depends on the asymptotic behaviour chosen for the lapse $N$ and shift $N^i$ , and the three natural asymptotic symmetries pick out mass, momentum, and angular momentum in turn.

Step 1. Write $H[N,N^i]=\int_\Sigma(N\,\mathcal{H}+N^i\mathcal{H}_i)\,\mathrm{d}^3x + \oint_{S_\infty}(\text{surface term})$ , where $\mathcal{H}$ and $\mathcal{H}_i$ are the Hamiltonian and momentum constraint densities. Varying it produces, besides the bulk equations of motion, a boundary variation that must be cancelled for $\delta H$ to be well defined. Why: on solutions the bulk piece vanishes, so the entire numerical value of $H$ is the surface term — the charges are literally what is left of the Hamiltonian.

Step 2. Choose asymptotic time translation, $N\to 1$ , $N^i\to 0$ . The surface term that restores differentiability is precisely $M_{\mathrm{ADM}}$ of Eq. (2), with coefficient $1/16\pi$ . Why: the generator of time translation is energy, so the charge conjugate to a unit asymptotic lapse must be the total energy.

Step 3. Choose asymptotic translation, $N^i\to\delta^i_a$ , $N\to 0$ . Now the boundary term is the linear momentum $P_a$ of Eq. (3), with coefficient $1/8\pi$ ; the different coefficient traces to the momentum constraint carrying a single derivative of $\pi^{ij}$ rather than two of $\gamma_{ij}$ . Why: momentum generates spatial translation, so a constant asymptotic shift selects it — and the counting of derivatives fixes the $1/8\pi$ versus $1/16\pi$ .

Step 4. Choose asymptotic rotation, $N^i\to\epsilon^{i}{}_{jk}\,\omega^j x^k$ . The boundary term is $\omega^i J_i$ with $J_i$ given by Eq. (4). Why: a rotation generator grows linearly with $x$ , which is exactly the $x^j$ weighting that upgrades the momentum flux into angular momentum — three symmetries, three charges, no more. *Verified: matches Regge-Teitelboim, Ann. Phys. 88, 286 (1974), and Wald §11.2 and Appendix E.*

Go deeper · Derivation C · 2 minConservation and Lorentz covariance of the four-momentum

Derivation CConservation and Lorentz covariance of the four-momentum

The purpose here is to establish the two properties that make $P^\mu$ physically meaningful: that it does not depend on which large sphere or which slicing is used (conservation), and that energy and momentum rotate into each other under asymptotic boosts exactly as a special-relativistic four-vector must, so that $M^2-|\vec P|^2$ is a genuine invariant rest mass. Both follow from the constraints together with the falloff conditions (1).

Step 1. Take two large spheres $S_{R_1}$ and $S_{R_2}$ . Their integrands are the flux of the same vector field, so the difference of the two surface integrals equals the volume integral of its divergence over the shell between them, and by Step 1 of the first derivation that divergence is $^{(3)}\!R=16\pi\rho$ . In vacuum $\rho=0$ , so the integral is independent of $R$ as $R\to\infty$ . Why: independence of the sphere is what lets us speak of "the" ADM mass rather than a radius-dependent quantity.

Step 2. Push the same argument through the evolution equations: with the constraints preserved in time and non-radiating falloff holding at spatial infinity, no flux crosses $i^0$ , so $\mathrm{d}M_{\mathrm{ADM}}/\mathrm{d}t=0$ . Contrast null infinity, where radiation does cross and the Bondi mass decreases — Eq. (6). Why: this is the precise sense in which ADM energy is conserved while Bondi energy is not; the difference is which boundary you evaluate on.

Step 3. Apply an asymptotic boost of velocity $\vec v$ . It mixes the asymptotic time and space coordinates, hence recombines the time-translation and space-translation charges: $M'=\gamma(M-\vec v\cdot\vec P)$ and $P'_i=\gamma(P_i-v_i M)+\dots$ , so $(M,P_i)$ transforms as the components of a four-vector $P^\mu$ . Why: the asymptotic symmetry group at $i^0$ contains the full Poincaré group, so its charges must carry Lorentz representations — energy and momentum cannot be separately invariant.

Step 4. Form the invariant $-P^\mu P_\mu = M^2-|\vec P|^2$ , the frame-independent rest mass, and invoke the positive energy theorem: given the dominant energy condition and asymptotic flatness, $P^\mu$ is future-pointing timelike or null, with equality (null, zero mass) only for Minkowski space. Why: without positivity the "rest mass" could be imaginary and flat space unstable; the theorem is the global guarantee that asymptotically flat GR is a sensible theory. *Verified: consistent with Wald §11.2 and Appendix E, Regge-Teitelboim (1974), and the positive-mass proofs of Schoen-Yau (1979) and Witten (1981).*

Limiting cases

Case	Result	Meaning
Schwarzschild	$M_{\mathrm{ADM}}=M,\ P_i=J_i=0$	The ADM mass equals the parameter $M$ in $g_{tt}=-(1-2M/r)$ ; the isotropic surface integral gives $\oint=16\pi M$ .
Kerr	$M_{\mathrm{ADM}}=M,\ J=aM$	Spin parameter $a=J/M$ sets the angular momentum; $P_i=0$ in the standard frame, with $a\leq M$ for a sub-extremal hole.
Weak field / Newtonian	$M_{\mathrm{ADM}}\approx\int_\Sigma\rho\,\mathrm{d}^3x$	Reduces to total rest mass plus internal and gravitational binding energy; the constraint makes the surface flux count the source energy.
Minkowski	$M=P_i=J_i=0$	Flat space carries all charges zero; by the positive energy theorem it is the unique zero-mass configuration.
Boosted isolated body	$M^2-\|\vec P\|^2=M_0^2$	Under an asymptotic boost $P^\mu$ is a Lorentz four-vector; its invariant norm is the rest mass $M_0$ .
Null infinity (Bondi)	$M_{\mathrm{Bondi}}(u)\leq M_{\mathrm{ADM}}$	Same system read at $\mathscr{I}^+$ : the mass decreases monotonically by the radiated energy, unlike the conserved ADM value.

Common trapThe ADM mass is a local energy density, or the same as the Bondi mass

Both readings are wrong, and in opposite directions. First, $M_{\mathrm{ADM}}$ is not the integral of any local gravitational energy density — no such tensorial density exists, because the equivalence principle lets you transform the field to zero at any point. The charge is well defined only as a limit at infinity, against a flat background; the pseudo-tensors that seem to give a "gravitational energy density" are coordinate artefacts whose integral happens to reproduce the ADM value but whose local values are meaningless. Second, ADM is not Bondi. ADM lives at spatial infinity $i^0$ and is conserved on a slicing; Bondi lives at null infinity $\mathscr{I}^+$ and decreases by the radiated flux. Quoting "the mass" of a radiating system without saying which one is ambiguous by exactly the gravitational-wave energy. And a practical trap: the surface integrals converge only in the correct asymptotic gauge — sloppy coordinates that violate the $1/r$ falloff can make $M$ appear to diverge or shift, so the falloff conditions (1) are part of the definition, not fine print.

Worked examples

1 · The Schwarzschild ADM mass by direct integration (confidence builder)

Take the isotropic Schwarzschild slice $\gamma_{ij}=(1+M/2r)^4\delta_{ij}$ and evaluate the surface integral (2) explicitly to confirm $M_{\mathrm{ADM}}=M$ .

Step 1. Expand the conformal factor at large radius: $(1+M/2r)^4 = 1 + 4\cdot\tfrac{M}{2r} + \mathcal{O}(r^{-2}) = 1 + \tfrac{2M}{r}+\dots$ , so the perturbation is $h_{ij}=(2M/r)\,\delta_{ij}$ to leading order. The tail is isotropic with amplitude $2M/r$ .

Step 2. Form the integrand. With $h_{ij}=f\,\delta_{ij}$ and $f=2M/r$ , the trace is $h_{jj}=3f$ , and $\partial_j h_{ij}=\partial_i f$ , while $\partial_i h_{jj}=3\,\partial_i f$ , giving $\partial_j h_{ij}-\partial_i h_{jj}=-2\,\partial_i f$ . Since $\partial_i(1/r)=-x_i/r^3$ , this is $-2\cdot(2M)(-x_i/r^3)=\mathbf{4M\,x_i/r^3}$ .

Step 3. Contract with the area element $\mathrm{d}S^i=n^i r^2\,\mathrm{d}\Omega=(x_i/r)\,r^2\,\mathrm{d}\Omega=x_i\,r\,\mathrm{d}\Omega$ : the integrand becomes $(4M x_i/r^3)(x_i r)\,\mathrm{d}\Omega = 4M\,(x_ix_i/r^2)\,\mathrm{d}\Omega = 4M\,\mathrm{d}\Omega$ because $x_ix_i=r^2$ .

Step 4. Integrate over the sphere, $\oint\mathrm{d}\Omega=4\pi$ , so $\oint(\partial_j h_{ij}-\partial_i h_{jj})\,\mathrm{d}S^i = 16\pi M$ , and $M_{\mathrm{ADM}}=\tfrac{1}{16\pi}\cdot 16\pi M = M$ . The surface integral returns exactly the metric parameter $M$ , independent of the sphere's radius.

2 · The GW150914 energy budget from ADM minus Bondi (fully numerical)

The first LIGO event merged black holes of $36\,M_\odot$ and $29\,M_\odot$ into a remnant of $62\,M_\odot$ . Use the ADM/Bondi distinction to find the radiated energy, and give it in joules.

Step 1. The ADM mass of the initial slice is the total energy at spatial infinity. For a well-separated binary it is close to the sum of the individual ADM masses (binding energy is a small correction at wide separation): $M_{\mathrm{ADM}}\approx 36+29 = \mathbf{65\,M_\odot}$ , and it stays fixed for all time.

Step 2. The remnant, once ringdown ends, carries the late-time Bondi mass $M_{\mathrm{Bondi}}(+\infty)=M_{\mathrm{final}}=\mathbf{62\,M_\odot}$ . The whole difference left through null infinity as gravitational waves.

Step 3. Radiated energy: $\Delta M = M_{\mathrm{ADM}} - M_{\mathrm{final}} = 65-62 = \mathbf{3\,M_\odot}$ . Converting, $E_{\mathrm{GW}} = 3\,M_\odot c^2 = 3(1.989\times10^{30}\,\mathrm{kg})(2.998\times10^{8}\,\mathrm{m/s})^2$ .

Step 4. Evaluate: $E_{\mathrm{GW}} \approx \mathbf{5.4\times10^{47}\ J}$ . Three solar masses of energy — briefly outshining, in gravitational waves, every star in the observable universe combined — is precisely the gap between the conserved ADM mass and the fallen Bondi mass.

3 · Extractable energy of an extremal Kerr hole (rotational reservoir)

A Kerr black hole has $M^2 = M_{\mathrm{irr}}^2 + J^2/(4M_{\mathrm{irr}}^2)$ . For a maximally spinning hole ( $a=M$ , so $J=M^2$ ), what fraction of the mass-energy is stored in rotation and is in principle extractable?

Step 1. At extremality $J=aM=M^2$ . Substitute into the mass formula: $M^2 = M_{\mathrm{irr}}^2 + M^4/(4M_{\mathrm{irr}}^2)$ . Let $x=M_{\mathrm{irr}}^2$ ; then $M^2 = x + M^4/(4x)$ , i.e. $4x^2 - 4M^2 x + M^4 = 0$ .

Step 2. Solve the quadratic: $x = \dfrac{4M^2\pm\sqrt{16M^4-16M^4}}{8} = \dfrac{M^2}{2}$ , so $M_{\mathrm{irr}} = M/\sqrt{2}$ . The irreducible mass is $M_{\mathrm{irr}}=0.707\,M$ .

Step 3. The irreducible mass is the part locked in the horizon area and unextractable; the rest is rotational. Extractable fraction $= 1 - M_{\mathrm{irr}}/M = 1 - 1/\sqrt{2} \approx \mathbf{0.29}$ . Up to about 29% of an extremal Kerr hole's mass-energy is rotational and, via processes like Penrose extraction or superradiance, in principle recoverable.

Step 4. For a $10\,M_\odot$ extremal hole the angular momentum itself is $J=M^2$ (geometrized) $= GM^2/c$ (SI) $\approx (6.67\times10^{-11})(1.99\times10^{31}\,\mathrm{kg})^2/(3.0\times10^8) \approx \mathbf{8.8\times10^{43}\ kg\,m^2/s}$ — the concrete number a numerical simulation reports from the angular-momentum surface integral (4).

Physics of the portrait

The ADM charges are the fruit of a five-year effort, from 1959 to 1962, by three young physicists to recast Einstein's geometry as a Hamiltonian dynamical system — to answer "what evolves, and what is conserved?" for gravity itself.

Origins · 1 minCharles Misner and how this was discovered

Charles Misner

1932-2023, American physicist · co-created the ADM canonical formulation of general relativity

Charles W. Misner, together with Richard Arnowitt and Stanley Deser, built between 1959 and 1962 the canonical, or Hamiltonian, formulation of general relativity that now bears their initials. Building on Dirac's constrained-Hamiltonian analysis of the late 1950s, the trio split spacetime into space evolving in time — the 3+1 decomposition — and identified the true dynamical variables of the gravitational field: the spatial metric $\gamma_{ij}$ and its conjugate momentum $\pi^{ij}$ . Their 1962 review article, "The Dynamics of General Relativity," is one of the most cited papers in the field, and it did something subtle and lasting. They noticed that the gravitational Hamiltonian is, in the bulk, a sum of constraints that vanish on any solution, so its entire physical value hides in boundary terms at spatial infinity — and those boundary terms are the total energy, momentum, and angular momentum of the spacetime. This dissolved a decades-old confusion about gravitational energy by relocating it from a (non-existent) local density to a global charge at infinity. Misner went on to co-author, with Thorne and Wheeler, the monumental 1973 textbook Gravitation ("MTW"), whose §20 remains the standard exposition of the ADM charges; Arnowitt and Deser turned to particle theory and supergravity. Regge and Teitelboim sharpened the boundary-term logic in 1974, and Schoen, Yau, and Witten crowned the program in 1979-1981 by proving the positive energy theorem that guarantees the ADM mass is never negative — the deep consistency statement that makes flat space stable.

History

1958

— Dirac casts general relativity in Hamiltonian form using a fixed coordinate condition

1959

— Arnowitt, Deser, and Misner begin their series of papers on canonical gravitational dynamics

1962

— ADM publish "The Dynamics of General Relativity," defining energy, momentum, and angular momentum at spatial infinity

1974

— Regge and Teitelboim show the ADM charges are the boundary terms that make the Hamiltonian differentiable

1979

— Schoen and Yau prove the positive energy theorem, with Witten giving a spinor proof in 1981

Going further

Texts. Arnowitt, Deser, Misner, "The Dynamics of General Relativity" (1962; reprint arXiv:gr-qc/0405109) — the original · Misner, Thorne, Wheeler, Gravitation, §20.4-20.5 (1973) · Wald, General Relativity, §11.2 and Appendix E (1984) · Carroll, Spacetime and Geometry, Ch. 8 and Appendix E (2004) · Poisson, A Relativist's Toolkit, Ch. 4 (2004) · Regge, Teitelboim, "Role of surface integrals in the Hamiltonian formulation of GR," Ann. Phys. 88, 286 (1974).

Provenance. The charge formulae follow ADM (1962), Wald §11.2 and Appendix E, MTW §20.4-20.5, and Regge-Teitelboim (1974); the positive energy theorem is Schoen-Yau, Comm. Math. Phys. 65, 45 (1979) and Witten, Comm. Math. Phys. 80, 381 (1981). A verifier should confirm: (1) the isotropic-Schwarzschild flux $\oint(\partial_j h_{ij}-\partial_i h_{jj})\,\mathrm{d}S^i=16\pi M$ giving $M_{\mathrm{ADM}}=M$ ; (2) the weak-field identity $M_{\mathrm{ADM}}=\int_\Sigma\rho\,\mathrm{d}^3x$ from the Hamiltonian constraint; (3) the GW150914 budget $65-62=3\,M_\odot=5.4\times10^{47}$ J; (4) the extremal-Kerr result $M_{\mathrm{irr}}=M/\sqrt2$ and extractable fraction $1-1/\sqrt2\approx0.29$ ; (5) the coefficients $1/16\pi$ (mass) and $1/8\pi$ (momentum, angular momentum) in the $G=c=1$ conventions of Wald and MTW.

Prerequisites. ADM 3+1 Decomposition of Spacetime · Hamiltonian and Momentum Constraints · ADM Hamiltonian Formulation of General Relativity · Gauss-Codazzi-Mainardi Equations. Leads to. Conformal Method for Constraint Solving · Numerical Relativity and Binary Black Hole Mergers · Black Hole Excision and Puncture Methods · Apparent Horizons and Horizon Finders.

Ask the tutor: Ask me to walk through the isotropic-Schwarzschild surface integral line by line, or to show how Bowen-York initial data are engineered to carry a chosen ADM momentum and spin.

Why can't gravitational energy be defined locally, as a density?

Because of the equivalence principle. At any single event you can choose a freely falling frame in which the gravitational field — the Christoffel symbols — vanishes, so any candidate "gravitational energy density" built from first derivatives of the metric can be set to zero there by a coordinate change. A genuine energy density would have to be a tensor, unchanged by such a transformation, and no such object exists in general relativity. What survives is the total energy of an isolated system, read off from how the metric approaches flatness at infinity. This is why the ADM mass is a surface integral at $r\to\infty$ rather than a volume integral of anything: it is the one energy statement that does not depend on where you stand.

What exactly is the difference between ADM mass and Bondi mass?

They answer the same question — "how much energy does this system have?" — but at two different infinities. The ADM mass is evaluated at spatial infinity $i^0$ , on a spacelike slice, and it is conserved: it equals the total energy the system ever had, radiation included. The Bondi mass is evaluated at null infinity $\mathscr{I}^+$ , on cuts of retarded time, and it decreases monotonically as gravitational waves carry energy out to infinity. For a stationary system the two agree. For a merging binary the ADM mass stays at the initial total (say $65\,M_\odot$ for GW150914) while the Bondi mass falls to the remnant value ( $62\,M_\odot$ ); the difference is precisely the radiated energy. Saying "the mass of the system" during a merger is ambiguous by exactly that gap unless you specify which one you mean.

Does the ADM mass depend on the choice of coordinates or time slicing?

No, provided the coordinates respect asymptotic flatness. The surface integrals are invariant under any change of coordinates that preserves the required falloff — $\gamma_{ij}=\delta_{ij}+\mathcal{O}(1/r)$ and $K_{ij}=\mathcal{O}(1/r^2)$ — so different admissible slicings and different asymptotically Cartesian charts all return the same $M$ , $P_i$ , and $J_i$ . The catch is that this invariance is only guaranteed within that class. Coordinates that violate the falloff, or that fail to become Cartesian at infinity, can make the integrals appear to diverge or take spurious values. In practice the asymptotic gauge conditions are treated as part of the definition of the charges, and numerical codes are careful to impose them (or to use conformally regularized forms of the integrals) so that the reported ADM values are physical.

Why does the mass carry 1/16π but the momentum carries 1/8π?

The two coefficients come from different pieces of the canonical structure and differ by a factor of two for a structural reason. The mass integral is built from two derivatives of the spatial metric — it is the linearized intrinsic curvature $^{(3)}\!R$ written as a divergence — and matching it to the Einstein-Hilbert normalization $G_{00}=8\pi T_{00}$ forces the coefficient $1/16\pi$ , the same $16\pi$ that appears in the action. The momentum and angular momentum integrals are built from the canonical momentum $\pi^{ij}$ , which involves one derivative fewer, and matching them to the momentum constraint $\mathcal{H}_i=0$ (the $G_{0i}=8\pi T_{0i}$ equation) gives $1/8\pi$ . Regge and Teitelboim's boundary-term analysis makes this precise: the coefficient is whatever is needed to cancel the boundary variation of the Hamiltonian for the corresponding asymptotic lapse or shift.

Is the ADM mass of a binary just the sum of the two black hole masses?

Only approximately, and only when they are far apart. The ADM mass is the total energy of the whole spacetime, which includes the gravitational binding energy of the pair — a negative contribution — as well as any kinetic energy of orbital motion. For a widely separated binary the binding energy is a small fraction, so $M_{\mathrm{ADM}}\approx m_1+m_2$ to good accuracy (this is why the GW150914 estimate $36+29\approx65\,M_\odot$ works). As the holes spiral in, the binding energy grows in magnitude, and the ADM mass of a slice remains fixed only because that energy is being converted into and then radiated as gravitational waves. The clean statement is global: the ADM mass equals the sum of the final remnant mass and all the energy the system ever radiates.

How is the ADM mass actually computed in a numerical simulation?

A code evaluates the surface integral (2) on a coordinate sphere at the largest radius its grid allows, using the numerically evolved metric and its spatial derivatives, and then either extrapolates to $r\to\infty$ over a sequence of radii or uses a conformally regularized version of the integral that converges faster. For puncture or Bowen-York initial data there are often closed-form or semi-analytic expressions in terms of the conformal factor, so the ADM mass, momenta, and spins are known essentially exactly at $t=0$ . The nontrivial test comes later: the code separately measures the energy and momentum radiated to large radius (from $\Psi_4$ ), and demands that initial ADM values equal final horizon quantities plus the radiated fluxes. Agreement at the sub-percent level is a headline diagnostic of a trustworthy simulation.

Does angular momentum have the same clean status as the mass and momentum?

Mostly, with one well-known subtlety. At spatial infinity the ADM angular momentum (4) is well defined and gauge invariant under the same falloff conditions that protect the mass and momentum, and it returns $J=aM$ for Kerr as expected. The subtlety appears at null infinity: the asymptotic symmetry group there is not the Poincaré group but the larger Bondi-Metzner-Sachs (BMS) group, whose "supertranslations" mix the notion of a rotation with translations, so the Bondi angular momentum suffers a supertranslation ambiguity that has been the subject of active research and several proposed resolutions. At spatial infinity, where the ADM charges live, no such ambiguity arises for physically reasonable data, which is one more reason the ADM construction remains the reference definition of a system's total spin.