Canonical Constraints and Total Hamiltonian

The Legendre transform of the ADM action does not produce an unconstrained Hamiltonian system. Lapse and shift multiply, respectively, a scalar and a spatial-vector constraint, while their own conjugate momenta vanish. The resulting total Hamiltonian separates bulk gauge evolution from boundary generators and makes the two local gravitational degrees of freedom visible.

Required background. Presymplectic Potentials and Currents supplies the canonical meaning of the action’s boundary variation, while The Gibbons-Hawking-York Boundary Term supplies the differentiable spacetime action from which the split begins.

Helpful background. The Einstein Equation supplies the covariant constraints being recast, Hypersurface Normals and Tangential Projectors supplies normal and tangential projections, and Formulations of General Relativity locates the Hamiltonian system among equivalent formulations.

Legendre transform of the gravitational action

The velocity of the spatial metric is

$$\dot\gamma_{ij}=-2\alpha K_{ij}+D_i\beta_j+D_j\beta_i,$$

and the momentum appropriate to this sign of $K_{ij}$ is

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

Consequently,

$$-2\alpha\pi^{ij}K_{ij} =\frac{\alpha\sqrt\gamma}{8\pi G} \left(K_{ij}K^{ij}-K^2\right).$$

The shift term can be integrated by parts:

$$2\pi^{ij}D_i\beta_j =-2\beta^iD_j\pi^j{}_i +2D_i(\pi^{ij}\beta_j).$$

Subtracting the ADM Lagrangian and collecting the divergence into a boundary term gives

$$H_{\mathrm{grav}} =\int_\Sigma d^3x \left(\alpha\mathcal C_\perp^{\rm g} +\beta^i\mathcal C_i^{\rm g}\right)+B,$$

where

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

and

$$\mathcal C_i^{\rm g}=-2D_j\pi^j{}_i.$$

This Legendre transform and its surface contribution are developed in Arnowitt, Deser, and Misner 1962/2004, §§3–5, PDF; the same sign-consistent construction is given in Gourgoulhon 2007, §4.5.2, pp. 67–69, PDF.

Matter and the secondary constraints

Define the projections of stress energy by

$$\rho=T_{\mu\nu}n^\mu n^\nu, \qquad j_i=-\gamma_i{}^\mu n^\nu T_{\mu\nu}.$$

For minimally coupled matter with no metric-velocity coupling, its Legendre transform adds the corresponding matter canonical terms and changes the gravitational constraints to

$$\mathcal C_\perp =\mathcal C_\perp^{\rm g}+\sqrt\gamma\,\rho, \qquad \mathcal C_i =-2D_j\pi^j{}_i-\sqrt\gamma\,j_i.$$

Varying lapse and shift gives

$$\mathcal C_\perp=0, \qquad \mathcal C_i=0.$$

In geometric variables these are exactly

$${}^{(3)}R+K^2-K_{ij}K^{ij}-2\Lambda=16\pi G\rho,$$ $$D_jK^j{}_i-D_iK=8\pi G j_i.$$

The signs in the vector constraint are linked: with $j_i=-\gamma_i{}^\mu n^\nu T_{\mu\nu}$, the identity $-2D_j\pi^j{}_i=\sqrt\gamma j_i$ holds on the momentum constraint.

A real scalar field supplies a useful independent sign check. For $\mathcal L_{\rm m}=-\sqrt{-g}[(\nabla\phi)^2/2+V(\phi)]$, define $v=n^\mu\nabla_\mu\phi=(\dot\phi-\beta^iD_i\phi)/\alpha$. Then

$$p_\phi=\sqrt\gamma\,v, \qquad \rho=\frac12v^2+\frac12D_i\phi D^i\phi+V, \qquad j_i=-vD_i\phi.$$

The matter Legendre transform is therefore

$$p_\phi\dot\phi-\mathcal L_{\rm m} =\alpha\sqrt\gamma\rho+\beta^i p_\phi D_i\phi =\alpha\sqrt\gamma\rho-\beta^i\sqrt\gamma j_i.$$

The minus sign multiplying $j_i$ is thus required by the scalar’s ordinary canonical momentum, not a discretionary choice. Gauge fields can also have their own Gauss constraints; the full total Hamiltonian then includes those additional multipliers.

Primary constraints and the total Hamiltonian

Because neither $\dot\alpha$ nor $\dot\beta^i$ occurs in the action,

$$p_\alpha=0, \qquad p_i=0$$

are primary constraints. The Dirac total Hamiltonian is

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

The arbitrary multipliers $u$ and $u^i$ determine the velocities of lapse and shift. Preserving their primary constraints in time produces the scalar and vector constraints. The metric Hamilton equation begins

$$\dot\gamma_{ij} =\frac{32\pi G\alpha}{\sqrt\gamma} \left(\pi_{ij}-\frac12\pi\gamma_{ij}\right) +\mathcal L_\beta\gamma_{ij},$$

which returns $\dot\gamma_{ij}=-2\alpha K_{ij}+\mathcal L_\beta\gamma_{ij}$. The momentum equation is equivalent to the spatial projection of the Einstein equation.

At a regular point of the pure-gravity constraint surface on a closed slice, the twelve canonical fields $(\gamma_{ij},\pi^{ij})$ are reduced by four first-class constraints and four gauge directions, leaving four phase-space functions, or two configuration degrees of freedom, per point. On a region with boundary, $B$ is not optional bookkeeping: it is chosen so that $H_T$ has well-defined functional derivatives under the stated boundary conditions.

Flat-spacetime checks

Inertial Minkowski data have $\gamma_{ij}=\delta_{ij}$, $\pi^{ij}=0$, and ${}^{(3)}R=0$, so both vacuum constraints vanish. For Milne slices,

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

The momentum constraint vanishes by spatial homogeneity. The curvature and momentum terms in $\mathcal C_\perp$ cancel, confirming that a nonzero canonical momentum can describe flat spacetime.

Exercise: verify the Milne scalar constraint

Using the Milne data above with $\Lambda=0$ and no matter, compute $\pi_{ij}\pi^{ij}-\tfrac12\pi^2$ and verify $\mathcal C_\perp=0$.

Solution

Let $A=\sqrt\gamma/(8\pi G\tau)$, so $\pi^{ij}=-A\gamma^{ij}$ and $\pi=-3A$. Then

$$\pi_{ij}\pi^{ij}-\frac12\pi^2 =3A^2-\frac92A^2=-\frac32A^2.$$

The kinetic contribution is therefore

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

The curvature contribution is

$$-\frac{\sqrt\gamma}{16\pi G} \left(-\frac6{\tau^2}\right) =\frac{3\sqrt\gamma}{8\pi G\tau^2}.$$

Their sum vanishes. Milne data satisfy the vacuum scalar constraint even though neither term vanishes separately.

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.