Hamiltonian and Momentum Constraints

Initial geometry in general relativity cannot be chosen freely. On a spacelike hypersurface, one scalar and three vector equations constrain the spatial metric, extrinsic curvature, and matter data before evolution begins. They are the normal–normal and normal–tangential projections of Einstein’s equation. Their absence of second normal derivatives makes them restrictions on a snapshot rather than evolution equations for that snapshot.

Required background. Hypersurface Normals and Tangential Projectors supplies the induced geometry and projections, while PDEs and Green Functions for Gravitation supplies the distinction between constraint and evolution equations.

Helpful background. The Einstein Equation supplies the covariant dynamics; Formulations of General Relativity orients the later evolution systems; and Presymplectic Potentials and Currents prepares the canonical interpretation of first-class constraints.

Geometric data on a slice

Let $\Sigma$ be a spacelike hypersurface with future unit normal $n^\mu$,

$$n^\mu n_\mu=-1, \qquad \gamma_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu.$$

This volume fixes

$$K_{\mu\nu} =-\gamma_\mu{}^\rho\gamma_\nu{}^\sigma \nabla_\rho n_\sigma.$$

The matter projections measured by normal observers are

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

Thus an initial data set is

$$(\Sigma,\gamma_{ij},K_{ij};\text{matter positions and momenta}),$$

with the matter variables also satisfying their own constraints. The lapse and shift are not part of this intrinsic geometric data; they specify how a development is coordinatized.

The Hamiltonian constraint

The contracted Gauss relation for the selected sign of $K_{ij}$ is

$${}^{(3)}R+K^2-K_{ij}K^{ij} =2G_{\mu\nu}n^\mu n^\nu.$$

Contracting the Einstein equation

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$$

twice with $n^\mu$ gives

$$G_{\mu\nu}n^\mu n^\nu-\Lambda=8\pi G\rho.$$

Combining them yields the scalar or Hamiltonian constraint

$$\boxed{ \mathcal H \equiv{}^{(3)}R+K^2-K_{ij}K^{ij} -16\pi G\rho-2\Lambda=0. }$$

The equation balances intrinsic spatial curvature, the two independent quadratic contractions of extrinsic curvature, matter energy density, and the cosmological term. Reversing the sign of $K_{ij}$ changes neither quadratic term and therefore leaves this displayed constraint unchanged.

In time-symmetric data, $K_{ij}=0$. Vacuum data with $\Lambda=0$ must then satisfy

$${}^{(3)}R=0.$$

Scalar flatness does not mean that $\gamma_{ij}$ is flat: in three dimensions the Ricci tensor can be nonzero while its trace vanishes.

The momentum constraints

The contracted Codazzi relation gives

$$\gamma_i{}^\mu n^\nu G_{\mu\nu} =-D_jK^j{}_i+D_iK.$$

The same projection of Einstein’s equation is

$$\gamma_i{}^\mu n^\nu G_{\mu\nu} =8\pi G\,\gamma_i{}^\mu n^\nu T_{\mu\nu} =-8\pi G j_i,$$

because the cosmological term has zero mixed projection. Hence

$$\boxed{ \mathcal M_i \equiv D_jK^j{}_i-D_iK-8\pi G j_i=0. }$$

This sign is sensitive to both conventions: changing $K_{ij}\mapsto-K_{ij}$ reverses the geometric left-hand side, while defining matter momentum without the minus sign reverses its source. Quoting a momentum constraint without both definitions is incomplete.

The three equations state that the longitudinal part of the extrinsic curvature is fixed by matter momentum and the gradient of its trace. They do not force the trace-free transverse part to vanish.

Gourgoulhon derives the Gauss–Codazzi projections in Gourgoulhon 2007, §§2.5 and 4.1, pp. 34–38 and 51–54, PDF. Bartnik and Isenberg formulate the same geometric constraint map and its role in Cauchy data in Bartnik and Isenberg 2004, §2, pp. 3–7, PDF. Their second-fundamental-form convention is opposite to the one used here, so their mixed projection must be translated.

Why these are constraints

The spacetime Einstein equation has ten symmetric components. Relative to a slice:

• one normal–normal projection gives $\mathcal H$;
• three normal–tangential projections give $\mathcal M_i$; and
• six tangential–tangential projections contain the normal evolution of $K_{ij}$.

The Gauss and Codazzi equations express the first four projections entirely in terms of $\gamma_{ij}$, $K_{ij}$, matter data, and spatial derivatives. No second derivative normal to $\Sigma$ occurs. Thus arbitrary values of the twelve functions in $(\gamma_{ij},K_{ij})$ do not define an admissible spacetime snapshot.

At the local counting level, twelve gravitational phase-space functions minus four constraints and four diffeomorphism gauge directions leave four physical phase-space functions, or two configuration degrees of freedom. This count assumes a regular first-class system and appropriate boundary behavior; it is not an existence theorem for the constraint equations.

Constraint propagation

The contracted Bianchi identity and matter conservation imply a homogeneous subsidiary system for $\mathcal H$ and $\mathcal M_i$ once the six evolution equations hold. Schematically,

$$\partial_\perp\mathcal H =\text{spatial derivatives of }\mathcal M +\text{lower-order constraints},$$ $$\partial_\perp\mathcal M_i =\text{spatial derivative of }\mathcal H +\text{lower-order constraints}.$$

Therefore exactly vanishing constraints propagate in the continuum Cauchy problem under the hypotheses of a well-posed formulation. This does not make the initial solve optional. Nor does it guarantee that small violations remain small: off-constraint behavior depends on the chosen evolution system, constraint additions, gauge, and boundaries.

For a finite domain, incoming characteristic fields of the subsidiary system must also be controlled. Initial constraint satisfaction alone cannot prevent a timelike boundary from injecting violations.

Independent checks

Minkowski slice. With $\gamma_{ij}=\delta_{ij}$, $K_{ij}=0$, $\rho=j_i=0$, and $\Lambda=0$, both constraints vanish.

Milne hyperboloid. Flat spacetime admits slices with

$${}^{(3)}R=-\frac6{\tau^2}, \qquad K_{ij}=-\frac1\tau\gamma_{ij}.$$

Then $K=-3/\tau$ and $K_{ij}K^{ij}=3/\tau^2$, so

$${}^{(3)}R+K^2-K_{ij}K^{ij} =-\frac6{\tau^2}+\frac9{\tau^2}-\frac3{\tau^2}=0.$$

The check shows why intrinsic curvature alone does not decide whether initial data can be embedded in flat spacetime.

Common pitfalls

A constraint is not a gauge condition. Constraints restrict admissible geometric data; lapse and shift select a coordinate development of those data.

Propagation is not satisfaction. A homogeneous subsidiary system preserves zero data but does not turn arbitrary nonzero initial residuals into zero.

Time symmetry is stronger than maximality. $K_{ij}=0$ implies $K=0$, but a maximal slice can have nonzero trace-free $K_{ij}$.

Exercise: pure-trace vacuum data

Assume vacuum, $\Lambda=0$, and

$$K_{ij}=\frac13K\gamma_{ij}, \qquad D_iK=0.$$

Show that the momentum constraints hold and determine the scalar curvature required by the Hamiltonian constraint.

Solution

Since $K$ is spatially constant,

$$D_jK^j{}_i-D_iK =\frac13D_iK-D_iK=0.$$

Also

$$K_{ij}K^{ij}=\frac13K^2.$$

The Hamiltonian constraint becomes

$${}^{(3)}R+K^2-\frac13K^2=0,$$

so

$${}^{(3)}R=-\frac23K^2.$$

For $K=-3/\tau$, this gives $-6/\tau^2$, precisely the Milne-hyperboloid check.

References

• Bartnik, Robert, and Jim Isenberg. “The Constraint Equations.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 1–38. Basel: Birkhäuser, 2004. arXiv:gr-qc/0405092.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.