Constraint-Preserving Boundary Conditions

Constraint-preserving boundary conditions control every characteristic field that enters the domain in the subsidiary system obeyed by the Einstein constraints. They do not set all constraints pointwise on the boundary, and they do not determine the physical or gauge incoming fields of the main evolution system. The continuum implication is conditional: if the subsidiary IBVP has an energy estimate, its initial data vanish, and its incoming boundary data vanish homogeneously, then the constraints remain zero. Controlled nonzero incoming data instead give a bound on constraint violations.

Required background. The Gibbons–Hawking–York Boundary Term supplies the boundary geometry, while PDEs and Green Functions for Gravitation supplies characteristic decompositions, subsidiary equations, and energy uniqueness.

Helpful background. The Einstein Equation supplies Hamiltonian, momentum, and gauge constraints; Hypersurface Normals and Tangential Projectors supplies outward normals; Formulations of General Relativity orients the volume; and Causality and Global Structure supplies the domain in which uniqueness propagates zero data.

Main and subsidiary characteristic fields

Write a hyperbolic evolution system schematically as

$$\partial_tU=A^i(U)\partial_iU+F(U).$$

The formulation also has constraints $C(U,\partial U)=0$. Using the evolution equations and the Bianchi identities, derive a closed subsidiary system

$$\partial_tC=B^i(U)\partial_iC+M(U,\partial U)C.$$

At a timelike boundary with outward spacelike normal $s_i$, diagonalize the normal symbol $B^s=B^is_i$. With the evolution equation written as $\partial_tC=B^s\partial_sC+\cdots$, the propagation speed is minus the eigenvalue of $B^s$; moving the spatial term to the left changes that sign convention. A subsidiary characteristic field is incoming when its future characteristic points from the boundary into the computational domain. Denote the incoming, outgoing, and tangential or zero-speed fields by $C_{\mathrm{in}}$, $C_{\mathrm{out}}$, and $C_0$.

Constraint preservation requires a condition on every member of $C_{\mathrm{in}}$, for example

$$C_{\mathrm{in}}=H_C C_{\mathrm{out}}+q_C, \qquad \|H_C\|\le1.$$

For exact constraint preservation one usually chooses homogeneous compatible data $q_C=0$ and a contraction compatible with the subsidiary energy. No independent condition is assigned to outgoing fields; doing so generally overdetermines the problem. Zero-speed fields obey intrinsic boundary or bulk transport equations and must be handled according to the named formulation.

The norms here and below use characteristic amplitudes weighted by the symmetrizer and the absolute normal speeds. With unequal speeds, an unweighted component norm would not give the displayed contraction test.

Energy implication

Suppose the subsidiary system is symmetric hyperbolic with positive symmetrizer $S$. Its energy

$$E_C(t)=\frac12\int_{\Sigma_t}C^TS C\,dV$$

obeys an estimate of the form

$$\frac{dE_C}{dt} \le cE_C -\frac12\int_{\partial\Sigma_t} \left( \lVert C_{\mathrm{out}}\rVert^2 -\lVert C_{\mathrm{in}}\rVert^2 \right)dA.$$

If $C_{\mathrm{in}}=H_CC_{\mathrm{out}}$ with contraction norm at most one, the boundary term cannot inject subsidiary energy. With $E_C(0)=0$, Grönwall’s inequality gives $E_C(t)=0$, hence $C=0$ in the domain of dependence. If $q_C\ne0$, the estimate instead bounds constraint energy by the boundary forcing; it does not make the constraints vanish.

Friedrich and Nagy construct homogeneous symmetric-hyperbolic systems for their geometric “zero quantities” and prove that vanishing initial constraints imply the full vacuum equations in the relevant domain in Friedrich and Nagy 1999, Theorem 6.1 and proof, pp. 636–644. Their special frame reduction has the remarkable property that the subsidiary estimate needs no extra boundary condition; this formulation-specific feature must not be promoted to a universal rule.

Harmonic constraint hierarchy

For generalized harmonic evolution,

$$C^\mu=\Gamma^\mu-H^\mu, \qquad \Gamma^\mu=g^{\rho\sigma}\Gamma^\mu{}_{\rho\sigma},$$

satisfies a wave-type subsidiary equation when the chosen reduced equations hold. Here $H^\mu$ prescribes the contracted Christoffel symbol, so $\Box_gx^\mu=-H^\mu$ in the target gauge. Zero initial data for this second-order subsidiary equation means both $C^\mu=0$ and its initial normal derivative vanish; the latter follows from the Einstein initial constraints with compatible gauge initialization. At a frozen planar boundary, its incoming combinations are built from normal and time derivatives of $C^\mu$. A useful boundary hierarchy first fixes combinations of metric derivatives that enforce the four harmonic constraints, then assigns gauge and radiative data to the remaining incoming fields. This triangular order avoids using the same incoming component twice.

Kreiss and collaborators analyze harmonic boundary conditions whose constraint data vanish and prove strong well-posedness when the boundary coefficient matrices meet their stated spectral conditions in Kreiss et al. 2007, §3, eqs. (11)–(14) and Theorem 3, pp. 7–10, PDF. The proof exploits the sequential structure; an arbitrary mixing of tangential derivatives does not inherit the theorem.

Boundary implementation checklist

For a named formulation:

1. derive the subsidiary system from the continuum evolution equations;
2. compute its normal characteristic fields using the actual boundary motion;
3. identify every incoming constraint mode, including gauge-constraint modes;
4. express the desired homogeneous subsidiary condition in the main characteristic variables;
5. verify that the remaining main incoming fields are available for physical and gauge data;
6. close the main and subsidiary energy estimates and check the initial corner.

A condition called “constraint preserving” because it sets one familiar scalar constraint to zero is incomplete unless this full incoming inventory has been performed.

Continuum and numerical meanings

The theorem is a continuum uniqueness statement. A discretization produces truncation and roundoff errors throughout the bulk, so numerical constraints do not remain identically zero even with a correct boundary condition. Boundary treatment should prevent additional incoming violations and support a discrete energy estimate; damping may reduce bulk violations but does not replace the incoming-mode condition.

Common pitfalls

Do not impose all constraints as independent boundary values. Only incoming subsidiary fields accept boundary data.

Main-system stability does not prove subsidiary stability. Both boundary symbols and estimates must be checked.

Constraint preservation does not choose radiation. Two physical incoming polarizations and gauge modes still require separate prescriptions in common Einstein reductions.

Exercise: one incoming subsidiary mode

Suppose a subsidiary energy satisfies

$$\frac{dE}{dt}\le cE-\frac12(C_{\mathrm{out}}^2-C_{\mathrm{in}}^2)$$

at the boundary. Determine the range of constant $a$ for which $C_{\mathrm{in}}=aC_{\mathrm{out}}$ cannot inject constraint energy.

Solution

Substitution gives the boundary contribution

$$-\frac12(1-a^2)C_{\mathrm{out}}^2.$$

It is nonpositive precisely when $\lvert a\rvert\le1$. With homogeneous initial data, the resulting estimate and Grönwall imply $E=0$. If $\lvert a\rvert\gt 1$, the boundary term can be positive and the energy argument does not establish constraint preservation.

References

• Friedrich, Helmut, and Gabriel Nagy. “The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.” Communications in Mathematical Physics 201 (1999): 619–655. DOI.
• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Well-Posed Initial-Boundary Value Problem for the Harmonic Einstein Equations Using Energy Estimates.” Classical and Quantum Gravity 24, no. 23 (2007): 5973–5984. DOI; Open PDF.