ADM Phase Space and Constraints

The ADM formulation turns spacetime geometry into constrained Hamiltonian data on a spatial slice. Its decisive result is that lapse and shift are Lagrange multipliers: in a closed universe the bulk Hamiltonian vanishes on solutions, while with a boundary the surviving generator is a charge such as ADM energy. This page derives that statement and the hypersurface-deformation algebra; quantization and the physical inner product come later.

Required background. Variational and Symplectic Geometry supplies action variations, canonical pairs, presymplectic degeneracies, and boundary generators.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical field variables; Einstein Equations supplies the covariant equations to recover; Quantum Information supplies the later state-space language; Formulations of General Relativity supplies the $3+1$ decomposition; and Quantum Gravity supplies the scale and evidence conventions used here.

Geometry of a spatial slice

Let spacetime have signature $(-,+,+,+)$ and let $\Sigma_t$ have future-directed unit normal $n^\mu$. We choose

$$K_{ij}=h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu, \qquad \dot h_{ij}=2NK_{ij}+D_iN_j+D_jN_i,$$

where $h_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$, $N$ is the lapse, and $N^i$ is the shift. This explicit sign choice is essential: changing the sign of $K_{ij}$ also changes the momentum convention. For the Einstein–Hilbert action with its required boundary term, the momentum conjugate to $h_{ij}$ is

$$\pi^{ij}=\frac{\sqrt h}{16\pi G}\left(K^{ij}-Kh^{ij}\right), \qquad \pi=h_{ij}\pi^{ij}=-\frac{\sqrt h}{8\pi G}K.$$

The lapse and shift have no velocities in the Lagrangian, so their conjugate momenta vanish as primary constraints. The Legendre transform with respect to $h_{ij}$ gives the canonical action and constraint structure derived in Arnowitt, Deser, and Misner 2008, §§3–4.

Hamiltonian and momentum constraints

With vacuum matter terms suppressed for clarity, the canonical action is

$$S=\int dt\int_\Sigma d^3x \left(\pi^{ij}\dot h_{ij}-N\mathcal H_\perp-N^i\mathcal H_i\right) -\int dt\,H_{\partial\Sigma},$$

where

$$\begin{aligned} \mathcal H_\perp &=\frac{16\pi G}{\sqrt h} \left(\pi_{ij}\pi^{ij}-\frac12\pi^2\right) -\frac{\sqrt h}{16\pi G}\left({}^{(3)}R-2\Lambda\right),\\ \mathcal H_i&=-2D_j\pi^j{}_i. \end{aligned}$$

Varying $N$ and $N^i$ imposes $\mathcal H_\perp=0$ and $\mathcal H_i=0$. The first is the normal-deformation, or Hamiltonian, constraint; the second generates spatial diffeomorphisms. Matter adds its energy and momentum densities with matching signs. Direct substitution of $\pi^{ij}$ recovers the Gauss–Codazzi constraint

$${}^{(3)}R+K^2-K_{ij}K^{ij}-2\Lambda=0,$$

which is an independent sign check for the chosen extrinsic-curvature convention. Gourgoulhon develops the constraints and Hamiltonian in Gourgoulhon 2007, §§4.2 and 4.5, Eqs. (4.19) and (4.22). His extrinsic curvature is the negative of the $K_{ij}$ used here; restoring $G$ and reversing $K_{ij}$ together translates his formulas to the present momentum convention.

The degree count is a useful first interpretation. The six independent components of $h_{ij}$ and their six momenta give twelve phase-space functions per point. Four independent first-class constraints remove four constrained directions and four gauge directions, leaving four phase-space functions: two canonical pairs, or the two local gravitational polarizations. This count applies at regular points of the vacuum constraint surface and does not count boundary charges or special global modes.

Why the algebra has structure functions

Smear the constraints as $H[N]=\int N\mathcal H_\perp$ and $D[\vec N]=\int N^i\mathcal H_i$. Ignoring boundary contributions and using compatible falloff or compact support,

$$\begin{aligned} \{D[\vec N],D[\vec M]\}&=D[[\vec N,\vec M]],\\ \{D[\vec N],H[M]\}&=H[\mathcal L_{\vec N}M],\\ \{H[N],H[M]\}&=D\!\left[h^{ij}(N\partial_jM-M\partial_jN)\right]. \end{aligned}$$

Two successive normal deformations do not simply commute: their commutator is a tangential deformation. Because $h^{ij}$ appears on the right, this is not a Lie algebra with constant structure coefficients. It is the canonical encoding of foliation independence. A proposed quantum constraint algebra must reproduce this relation in a suitable limit or explain a controlled deformation; merely finding states killed by separate regulated operators is not enough.

Boundary charges and the meaning of zero

Functional differentiability fixes $H_{\partial\Sigma}$. On compact $\Sigma$ without boundary it is absent, and the Hamiltonian is a sum of constraints. For asymptotically flat data, integration by parts produces a surface variation that is cancelled by adding the ADM charge. With asymptotic lapse approaching one and shift approaching zero, the on-shell Hamiltonian is the ADM energy, not zero Arnowitt, Deser, and Misner 2008, §§5–7.

Thus “the Hamiltonian vanishes” is a statement about bulk gauge generators on the constraint surface, not a statement that isolated gravitating systems have no energy. Boundary conditions choose the allowable gauge transformations: transformations that act nontrivially at the boundary can carry charges rather than being redundancies. Classical alternative gravities and their modified constraints are outside this page; the present result is the canonical form of general relativity.

The same distinction appears for a finite timelike boundary. One must fix a boundary variational problem, add its corresponding surface term, and identify which lapse–shift deformations preserve the boundary data. The resulting quasi-local generator need not equal the asymptotically flat ADM expression. A boundary charge is therefore not an optional correction to an otherwise complete bulk formula; it is part of the definition of the canonical problem.

Checks and limiting cases

Dimensions provide a quick check. In natural units, with coordinates carrying length and dimensionless $h_{ij}$, $[G]=L^2$, $[K]=L^{-1}$, and $[\pi^{ij}]=L^{-3}$. Both terms in $\mathcal H_\perp$ therefore have density dimension $L^{-4}$. For time-symmetric vacuum data, $K_{ij}=0$ reduces the Hamiltonian constraint to ${}^{(3)}R=2\Lambda$. For a weak perturbation about flat space, its linearization gives the familiar scalar and vector constraints, while the transverse-traceless modes survive as propagating data.

Exercises

Take vacuum, asymptotically flat initial data with $K_{ij}=0$ and $\Lambda=0$. What do the two ADM constraints require, and why can the ADM energy nevertheless be nonzero?

Solution

Time symmetry gives $\pi^{ij}=0$, so the momentum constraint $D_j\pi^j{}_i=0$ holds identically. The Hamiltonian constraint becomes ${}^{(3)}R=0$ pointwise. Scalar-flat does not mean that the full three-metric is globally Euclidean: the Schwarzschild time-symmetric slice, for example, has nontrivial geometry and an asymptotic $1/r$ term. The bulk constraint therefore vanishes while the differentiable generator retains its surface term. That surface integral measures ADM mass, showing explicitly why “constraint Hamiltonian equals zero” and “physical energy equals zero” are different claims.

References

• Arnowitt, Richard, Stanley Deser, and Charles Misner. “The Dynamics of General Relativity.” Reprint of the 1962 chapter, 2008. arXiv:gr-qc/0405109.
• Gourgoulhon, Éric. “3+1 Formalism and Bases of Numerical Relativity.” Lecture notes, 2007. arXiv:gr-qc/0703035.