Gluing and Deformation of Initial Data

Gluing does not interpolate two metrics and hope that Einstein’s constraints survive. A cutoff interpolation creates a compactly supported constraint error; one then solves a localized nonlinear correction problem. The decisive issue is the adjoint kernel of the linearized constraint map. Where that kernel contains Killing initial data, a local correction has finite-dimensional obstructions that must be eliminated, balanced by external parameters, or excluded by a nondegeneracy hypothesis.

Required background. Hypersurface Normals and Tangential Projectors supplies the initial-data tensors, while PDEs and Green Functions for Gravitation supplies elliptic adjoints, weighted spaces, kernels, and localized estimates.

Helpful background. The Einstein Equation supplies the vacuum constraints; Formulations of General Relativity locates gluing among the formulations in this volume; and Presymplectic Potentials and Currents prepares the role of lapse–shift generators and boundary terms.

Constraint map and local obstruction

Use $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$. For vacuum data $(\gamma_{ij},K_{ij})$, define

$$\Phi(\gamma,K) =\begin{pmatrix} {}^{(3)}R+K^2-K_{ij}K^{ij}-2\Lambda\\[2pt] D_jK^j{}_i-D_iK \end{pmatrix}.$$

A valid data set satisfies $\Phi=0$. If a trial splice $(\gamma_0,K_0)$ has residual $f=-\Phi(\gamma_0,K_0)$ in a transition region $\Omega$, a first correction $(h,p)$ should solve

$$D\Phi_{(\gamma_0,K_0)}(h,p)=f.$$

The formal adjoint acts on a lapse–shift pair $\xi=(N,X^i)$. Elements of $\ker D\Phi^*$ are Killing initial data: under the appropriate development hypotheses they encode spacetime Killing fields. For a compactly supported correction, integration by parts gives

$$\int_\Omega\langle f,\xi\rangle\,dV =\int_\Omega\langle(h,p),D\Phi^*\xi\rangle\,dV,$$

because the boundary term vanishes. Hence every adjoint-kernel element imposes the compatibility condition $\int_\Omega\langle f,\xi\rangle dV=0$. A generic cutoff error will not satisfy it automatically.

This is the essential independent check on any claimed local gluing scheme: write its residual, identify the operator domain, and test the residual against the full adjoint kernel before asserting that a compactly supported correction exists.

Localized connected-sum gluing

Two gluing statements must be separated. The Isenberg–Mazzeo–Pollack conformal connected-sum construction, summarized in Bartnik and Isenberg 2004, §4.5, pp. 25–27, PDF, joins sufficiently regular constant-mean-curvature vacuum data by a long $S^2\times I$ neck and makes the result arbitrarily close to the original data away from the neck. That conformal stage alone does not give exact agreement.

Exact localization is supplied by the Chruściel–Isenberg–Pollack deformation theorem. In its smooth vacuum, $\Lambda=0$, three-dimensional version, choose two distinct points $p_1,p_2$, possibly in different connected components, and open neighborhoods $\Omega_1,\Omega_2$ with compact closure and smooth boundary on which the local Killing-initial-data spaces are trivial. For any sufficiently small $\epsilon\gt0$ and fixed finite $k$, the construction removes balls around the points and joins their sphere boundaries by a neck. The new vacuum data are $\epsilon$-close in $C^k\times C^k$ away from $B(p_1,\epsilon)\cup B(p_2,\epsilon)$ and agree exactly with the originals outside $\Omega_1\cup\Omega_2$. No closeness is asserted on the new neck, where the topology itself has changed. See Chruściel, Isenberg, and Pollack 2005, Theorem 1.1, preprint p. 3, PDF.

This local result does not require a particular asymptotic end: exact agreement outside the neighborhoods preserves whichever ends were already present. Finite-regularity variants need their own differentiability hypotheses; matter systems need compatible matter constraints; and nontrivial local KIDs invalidate the stated nondegenerate version. It is not a theorem that arbitrary incompatible data can be patched across a prescribed interface.

The proof architecture has several distinct stages:

1. conformally blow up small balls around the chosen points to create half cylinders;
2. identify the cylindrical ends and splice the fields with cutoffs, producing an approximate solution on a long neck;
3. solve the projected momentum and Lichnerowicz-type equations, using estimates uniform in the neck length, to obtain an exact but generally nonlocalized conformal gluing;
4. use the no-local-KID hypothesis to solve a compactly supported deformation problem on surrounding annuli;
5. match the localized correction to the conformal solution so the final data equal the originals—not merely approach them—outside the chosen neighborhoods. Chruściel, Isenberg, and Pollack give this two-stage proof in Chruściel, Isenberg, and Pollack 2005, §§2–4, pp. 34–40.

The construction can connect different manifolds or two points of one manifold. It changes topology and may create minimal surfaces on the bridge, which makes it useful for multi-black-hole and wormhole existence constructions. Those geometric applications do not by themselves predict the long-time evolution of the neck.

Exact model ends by asymptotic deformation

Corvino–Schoen gluing preserves an interior while replacing a distant end. For the exact-preservation result, take smooth vacuum data with an asymptotically Cartesian chart satisfying

$$\begin{aligned} \gamma_{ij}(x)-\delta_{ij}&=O(r^{-1}),& K_{ij}(x)&=O(r^{-2}),\\ \gamma_{ij}(x)-\gamma_{ij}(-x)&=O(r^{-2}),& K_{ij}(x)+K_{ij}(-x)&=O(r^{-3}). \end{aligned}$$

Include differentiated estimates: each derivative adds one inverse power of $r$. The leading metric perturbation is even and the leading extrinsic curvature is odd. Choose a smooth model family satisfying these estimates locally uniformly, whose map to the ten asymptotic charges $(E,P_i,J_i,C_i)$ is a homeomorphism onto an open set containing the initial charges; $C_i$ denotes the center-of-mass integrals. Such a family is called admissible. Kerr slices with suitable translations, rotations, boosts, mass, and spin provide the model when this matching condition holds.

For sufficiently large $R$, the construction gives vacuum data equal to the originals inside $r=R$ and to a suitably selected model beyond $r=2R$. These are the parity condition and exact gluing theorem of Corvino and Schoen 2006, arXiv v1, equation (5), p. 3, and § 5, Theorem 4, p. 23, PDF. The exterior parameters are adjusted, not prescribed arbitrarily. Their Theorem 5 instead approximates general asymptotically flat data; it does not assert exact interior preservation without these hypotheses.

The time-symmetric scalar-flat specialization produces an exact Schwarzschild exterior. Bartnik and Isenberg state this application and sketch its proof in Bartnik and Isenberg 2004, §4.6, pp. 28–29, PDF.

One first cuts between the original end and a model end across a large annulus. The residual constraint error lies in that annulus. Weighted estimates for an operator of the schematic form

$$D\Phi\bigl(\rho D\Phi^*\xi\bigr)$$

produce corrections that vanish to high order at the annulus boundary, so the corrected fields join smoothly to the unchanged pieces. Near Euclidean data, however, the adjoint kernel is nontrivial. In the time-symmetric scalar-curvature problem it contains the affine lapses

$$\ker DR_\delta^*=\operatorname{span}\{1,x^1,x^2,x^3\}.$$

Rather than pretending this kernel is absent, the construction solves the projected equation and varies the mass and center parameters of the Schwarzschild model to cancel the four obstruction components. The full constraint problem similarly uses the finite-dimensional parameters of the stationary model end.

What gluing establishes

Gluing demonstrates flexibility of the constraint set. It can preserve local geometry while altering topology or asymptotics, construct data with exact stationary exteriors, and separate questions about an interior strong-field region from questions about infinity. It does not show that the resulting data are dynamically stable, astrophysically natural, free of radiation, or unique.

The word “localized” must also be read precisely. Exact agreement holds in regions specified by the theorem, while a transition region necessarily contains the deformation. Elliptic equations remain globally sensitive even when weighted techniques confine the final correction’s support.

Common pitfalls

Cutoff interpolation is only an approximate solution. The nonlinear constraint residual must still be removed.

No-KID is not a cosmetic genericity phrase. It is the condition that removes the cokernel obstructing a compactly supported correction in the localized theorem.

Connected-sum and exact-end gluing are different constructions. The former creates a neck and changes topology; the latter deforms an asymptotic annulus to match a stationary exterior.

Exercise: an obstructed scalar-curvature correction

On a Euclidean annulus, suppose one seeks a compactly supported tensor $h$ with $DR_\delta h=f$. Show why a source with nonzero integral cannot be corrected without changing a model parameter or allowing a boundary contribution.

Solution

The constant function $N=1$ lies in $\ker DR_\delta^*$. Because $h$ is compactly supported in the annulus, integration by parts has no boundary term:

$$\int_\Omega f\,dV =\int_\Omega N\,DR_\delta h\,dV =\int_\Omega\langle DR_\delta^*N,h\rangle\,dV =0.$$

Thus a source with nonzero integral is outside the range of the compactly supported linearized operator. In exact-end gluing, changing the model mass changes precisely such a finite-dimensional residual component; in a nondegenerate local theorem, the corresponding adjoint kernel is assumed trivial.

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.
• Chruściel, Piotr T., James Isenberg, and Daniel Pollack. “Initial Data Engineering.” Communications in Mathematical Physics 257 (2005): 29–42. DOI. Open PDF.
• Corvino, Justin, and Richard M. Schoen. “On the Asymptotics for the Vacuum Einstein Constraint Equations.” Journal of Differential Geometry 73, no. 2 (2006): 185–217. DOI. Open PDF.