Poincare Symmetry and ADM Charges

At an asymptotically Euclidean end, the ADM charges are the Hamiltonian generators of asymptotic translations, rotations, and boosts. Their formulas are surface limits, but their existence is not guaranteed by writing a metric as “flat plus corrections.” The decay rate, derivative control, parity, constraint equations, asymptotic coordinates, and reference normalization determine which Poincaré generators have finite, invariant charges.

Required background. Variational and Symplectic Geometry supplies Hamiltonian generators and boundary terms. Helpful background. Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply the asymptotic geometry; Causality and Global Structure supplies spatial infinity; Formulations of General Relativity develops general charge derivations; Waves, Green Functions, and Geometric Optics supplies the later radiative comparison.

Asymptotically Euclidean initial data

Let $(\Sigma,h_{ij},K_{ij})$ be a three-dimensional initial-data set with one end diffeomorphic to the complement of a ball in $\mathbb R^3$. If $u^a$ is the future-directed unit normal to $\Sigma$, with $u^a u_a=-1$, we use

$$K_{ij}=-h_i{}^a h_j{}^b\nabla_a u_b =-\frac12\mathcal L_u h_{ij}.$$

The spacetime normal $u^a$ is distinct from the outward Euclidean sphere normal $n^i$ below. In asymptotically Cartesian coordinates $x^i$, a standard four-dimensional starting point is

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

with stronger derivative and integrability conditions as needed. The vacuum constraint violations vanish; with matter, energy and momentum densities must decay sufficiently fast. These are not the weakest possible hypotheses, but they make the leading surface formulas transparent.

To control angular momentum and center of mass, impose Regge–Teitelboim parity in one common phase space:

$$q_{ij}^{(1)}(-\widehat x)=q_{ij}^{(1)}(\widehat x), \qquad K_{ij}^{(2)}(-\widehat x)=-K_{ij}^{(2)}(\widehat x),$$

where superscripts label the leading inverse-radius coefficients. Odd integrands that would otherwise diverge then integrate to zero. Alternative twisted parity conditions retain supertranslations and lead to a larger BMS-type asymptotic group at spatial infinity. Henneaux and Troessaert compare these choices in Henneaux and Troessaert 2019, §§2–5.

Allowed lapse–shift pairs approach Minkowski Killing data:

$$N=a+b_ix^i+o(r), \qquad N^i=a^i+\omega^i{}_jx^j+o(r).$$

Constants generate time and space translations; antisymmetric $\omega_{ij}$ generates rotations; the linear lapse generates boosts. The displayed $o(r)$ terms record only the leading Killing decomposition: an admissible phase space must also control their derivatives and parity, require preservation of every boundary condition, and distinguish zero-charge proper gauge transformations from large symmetries. After proper gauge transformations are quotiented, the remaining generators form the Poincaré algebra in the standard parity phase space.

Energy and linear momentum

The gravitational Hamiltonian contains bulk constraints plus a boundary term. For an asymptotic unit time translation, differentiability of the Hamiltonian fixes the ADM energy

$$E_{\rm ADM} =\frac{1}{16\pi G} \lim_{r\to\infty} \int_{S_r} \left(\partial_jh_{ij}-\partial_ih_{jj}\right)n^i\,dS.$$

Here $S_r$ is a large coordinate sphere, $n^i$ and $dS$ are computed with the asymptotic Euclidean metric at leading order, and indices in the integrand are contracted with $\delta_{ij}$. The formula is meaningful only under coordinate transformations preserving the asymptotic class. An arbitrary slow or angle-dependent change of coordinates can alter the apparent coefficient.

Define

$$\Pi_{ij}=K_{ij}-Kh_{ij}.$$

It is related to the canonical momentum density by

$$\pi^{ij}=-\frac{\sqrt h}{16\pi G}\Pi^{ij} =\frac{\sqrt h}{16\pi G}(Kh^{ij}-K^{ij}).$$

For an asymptotic spatial translation, the momentum is

$$P_i =\frac{1}{8\pi G} \lim_{r\to\infty} \int_{S_r}\Pi_{ij}n^j\,dS.$$

With this definition of $K_{ij}$ and the outward orientation of $S_r$, the displayed momentum sign is fixed. When comparing a source that defines extrinsic curvature with the opposite sign, translate $K_{ij}$ and its Hamiltonian boundary term together. Asymptotically Flat Initial Data states the same convention-matched surface term and its primary-source locator. The four quantities $(E,P_i)$ transform as the ADM four-momentum under the asymptotic Poincaré transformations.

For time-symmetric data $K_{ij}=0$, the momentum vanishes. On an isotropic Schwarzschild slice,

$$h_{ij}=\left(1+\frac{GM}{2r}\right)^4\delta_{ij} =\left(1+\frac{2GM}{r}+O(r^{-2})\right)\delta_{ij}.$$

Substitution gives $E_{\rm ADM}=M$: the derivative produces a factor of $GM/r^2$, the sphere integral supplies $4\pi r^2$, and $1/(16\pi G)$ restores mass. Minkowski data give zero. These tests fix the energy normalization and sign, but $K_{ij}=0$ makes them blind to the momentum sign.

A non-time-symmetric surface-integral check retains the missing information. Take trace-free leading data

$$K^{(p)}_{ij} =\frac{3G}{2r^2} \left[p_i n_j+p_j n_i -(\delta_{ij}-n_i n_j)(p_kn^k)\right] +O(r^{-3}).$$

Then

$$\Pi_{ij}n^j =\frac{3G}{2r^2}\left[p_i+(p_kn^k)n_i\right]+O(r^{-3}).$$

Using $\int_{S^2}n_in_j\,d\Omega=(4\pi/3)\delta_{ij}$ in the momentum integral gives $P_i=p_i$. Reversing the sign of $K_{ij}$ without translating the charge formula would instead give $-p_i$, so this check fixes the convention-sensitive sign.

Rotations, boosts, and parity

For an asymptotic rotation vector $\varphi^i=\omega^i{}_jx^j$, the ADM angular momentum is schematically

$$J[\varphi] =\frac{1}{8\pi G} \lim_{r\to\infty} \int_{S_r}\Pi_{ij}\varphi^i n^j\,dS.$$

Because $\varphi^i$ grows like $r$, naive power counting gives a possible linear divergence. Parity removes it and controls changes under asymptotic translations. Without selecting an origin, angular momentum also has the ordinary special-relativistic shift

$$J^{ij}\mapsto J^{ij}+a^iP^j-a^jP^i.$$

On a selected asymptotic $t=0$ slice, write $q_{ij}=h_{ij}-\delta_{ij}$. The boost or mass-dipole charge is

$$B^k=\frac{1}{16\pi G}\lim_{r\to\infty}\int_{S_r} \left[ x^k(\partial_jq_{ij}-\partial_iq_{jj})n^i -q_{ik}n^i+q_{ii}n^k \right]dS.$$

The algebraic terms cancel the leading coordinate artifact in the $x^k$-weighted energy integrand. This charge requires one more asymptotic moment than energy and is correspondingly more sensitive to parity, origin, and coordinate choice. In a selected asymptotic frame with $E_{\rm ADM}\ne0$, the center-of-mass coordinate is $C^k=B^k/E_{\rm ADM}$; when $E_{\rm ADM}=0$, that division is not available. Regge and Teitelboim derive the translation, rotation, and boost surface generators in Regge and Teitelboim 1974, §§4–5, pp. 303–311.

Thus “ten ADM charges” is shorthand for a phase space on which all Poincaré generators are differentiable and finite. Energy–momentum can exist under weaker conditions than angular momentum and center of mass. If supertranslations are retained, the Lorentz subgroup and associated angular momentum require additional frame choices rather than emerging uniquely.

Invariance, constraints, and comparisons

The constraints are crucial for surface independence. Comparing integrals on two large spheres and applying Gauss’s theorem converts their difference to a volume integral involving constraint terms plus faster-decaying remainders. For constraint-satisfying data with adequate decay, the limit is independent of the chosen asymptotic spheres. This is the Hamiltonian version of charge conservation at spatial infinity.

ADM energy is not a local gravitational energy density. The equivalence principle prevents a tensorial density built only from first derivatives of the metric, and different pseudotensors redistribute the bulk contribution. The surface charge is invariant under the allowed asymptotic transformations because the boundary conditions and constraints control that redistribution.

The ADM charge at $i^0$ and a Bondi charge on $\mathscr I^+$ live in different asymptotic regimes. Relating them requires a matching prescription through spatial infinity, regularity near the corner, and flux control. Equality cannot be inferred from adjacency in a Penrose diagram.

General comparisons among Komar, ADM, and covariant charges are developed in Komar, ADM, and Covariant Charges Compared. Compère and Fiorucci supply the covariant normalization and integrability framework used to make those comparisons Compère and Fiorucci 2019, §§1.2–1.5, pp. 10–32.

Exercise

For conformally flat time-symmetric data $h_{ij}=(1+2a/r+O(r^{-2}))\delta_{ij}$ and $K_{ij}=0$, compute the ADM energy and momentum. Then evaluate the momentum of the trace-free leading data $K^{(p)}_{ij}$ displayed above.

Solution

Write $f=1+2a/r$. Then $h_{ij}=f\delta_{ij}$ at the required order,

$$\partial_jh_{ij}=\partial_if, \qquad \partial_ih_{jj}=3\partial_if,$$

so the energy integrand is $-2\partial_if\,n^i=4a/r^2$. Therefore

$$E_{\rm ADM} =\frac{1}{16\pi G}\lim_{r\to\infty} \int_{S_r}\frac{4a}{r^2}\,dS =\frac{a}{G}.$$

Because $K_{ij}=0$, $\Pi_{ij}=0$ and $P_i=0$. For isotropic Schwarzschild data $a=GM$, giving $E=M$ as required.

For the second part, $K^{(p)}=0$ and therefore $\Pi_{ij}=K^{(p)}_{ij}$ to leading order. Contracting with $n^j$ gives

$$\Pi_{ij}n^j =\frac{3G}{2r^2}\left[p_i+(p\mathbin{\cdot}n)n_i\right]+O(r^{-3}).$$

The angular average of the first term is $4\pi p_i$, and that of the second is $(4\pi/3)p_i$. Hence

$$P_i=\frac{1}{8\pi G}\frac{3G}{2} \left(4\pi+\frac{4\pi}{3}\right)p_i=p_i.$$

The Bondi–Metzner–Sachs Group replaces spatial-infinity parity data by the universal null structure of $\mathscr I^+$.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§1.2–1.5, pp. 10–32, Springer, 2019.
• Henneaux, Marc, and Cédric Troessaert, “The Asymptotic Structure of Gravity at Spatial Infinity in Four Spacetime Dimensions”, Proceedings of Science CORFU2018, article 154, §§2–5, 2019.
• Regge, Tullio, and Claudio Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity”, Annals of Physics 88, no. 1, §§4–5, pp. 286–318, 1974.