Constraint Equations and Initial Data

Initial data for general relativity are not an arbitrary spatial metric and extrinsic curvature. They must satisfy four nonlinear constraint equations on each spacelike slice before the remaining Einstein equations can evolve them. This chapter develops the geometric content of those constraints, the elliptic decompositions used to solve them, and the asymptotic, black-hole, and gluing choices that turn a formal solution into a well-defined physical data set.

Helpful background. The Einstein Equation supplies the parent field system; Hypersurface Normals and Tangential Projectors supplies induced geometry and extrinsic curvature; PDEs and Green Functions for Gravitation supplies elliptic methods and boundary conditions; Formulations of General Relativity orients the volume; and Presymplectic Potentials and Currents prepares the later phase-space interpretation.

From spacetime equations to admissible data

For the convention $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$, the normal and mixed projections of Einstein’s equation give

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

These relations restrict the twelve components of $(\gamma_{ij},K_{ij})$ but do not form an ordinary set of algebraic equations for four selected components. Spatial diffeomorphism freedom, elliptic kernels, topology, boundaries, and asymptotic conditions all enter the solution space. Bartnik and Isenberg explain this underdetermined structure and the main construction methods in Bartnik and Isenberg 2004, §§2 and 4, pp. 4–29, PDF.

Constraint satisfaction is necessary, not sufficient, for a useful model. One must also state which asymptotic end or finite boundary is intended, which matter variables are freely specified, what regularity is required, and how the free data encode the target physical system.

A coherent route through the chapter

The six pages form a construction sequence:

1. Hamiltonian and Momentum Constraints derives the two projections, fixes signs, and checks them on flat and hyperboloidal slices.
2. York Decomposition of Symmetric Tensors separates trace, TT, and longitudinal pieces and exposes conformal-Killing kernels.
3. Conformal Decomposition of the Einstein Constraints derives the vector and Lichnerowicz equations and checks every conformal exponent.
4. Asymptotically Flat Initial Data states end conditions, ADM energy-momentum, and the parity needed for higher charges.
5. Black-Hole Initial Data compares throats, punctures, excision, Bowen–York fields, and quasiequilibrium choices.
6. Gluing and Deformation of Initial Data explains localized connected sums, exact stationary exteriors, and adjoint-kernel obstructions.

Read the first three in order: the projection formulas determine what must be solved, the York split identifies a useful elliptic operator, and the conformal method assigns weights that couple the scalar and vector equations correctly. The last three then change the global question without changing those local constraints.

Choices that must stay explicit

Several distinctions recur across the chapter:

Choice	What it controls	What it does not guarantee
Mean curvature $K$	Coupling of the conformal equations and slicing geometry	Existence or uniqueness by itself
TT seed	Trace-free conformal curvature data	Gauge-invariant radiation at infinity
End falloff and parity	Finiteness and invariance of asymptotic charges	A unique interior geometry
Puncture or excision	Topology and elliptic domain	Astrophysical fidelity
Local no-KID condition	Surjectivity needed for localized correction	Dynamical stability of glued data

Gourgoulhon develops the constraints, conformal decomposition, asymptotic charges, and standard black-hole constructions with one consistent 3+1 notation in Gourgoulhon 2007, Chapters 5–8, pp. 71–150, PDF. Where another source defines $K_{ij}$ with the opposite sign, every term linear in $K_{ij}$ must be translated; quadratic constraint terms remain unchanged.

Checks before evolution

A trustworthy data set should pass more than a residual norm. Check the scalar and vector constraints independently; verify boundary and asymptotic conditions; project residuals against elliptic adjoint kernels; evaluate the intended charges at increasing radii; and confirm that topology and coordinate compactification are interpreted correctly. For numerical data, convergence of the constraints must be demonstrated separately from convergence of masses, spins, horizons, and wave content.

The chapter stops before evolution. Constraint propagation and continuum hyperbolicity belong to the next chapter, while production algorithms, discretization choices, and astrophysical parameter tuning belong to numerical relativity.

Review the chapter

Suppose vacuum conformal data have $\bar\gamma_{ij}=\delta_{ij}$, $K=0$, and a TT tensor $\bar A^{ij}_{\mathrm{TT}}=O(r^{-3})$. State the two determined fields and explain which equation decouples first.

Solution

The determined fields are a positive conformal factor $\psi$ and a vector potential $W^i$. Constant mean curvature makes

$$\bar\Delta_{\mathbb L}W^i=0.$$

On a complete domain without inner boundaries, with decay and regularity that eliminate the boundary term and with no allowed conformal-Killing kernel, $W^i=0$. If the domain has punctures or excision surfaces, their data must also be fixed: decay at the outer end alone does not select that solution. The Hamiltonian constraint then becomes

$$\Delta\psi +\frac18 \bar A^{\mathrm{TT}}_{ij}\bar A_{\mathrm{TT}}^{ij}\psi^{-7}=0, \qquad \psi\to1.$$

The $r^{-3}$ TT falloff is fast enough not to supply ADM linear momentum, although its angular structure may encode other information. Solving this scalar equation still requires positivity, regularity, and the correct behavior at every end or inner boundary.

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.