Komar, ADM, and Covariant Charges Compared

Komar, ADM, and covariant phase-space formulas are not competing universal definitions with identical domains. Komar integrals use an exact spacetime symmetry, ADM charges use asymptotically Euclidean Cauchy data and asymptotic Poincaré generators, and covariant charges give variations for allowed symmetries on a declared solution space. They agree only after matching the generator, orientation, normalization, boundary conditions, reference, and field equations—including matter contributions when present.

Required background. The Gibbons–Hawking–York Boundary Term supplies the Hamiltonian surface terms; Presymplectic Potentials and Currents supplies covariant charge variations; and Killing Fields and Isometry Groups supplies exact and asymptotic symmetries.

Helpful background. The Einstein Equation supplies the relations used to compare surfaces; Hypersurface Normals and Tangential Projectors supplies spatial infinity and surface orientation; and Formulations of General Relativity orients the chapter.

Three constructions and their domains

For a closed two-surface $S$ and a Killing field $\xi^a$, the Einstein Noether two-form is

$$(Q_\xi)_{ab} =-\frac{1}{16\pi G}\epsilon_{abcd}\nabla^c\xi^d.$$

A conventional Komar charge is a normalized integral of $*d\xi^\flat$. For a stationary timelike Killing field $t^a$ normalized to unit time at infinity,

$$M_{\mathrm K} =-\frac{1}{8\pi G}\int_S *dt^\flat,$$

whereas for an axial field $\varphi^a$ with $2\pi$ orbits,

$$J_{\mathrm K} =\frac{1}{16\pi G}\int_S *d\varphi^\flat.$$

The different coefficients are conventional but essential. A Komar integral requires the relevant exact symmetry in the region used to move $S$.

ADM energy and momentum instead use an asymptotically flat spatial slice with spatial metric $\gamma_{ij}$ and future-normal sign $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$. In asymptotically Cartesian coordinates, $s^i$ and $dS$ below are the Euclidean outward normal and area element at leading order:

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

No exact timelike Killing field is needed. Sufficient falloff, constraint decay, and an asymptotic translation frame are required. The usual even/odd parity conditions are a convenient sufficient phase-space prescription, especially when rotations and boosts are also included; they should not be confused with necessary conditions for every definition of ADM energy alone.

The covariant construction begins with

$$\not\!\delta H_\xi =\int_S (\delta Q_\xi-\xi\mathbin{\lrcorner}\Theta).$$

It applies to allowed exact or asymptotic symmetries and perturbations tangent to the specified phase space. A finite charge exists only after finiteness and integrability are checked and a reference constant is chosen.

Why the bare Noether integral gives half a mass

For stationary vacuum Einstein gravity, Iyer and Wald show that the integral of $Q_t$ at infinity is one half of the conventionally normalized Komar mass. The Hamiltonian is not simply $\int Q_t$: the boundary potential supplies the additional contribution. Their calculation yields

$$H_t =\int_{S_\infty}(Q_t-t\mathbin{\lrcorner}B) =M_{\mathrm{ADM}}.$$

The explicit comparison appears in Iyer and Wald 1994, §6, eqs. (83)–(90), pp. 15–16, PDF. This resolves the “factor-of-two problem”: it is an error only if a bare Noether-form integral is mistaken for the complete time-translation Hamiltonian.

For rotations, the pullback of $\varphi\mathbin{\lrcorner}\Theta$ can vanish at the asymptotic sphere when $\varphi^a$ is tangent to it, and the canonical angular momentum reduces to the properly normalized Komar expression. Time translations and rotations must therefore not be normalized by copying one coefficient to the other.

There is also a generator sign: with the displayed $Q_\xi$ and charge prescription, the Hamiltonian associated with $\varphi$ is $H_\varphi=-J$, after choosing the common reference. Thus $H_{t+\Omega_H\varphi}=M-\Omega_HJ$ for fixed $\Omega_H$. Calling $H_\varphi$ the positive angular momentum would introduce an extra sign error even with the correct Komar coefficient.

Conditions for agreement

The common stationary asymptotically flat vacuum comparison requires:

• an asymptotic end with ADM falloff and parity;
• a timelike Killing field normalized as the chosen unit asymptotic time translation, or an axial Killing field with fixed orbit period;
• the Einstein equations in the region between comparison surfaces;
• compatible orientations and the same Newton constant;
• a covariant charge integrated from a reference that fixes the same zero.

Under these hypotheses, the canonical covariant energy equals ADM energy and, for a stationary vacuum solution, equals Komar mass. The angular momenta agree under the corresponding axial assumptions. Gourgoulhon compares ADM and Komar mass in 3+1 language in Gourgoulhon 2007, §7.6.2, pp. 119–122, PDF.

When they differ or do not apply

In a nonstationary spacetime there is generally no timelike Killing field, so a Komar mass is unavailable even though ADM energy remains defined at spatial infinity. At null infinity, Bondi charges and radiative flux replace a conserved spatial-infinity energy on successive cuts. At a finite surface, a covariant charge depends on the boundary phase space and may not be conserved.

Matter also changes the comparison. The identity for a Killing field contains the Ricci tensor, so moving a Komar integral between surfaces picks up a bulk matter contribution. Long-range gauge fields can contribute additional terms to the canonical Hamiltonian. A vacuum equality must not be transferred to an electrovacuum or fluid region without including those terms.

Formula	Needs exact Killing field?	Natural location	Principal limitation
Komar	Yes	Any homologous closed surface in a symmetry region	Matter bulk terms and normalization
ADM	No	Spatial infinity of a Cauchy slice	Requires asymptotic flatness and parity
Covariant phase space	Exact or allowed asymptotic symmetry	Declared boundary cut	Requires finite integrable variation and reference

Independent Schwarzschild check

For Schwarzschild with $f(r)=1-2GM/r$ and $t^a=(\partial_t)^a$, the normalized Komar mass on a sphere is

$$M_{\mathrm K} =\frac{r^2f'(r)}{2G}=M.$$

For example, $t^\flat=-f\,dt$ and $dt^\flat=f'\,dt\wedge dr$. With the inherited spacetime orientation and the sphere oriented by increasing $(\theta,\phi)$, $*(dt\wedge dr)=-r^2\sin\theta\,d\theta\wedge d\phi$. Therefore $\int_S *dt^\flat=-4\pi r^2f'$, fixing both the sign and the normalization in the result above.

It is independent of $r$ in the vacuum exterior. On the time-symmetric isotropic slice, $\gamma_{ij}=(1+GM/(2r))^4\delta_{ij}$, the ADM surface integral also returns $M$. The covariant Hamiltonian integrated from Minkowski along the Schwarzschild family gives $H_t=M$. By contrast, $\int Q_t=M/2$ with the Iyer–Wald Noether normalization, demonstrating why the potential boundary term cannot be omitted.

Common pitfalls

A Komar form is not automatically a Hamiltonian. The time-translation Hamiltonian contains a potential or boundary-action contribution.

Agreement is conditional, not definitional. Vacuum field equations, symmetry, asymptotics, and normalization do the work.

ADM is not a finite-radius local mass. It is a charge at an asymptotic end; finite-surface prescriptions require additional structure.

Exercise: rescaling the generator

Let $t'^a=c\,t^a$ with constant $c\gt 0$. How do the three stationary charge constructions change, and why does this not signal a contradiction?

Solution

The Komar and covariant forms are linear in the generator, so $M_{\mathrm K}[t']=cM_{\mathrm K}[t]$ and $H_{t'}=cH_t$. ADM energy is defined relative to a unit asymptotic time translation. If one calls $t'^a$ the generator without renormalizing it, its Hamiltonian is likewise $cE_{\mathrm{ADM}}$; the numerical ADM mass corresponds to the normalized generator. Charges are conjugate to generators, so changing the unit of asymptotic time changes the charge component being evaluated.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50, no. 2 (1994): 846–864. DOI; Open PDF.