The Einstein Constraint Map

The Einstein constraint map packages the four equations that admissible Cauchy data must satisfy. Its linearization is underdetermined elliptic: its principal symbol is onto, although it has a large kernel. The kernel of the formal adjoint instead detects Killing initial data and the finite-dimensional obstructions that appear in deformation and gluing theorems. An operator’s kernel and its adjoint kernel play different roles here.

Required background. Analysis for Mathematical Relativity supplies elliptic operators and adjoints, The Einstein Equation supplies the field equation, Hypersurface Normals and Tangential Projectors supplies normal projections, and Differential Equations for Relativity supplies linearization and boundary-value reasoning.

Helpful background. Causality and Global Structure explains developments of initial data, while PDEs and Green Functions for Gravitation reviews elliptic solvability.

The nonlinear constraint operator

Let $\Sigma$ be a three-manifold, possibly with specified asymptotic ends or boundary. An initial data set consists of a Riemannian metric $h_{ij}$ and a symmetric tensor $K_{ij}$. We use

$$K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$$

for a future unit normal. With $K=h^{ij}K_{ij}$, define the constraint map

$$\Phi(h,K)= \begin{pmatrix} R(h)+K^2-K_{ij}K^{ij}-2\Lambda-16\pi G\rho\\[2mm] D^j(K_{ij}-Kh_{ij})-8\pi G j_i \end{pmatrix}.$$

The constraint surface is $\Phi^{-1}(0)$, together with any matter constraints and boundary or asymptotic conditions. The first component is scalar and the second is a one-form: four equations constrain the twelve local components of $(h,K)$. Counting components does not prove solvability, because diffeomorphism covariance, ellipticity, topology, and asymptotic charges create kernels and cokernels.

The map acts between weighted Sobolev or Hölder manifolds chosen for the problem. Its smoothness depends on the regularity of $R(h)$ and the multiplication properties of those spaces. A statement that “nearby data can be corrected” is meaningful only after these Banach spaces and boundary conditions are fixed. Bartnik and Isenberg set out the operator and its role in Bartnik and Isenberg 2004, §2, pp. 3–7 and §§3.1–3.4, pp. 7–13.

Linearization and adjoint obstructions

At time-symmetric flat vacuum data $(\delta,0)$ with $\Lambda=0$, write variations $(\gamma_{ij},p_{ij})$. The linearization is

$$D\Phi_{(\delta,0)}(\gamma,p) = \begin{pmatrix} \partial_i\partial_j\gamma^{ij}-\Delta\operatorname{tr}\gamma\\[1mm] \partial^j(p_{ij}-\operatorname{tr}p\,\delta_{ij}) \end{pmatrix}.$$

Pair its output with a scalar–vector pair $(N,X^i)$ using the Euclidean volume form and the ordinary tensor inner products. For compactly supported variations, integration by parts gives the exact formal adjoint of the displayed normalization:

$$D\Phi^*_{(\delta,0)}(N,X) = \left( \partial_i\partial_jN-\Delta N\,\delta_{ij}, -\partial_{(i}X_{j)}+\partial_kX^k\delta_{ij} \right),$$

Parentheses have weight $1/2$. On general vacuum data, curvature and $K$ couple $N$ and $X$. Solutions of

$$D\Phi^*_{(h,K)}(N,X)=0$$

are Killing initial data, or KIDs, after the fixed lapse–shift rescaling determined by the normalization of $\Phi$. For the unweighted scalar and momentum components above, $(N,X)$ corresponds to a Killing field with initial decomposition $-2Nn+X$; many sources instead normalize the pairing so the pair is the physical lapse and shift directly. For example, the $K$-variation in the adjoint equation gives $\mathcal L_Xh=-4NK$, which is the spatial Killing equation for that decomposition. The vacuum evolution extends these data to a spacetime Killing field. For matter, one must include its equations and symmetry conditions; the vacuum KID identification cannot simply be applied to an arbitrarily prescribed $(\rho,j)$.

The Fredholm alternative explains its role. To solve

$$D\Phi_{(h,K)}(\gamma,p)=f$$

with compact support or fixed boundary behavior, $f$ must be orthogonal to $\ker D\Phi^*$. If no local KIDs exist, the linearized map is surjective in the appropriate localized setting and an implicit-function or nonlinear correction argument can proceed. If KIDs are present, one must impose balancing conditions or vary finite-dimensional parameters such as asymptotic mass and momentum. Carlotto surveys this deformation mechanism and its hypotheses in Carlotto 2021, §§2–3.

Underdetermined ellipticity and gauge

The linearized constraint system has a surjective mixed-order symbol: second order in the metric for the scalar equation, first order in $K$ for the momentum equation. This is precisely underdetermined ellipticity, not a failure of ellipticity that must first be repaired by gauge fixing. At flat data, take a unit covector $\xi=dx^1$ and replace derivatives by $\xi_i$. The scalar symbol is $-\gamma_{22}-\gamma_{33}$; the three momentum symbols are $-p_{22}-p_{33}$, $p_{12}$, and $p_{13}$. These four outputs can be chosen independently. An invertible square symbol is neither possible nor required.

Its kernel remains large. Infinitesimal spatial diffeomorphisms act by

$$(\gamma,p)=(\mathcal L_Yh,\mathcal L_YK),$$

and lie along gauge directions of the vacuum constraint surface because $D\Phi(\mathcal L_Yh,\mathcal L_YK)=\mathcal L_Y\Phi=0$ at a solution. To construct a particular inverse or obtain uniqueness, one selects a complement, quotients gauge directions, or uses a tensor decomposition. The conformal method turns this freedom into chosen free data plus a vector elliptic equation and a scalar semilinear elliptic equation. The adjoint, in contrast, has injective symbol; this is the starting point of localized estimates in Bartnik and Isenberg 2004, §4.6, equations (90)–(91).

KIDs are not merely gauge. They are cokernel elements reflecting true symmetries and conserved charges. Minkowski data provide the discriminating example: constant and affine lapses together with Euclidean Killing shifts encode time translations, boosts, spatial translations, and rotations. A compactly supported deformation cannot change the associated asymptotic charges arbitrarily while leaving the exterior exactly flat.

This is the central limitation of naive local correction. Elliptic equations transmit information, and symmetry creates finite-dimensional conservation laws. Gluing and Localized Deformation shows how actual theorems accommodate both facts.

Check your understanding

At $(\delta,0)$, verify that $N=1$ and a constant shift $X^i=a^i$ lie in the adjoint kernel. What symmetries do they represent?

Solution

For $N=1$, all second derivatives vanish, so the metric component of $D\Phi^*$ is zero. For constant $X$, all first derivatives vanish, so its momentum component is zero. They are the initial-data representatives of a Minkowski time translation and spatial translations. Their presence warns that a compactly supported correction must satisfy the corresponding charge-balancing conditions.

References

• Bartnik, Robert, and Jim Isenberg. “The Constraint Equations.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 1–38. Birkhäuser, 2004. arXiv:gr-qc/0405092.
• Carlotto, Alessandro. “The General Relativistic Constraint Equations.” Living Reviews in Relativity 24 (2021): 2. DOI.