Hamiltonian General Relativity

Hamiltonian general relativity recasts spacetime dynamics as constrained evolution of a spatial metric and its conjugate momentum. The lapse and shift do not carry independent velocities; they enforce the scalar and vector constraints. Those constraints generate normal and tangential deformations of the slice, while boundary terms convert selected deformations into physical energy and momentum generators.

Helpful background. Presymplectic Potentials and Currents supplies the canonical two-form and its gauge degeneracies, The Gibbons-Hawking-York Boundary Term supplies the differentiable Dirichlet action, The Einstein Equation supplies the covariant dynamics, Hypersurface Normals and Tangential Projectors supplies the geometry of a slice, and Formulations of General Relativity locates the canonical description among equivalent formulations.

From the ADM action to phase space

For

$$(\partial_t)^\mu=\alpha n^\mu+\beta^\mu, \qquad K_{ij}=-\frac12\mathcal L_n\gamma_{ij},$$

the gravitational bulk action is

$$S_{\rm ADM} =\frac1{16\pi G}\int dt\,d^3x\, \alpha\sqrt\gamma \left({}^{(3)}R+K_{ij}K^{ij}-K^2-2\Lambda\right),$$

supplemented by the surface terms appropriate to the region. Since

$$K_{ij}=-\frac1{2\alpha} \left(\dot\gamma_{ij}-D_i\beta_j-D_j\beta_i\right),$$

the sign-fixed canonical momentum is

$$\pi^{ij} =\frac{\sqrt\gamma}{16\pi G} \left(K\gamma^{ij}-K^{ij}\right).$$

There are no $\dot\alpha$ or $\dot\beta^i$ terms, so their momenta vanish as primary constraints. The Legendre transform gives

$$H_T=\int_\Sigma d^3x \left( \alpha\mathcal C_\perp+\beta^i\mathcal C_i +u p_\alpha+u^ip_i \right)+B,$$

with vacuum secondary constraints

$$\mathcal C_\perp =\frac{16\pi G}{\sqrt\gamma} \left(\pi_{ij}\pi^{ij}-\frac12\pi^2\right) -\frac{\sqrt\gamma}{16\pi G} \left({}^{(3)}R-2\Lambda\right),$$ $$\mathcal C_i=-2D_j\pi^j{}_i.$$

Matter adds $+\sqrt\gamma\rho$ to $\mathcal C_\perp$ and $-\sqrt\gamma j_i$ to $\mathcal C_i$ for $j_i=-\gamma_i{}^\mu n^\nu T_{\mu\nu}$. These signs reproduce the normal and tangential projections of Einstein’s equation.

Deformations, gauge, and boundaries

Smeared constraints

$$H[N]=\int_\Sigma N\mathcal C_\perp, \qquad D[\vec N]=\int_\Sigma N^i\mathcal C_i$$

obey, on a closed slice or for compactly supported smearings,

$$\{D[\vec N],D[\vec M]\}=D[[\vec N,\vec M]],$$ $$\{D[\vec N],H[M]\}=H[\mathcal L_{\vec N}M],$$ $$\{H[N],H[M]\} =D\!\left[\gamma^{ij} \left(N\partial_jM-M\partial_jN\right)\partial_i\right].$$

The inverse metric in the last bracket is a phase-space variable, so the algebra has structure functions. It encodes the geometry of composing normal and tangential hypersurface deformations rather than a Lie algebra of four fixed translation directions.

Proper gauge smearings generate different canonical descriptions of the same spacetime. A Dirac observable is constant along those gauge orbits. Smearings that approach nonzero asymptotic symmetries are different: their generators must be completed by surface terms and can carry physical charges. On the constraint surface the bulk integral vanishes, but the completed Hamiltonian can remain equal to its boundary term. For asymptotically flat unit time translation, that term is the ADM energy.

The action, momentum, and constraint construction is developed in Gourgoulhon 2007, §4.5, pp. 65–69, PDF and Arnowitt, Deser, and Misner 1962/2004, §§3–6, PDF. The asymptotic charges are treated in Gourgoulhon 2007, Ch. 7, pp. 103–124, PDF.

A route through the chapter

Read the six pages in order, because every sign in the later constraints depends on the action and momentum conventions established first:

1. The ADM Action separates the Einstein–Hilbert action into spatial curvature, an extrinsic-curvature kinetic term, and boundary divergences.
2. Canonical Momentum of Spatial Geometry differentiates with respect to $\dot\gamma_{ij}$ and inverts the trace-reversed momentum relation in arbitrary spatial dimension.
3. Canonical Constraints and Total Hamiltonian performs the Legendre transform, adds matter, and distinguishes primary from secondary constraints.
4. The Hypersurface-Deformation Algebra derives the three smeared brackets and interprets their metric-dependent structure functions.
5. Gauge Orbits and Canonical Observables develops Dirac observables, gauge fixing, relational clocks, reduced phase space, and their global limitations.
6. Boundary Hamiltonians and Gravitational Energy makes the generators differentiable and relates their surface terms to ADM, quasilocal, and horizon energies.

The first three pages construct the canonical system. The fourth proves its geometric closure. The final two explain what is gauge and what becomes a physical charge once boundary conditions are specified.

Two checks that fix the interpretation

Inertial Minkowski slices have

$$\gamma_{ij}=\delta_{ij}, \qquad K_{ij}=0, \qquad \pi^{ij}=0.$$

Both bulk constraints vanish, and the standard asymptotic reference sets the ADM energy to zero. This checks normalization but not the trace reversal.

Milne slices describe the same flat spacetime with

$${}^{(3)}R=-\frac6{\tau^2}, \qquad K_{ij}=-\frac1\tau\gamma_{ij}, \qquad \pi^{ij}=-\frac{\sqrt\gamma}{8\pi G\tau}\gamma^{ij}.$$

The momentum and curvature pieces of $\mathcal C_\perp$ cancel. This confirms that canonical momentum measures the embedding of the slice, not four-dimensional curvature. It also checks the relative sign between the ADM action, $\pi^{ij}$, and the scalar constraint.

Scope and limitations

The canonical formulation is classically equivalent to Einstein dynamics on constraint-satisfying data with compatible boundary conditions. It does not by itself solve the constraints, select a gauge, prove well-posedness, or provide a quantum theory. Reduced phase space may exist only locally, and relational clocks can fail where their gauge-fixing matrix becomes singular.

There is likewise no boundary-independent gravitational energy density in this construction. ADM, Brown–York, and horizon charges use different boundary structures and generator normalizations. Comparisons are meaningful only after those choices have been stated.

Review the chapter

For vacuum Milne data with $\alpha=1$, $\beta^i=0$, and $\Lambda=0$:

1. derive $\pi^{ij}$ from $K_{ij}=-\gamma_{ij}/\tau$;
2. verify the scalar and vector constraints;
3. explain why the nonzero momentum is gauge-compatible with flat spacetime;
4. state whether the bulk Hamiltonian or a possible boundary Hamiltonian must vanish on shell.

Solution

Since $K=-3/\tau$ and $K^{ij}=-\gamma^{ij}/\tau$,

$$\pi^{ij} =\frac{\sqrt\gamma}{16\pi G} \left(-\frac3\tau+\frac1\tau\right)\gamma^{ij} =-\frac{\sqrt\gamma}{8\pi G\tau}\gamma^{ij}.$$

Writing $A=\sqrt\gamma/(8\pi G\tau)$ gives $\pi_{ij}\pi^{ij}-\tfrac12\pi^2=-\tfrac32A^2$. Hence

$$\frac{16\pi G}{\sqrt\gamma} \left(-\frac32A^2\right) =-\frac{3\sqrt\gamma}{8\pi G\tau^2},$$

while

$$-\frac{\sqrt\gamma}{16\pi G}{}^{(3)}R =\frac{3\sqrt\gamma}{8\pi G\tau^2}.$$

The scalar constraint vanishes. Homogeneity and metric compatibility give $D_j\pi^j{}_i=0$, so the vector constraint also vanishes. The momentum is nonzero because the hyperboloidal slices have nonzero extrinsic curvature; the constraints, not $\pi^{ij}=0$, characterize admissible flat-spacetime data. On shell the bulk constraint integral vanishes. A completed Hamiltonian can nevertheless equal a nonzero boundary charge when the boundary conditions admit a physical symmetry.

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.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.