Boundary Hamiltonians and Gravitational Energy

The bulk Hamiltonian of general relativity is a sum of constraints, but the generator of an allowed spacetime motion need not vanish. Integrations by parts leave surface variations, and canceling them adds a boundary Hamiltonian. For an asymptotic time translation this surface term is the ADM energy. Finite timelike boundaries and horizons lead to different charges because they impose different boundary data and normalize different generators.

Required background. Presymplectic Potentials and Currents supplies the differentiability criterion for Hamiltonian generators, while The Gibbons-Hawking-York Boundary Term supplies the boundary completion of the Dirichlet gravitational action.

Helpful background. The Einstein Equation supplies the on-shell constraints, Hypersurface Normals and Tangential Projectors supplies the normals to the slice and its boundary, and Formulations of General Relativity places boundary charges alongside covariant descriptions.

Differentiable canonical generators

Begin with the bulk expression

$$H_{\rm bulk}[N,\vec N] =\int_\Sigma d^3x \left(N\mathcal C_\perp+N^i\mathcal C_i\right).$$

Varying its curvature term differentiates $\delta\gamma_{ij}$, while varying the vector constraint differentiates $\delta\pi^{ij}$. After integration by parts,

$$\delta H_{\rm bulk} =\int_\Sigma d^3x \left(A^{ij}\delta\gamma_{ij}+B_{ij}\delta\pi^{ij}\right) +\int_{\partial\Sigma}\mathcal S[N,\vec N;\delta\gamma,\delta\pi].$$

A Hamiltonian vector field exists only if the surface variation is zero or is the variation of a boundary functional. In the latter case one chooses

$$H[N,\vec N]=H_{\rm bulk}[N,\vec N]+B[N,\vec N]$$

so that $\delta B=-\int_{\partial\Sigma}\mathcal S$. The choice depends on the allowed boundary fields, their falloff or fixed values, and the asymptotic behavior of $N$ and $N^i$. It is determined only up to a constant on each connected component of phase space; selecting a zero of energy is therefore part of the boundary prescription.

This logic is explicit in Arnowitt, Deser, and Misner 1962/2004, §§5–6, PDF. It also explains the crucial on-shell distinction

$$\mathcal C_\perp=0=\mathcal C_i \quad\Longrightarrow\quad H[N,\vec N]=B[N,\vec N],$$

not $H=0$, whenever the allowed generator has a nonzero boundary charge.

ADM energy at spatial infinity

For one asymptotically Euclidean end, choose asymptotically Cartesian coordinates in which

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

with corresponding differentiated falloffs and integrable spatial scalar curvature at the end. For the full asymptotic Poincaré generators also impose the usual parity conditions: the leading $1/r$ metric perturbation is even under $x^i\mapsto-x^i$, and the leading $1/r^2$ momentum is odd. Let $N\to1$ and $N^i\to0$. The curvature part of $\delta H_{\rm bulk}$ contains minus the variation of

$$\boxed{ E_{\rm ADM} =\frac1{16\pi G} \lim_{r\to\infty} \oint_{S_r} \left(\partial_j\gamma_{ij}-\partial_i\gamma_{jj}\right) s^i\,dS }.$$

Adding this term makes the time-translation generator differentiable. Minkowski data give zero with the conventional reference choice. The formula is not invariant under arbitrary coordinate changes; it is invariant under the asymptotic coordinate class that preserves the stated falloff, and it joins the ADM linear momentum into an asymptotic four-vector. Rotations and boosts require stronger parity control because their smearings grow at infinity.

The asymptotic assumptions, ADM mass, momentum, and angular-momentum ambiguities are treated in Gourgoulhon 2007, §§7.2–7.5, pp. 103–115, PDF. Thus “the gravitational energy” is not a local scalar density extracted from the bulk constraints; it is a charge associated with specified asymptotic structure.

Finite boundaries and horizon generators

At a finite timelike Dirichlet boundary, the induced boundary metric is held fixed and the Hamilton–Jacobi derivative of the action defines a boundary stress tensor. For a two-surface $S\subset\Sigma$, one common Brown–York energy is, for observers normal to slices that meet the timelike boundary orthogonally,

$$E_{\rm BY} =\frac1{8\pi G}\oint_S d^2x\sqrt\sigma\,(k_0-k),$$

where $s^i$ is the outward unit normal within $\Sigma$ and $k=\sigma^{ij}D_i s_j$; with this Riemannian mean-curvature sign, a round sphere of radius $R$ in flat space has $k=2/R$. The quantity $k_0$ is a chosen reference or subtraction term. Its value depends on the observers, boundary data, and reference embedding or counterterm. It is therefore a quasilocal energy for that variational problem, not a universally interchangeable version of the ADM mass. The construction is derived in Brown and York 1993, §§III–IV, especially equation (4.8), pp. 1410–1414, PDF. Their two-surface extrinsic curvature has the opposite sign, as defined in Brown and York 1993, equation (A10), p. 1418, PDF; translating it gives the $k_0-k$ order used here.

An inner black-hole boundary contributes another surface term. For a stationary horizon generator

$$\chi^\mu=t^\mu+\Omega_H\phi^\mu,$$

equality of the surface variations at infinity and at the horizon gives, for variations among stationary asymptotically flat vacuum solutions with fixed $\Lambda=0$ and the same asymptotic normalization,

$$\delta E_{\rm ADM}-\Omega_H\delta J =\frac{\kappa}{8\pi G}\delta A.$$

Matter adds the appropriate work terms, such as $\Phi_H\delta Q$. The energy and surface gravity scale if $\chi^\mu$ is rescaled, so its normalization—often fixed by unit time translation at infinity—is part of the statement. The general boundary-variation derivation of the stationary first law is given in Iyer and Wald 1994, §6, pp. 14–19, PDF.

These examples share one principle but not one numerical definition. ADM, Brown–York, and horizon energies answer questions posed with different boundaries, observers, reference terms, and allowed variations.

Common pitfalls

Setting the Hamiltonian to zero before checking the boundary. Only the bulk constraint integral vanishes on shell. A physical symmetry can leave a nonzero surface generator.

Quoting an energy without its boundary prescription. Falloff, parity, reference subtraction, and generator normalization can affect whether a charge exists and what value it takes.

Exercise: ADM energy of Schwarzschild initial data

In isotropic coordinates, the time-symmetric Schwarzschild spatial metric is

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

Use its leading large-$r$ form in the ADM surface integral and recover the mass $M$.

Solution

At large radius,

$$\gamma_{ij}=f\delta_{ij}+O(r^{-2}), \qquad f=1+\frac{2GM}{r}.$$

Therefore

$$\partial_j\gamma_{ij}-\partial_i\gamma_{jj} =\partial_i f-3\partial_i f=-2\partial_i f.$$

Since $s^i\partial_i f=\partial_r f=-2GM/r^2$,

$$\left(\partial_j\gamma_{ij}-\partial_i\gamma_{jj}\right)s^i =\frac{4GM}{r^2}.$$

The sphere integral is $16\pi GM$, and hence

$$E_{\rm ADM}=\frac{16\pi GM}{16\pi G}=M.$$

The time-symmetric condition sets $K_{ij}=\pi^{ij}=0$, but it does not set the boundary Hamiltonian to zero.

References

• Arnowitt, Richard, Stanley Deser, and Charles W. Misner. “The Dynamics of General Relativity.” In Gravitation: An Introduction to Current Research, edited by Louis Witten, 227–265. Wiley, 1962; reprinted as arXiv:gr-qc/0405109 (2004). arXiv.
• Brown, J. David, and James W. York Jr. “Quasilocal Energy and Conserved Charges Derived from the Gravitational Action.” Physical Review D 47, no. 4 (1993): 1407–1419. DOI. Open PDF.
• 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.