Geometric Uniqueness and Patching

Patching local Einstein developments requires more than taking a union of coordinate domains. Overlap maps must be initial-data-preserving isometries, must satisfy a cocycle condition, and must yield a Hausdorff quotient. Local geometric uniqueness supplies the maps; global hyperbolicity and maximal common developments exclude incompatible identifications and nonseparable boundary points.

Required background. Analysis for Mathematical Relativity supplies local uniqueness and continuation, The Einstein Equation supplies diffeomorphism covariance, and PDEs and Green Functions for Gravitation supplies domains of dependence.

Helpful background. Causality and Global Structure supplies causal convexity, Cauchy surfaces, and Hausdorff spacetime assumptions.

Common developments

Let $(M_a,g_a,\iota_a)$, $a=1,2$, be globally hyperbolic developments of the same data. A common development is a spacetime $(U,g_U,\iota_U)$ with isometric embeddings

$$\psi_a:U\hookrightarrow M_a, \qquad \psi_a\circ\iota_U=\iota_a.$$

Local wave-gauge uniqueness guarantees at least a neighborhood of the initial hypersurface with this property. Common developments are ordered by extension. A maximal common development collects all regions where the two spacetimes have already been identified compatibly.

On an overlap, the transition map

$$\psi_2\circ\psi_1^{-1}$$

is an isometry. If two such maps agree on initial data in a connected domain of dependence, geometric uniqueness makes them agree wherever both are defined. This gives the cocycle condition needed for patching three or more pieces. Ringström presents the local-to-geometric uniqueness framework in Ringström 2009, Chs. 13–16, pp. 147–183.

The quotient construction

Form the disjoint union $M_1\sqcup M_2$ and identify $p\in M_1$ with $q\in M_2$ when the maximal common-development isometry maps $p$ to $q$. The metric descends because the transition map is an isometry. Smooth compatibility follows from the same fact.

The subtle point is Hausdorffness. A quotient can fail to be Hausdorff if sequences $p_n\in M_1$ and $q_n\in M_2$ are identified for all $n$, converge to $p$ and $q$, but $p$ and $q$ are not identified. Their images in the quotient cannot then be separated by disjoint neighborhoods. Such a quotient is not an admissible spacetime manifold.

A simple topological model is two copies of the real line with all nonzero points identified but the two origins kept distinct. Every neighborhood of one origin meets every neighborhood of the other, although each copy was Hausdorff before gluing. Agreement of smooth local fields on an overlap therefore does not, by itself, make the quotient an acceptable manifold.

The maximal-development proof excludes this boundary-pair obstruction using causal control and local uniqueness. Roughly, choose causally well-behaved neighborhoods of the limiting points. Data on a suitable spacelike patch determine local developments around both limits. Hyperbolic geometric uniqueness extends the overlap isometry across the alleged boundary, contradicting maximality of the common development. The exact global construction is the core of Choquet-Bruhat and Geroch 1969, pp. 329–335.

After the manifold and causal properties have both been established, the quotient is a common globally hyperbolic extension into which both original developments embed. It may coincide with one of them when one already contains the other. Applying the corresponding construction consistently to chains supplies upper bounds for a maximality argument.

Second countability and connectedness must also be retained. One works with representatives generated from the fixed initial manifold and compatible developments, rather than taking an unrestricted proper class of all spacetimes. The quotient atlas is inherited from the pieces, while the common embedded $\Sigma$ selects the relevant connected component. These manifold conditions are part of the construction, not consequences of the field equation alone.

The causal statement needs verification as well: any inextendible causal curve in the quotient must meet the embedded initial surface exactly once. A curve may leave one piece and enter another, so global hyperbolicity of the pieces alone does not prove this assertion. The proof uses causal neighborhoods tied to the common Cauchy data, together with maximality of the common development and the causal control used to extend overlap maps. It excludes both a curve avoiding the initial surface and one returning to it after changing pieces. This is a separate part of the global theorem, summarized here rather than replaced by the false rule that every Hausdorff gluing preserves global hyperbolicity.

Gauge normalization on overlaps

Suppose $g_1$ and $g_2$ are written in wave coordinates. Equality of their component functions follows only if the wave coordinates have identical Cauchy data. Residual vector fields satisfying a homogeneous wave equation can otherwise change the representative while preserving wave gauge.

The invariant procedure is:

1. solve wave-map equations from each development to a common target;
2. prescribe the same map and normal derivative on the embedded $\Sigma$;
3. use reduced-system uniqueness in the common target coordinates;
4. compose inverse wave maps to obtain the overlap isometry.

The transition map fixes the embedded initial hypersurface and its normal data. Away from that normalization, arbitrary coordinate equality is neither expected nor needed.

The local comparison by wave maps is described in Friedrich and Rendall 2000, §5.2.

Boundaries of the argument

Global hyperbolicity is essential. Beyond a Cauchy horizon, data on $\Sigma$ no longer determine every event, and distinct extensions can agree on the entire maximal globally hyperbolic region. A timelike boundary similarly introduces incoming information not fixed by Cauchy data. Low-regularity solutions require enough regularity for the overlap maps and pullback metric to be defined in the asserted category.

Patching also does not select a preferred maximal coordinate gauge. It constructs a maximal geometry. Any gauge that ceases to cover that geometry must be replaced rather than interpreted as a spacetime singularity.

Check your understanding

Two wave-gauge developments have the same geometric initial data but different initial coordinate labels. Can reduced-system uniqueness be applied directly to their metric components?

Solution

No. Reduced uniqueness compares solutions with identical gauge-fixed Cauchy data. First construct wave maps with the same initial map and normal derivative into a common target. The metrics expressed on that target then have identical reduced data and agree; composing the wave maps and their local inverses yields the geometric overlap isometry.

References

• Choquet-Bruhat, Yvonne, and Robert Geroch. “Global Aspects of the Cauchy Problem in General Relativity.” Communications in Mathematical Physics 14 (1969): 329–335. DOI.
• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074.
• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.