The Maximal Globally Hyperbolic Development

Local solutions can be patched into a unique maximal globally hyperbolic development. Its defining property is universal: every globally hyperbolic development of the same initial data embeds isometrically into it while fixing the embedded initial hypersurface. “Maximal” here is restricted to globally hyperbolic developments; it does not imply geodesic completeness or forbid extensions beyond a Cauchy horizon.

Required background. Analysis for Mathematical Relativity supplies local existence and set-theoretic patching, The Einstein Equation supplies the evolution system, and PDEs and Green Functions for Gravitation supplies uniqueness and domains of dependence.

Helpful background. Causality and Global Structure supplies global hyperbolicity, Cauchy hypersurfaces, and Cauchy horizons.

Statement of the theorem

For a precise classical version, let $\Sigma$ be a smooth connected three-manifold without boundary, let $h$ be a smooth Riemannian metric, and let $K$ be a smooth symmetric two-tensor satisfying the vacuum constraints with $\Lambda=0$. Neither compactness nor completeness of $h$ is required. A smooth globally hyperbolic development is a triple $(M,g,\iota)$ such that

• $(M,g)$ is a time-oriented vacuum spacetime;
• $\iota:\Sigma\to M$ embeds $\Sigma$ as a Cauchy hypersurface;
• $\iota^*g=h$ and its second fundamental form is $K$, using $K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$.

The Choquet-Bruhat–Geroch theorem states that there exists a globally hyperbolic development $(M_{\max},g_{\max},\iota_{\max})$ such that every other globally hyperbolic development $(M,g,\iota)$ admits an initial-data-preserving isometric embedding

$$\Psi:(M,g)\hookrightarrow(M_{\max},g_{\max}), \qquad \Psi\circ\iota=\iota_{\max}.$$

Any two developments with this universal property are isometric by an isometry fixing the initial data. The original statement and construction are in Choquet-Bruhat and Geroch 1969, theorem and proof, pp. 329–335; the smooth version above is also stated in Wald 1984, Theorem 10.2.2, p. 264.

For finite Sobolev regularity or an Einstein–matter extension, one must supply the local existence and geometric-uniqueness theorem in that same solution class. All developments must use the same matter model, time orientation, and initial embedding data. These hypotheses are what make the partial order and overlap maps well defined; the smooth vacuum statement does not by itself establish a lower-regularity version. Ringström places the maximal-development construction after the corresponding local uniqueness theory in Ringström 2009, Chs. 13–16, pp. 147–183.

For noncompact $\Sigma$, the theorem still concerns a global Cauchy hypersurface rather than a union of unrelated pointwise germs. The local developments must be patched with compatible causal domains and regularity. If a timelike boundary is introduced, boundary data become part of the problem and the theorem above does not apply unchanged.

The universal property is stronger than saying that no proper extension has been found. It characterizes the object categorically within the class of globally hyperbolic developments and immediately gives uniqueness: each maximal development embeds into the other, and geometric uniqueness makes the embeddings inverse on their images.

From local developments to a maximal one

The construction begins with local geometric uniqueness. Any two developments of the same data contain isometric neighborhoods of their Cauchy surfaces. More generally, they have a common globally hyperbolic subdevelopment determined by compatible overlap maps.

Order developments by initial-data-preserving isometric embedding. A chain can be combined by taking its disjoint union and quotienting by the compatible embeddings. One must show that the quotient is a smooth Hausdorff spacetime, inherits a Lorentzian metric, contains the same Cauchy data, and remains globally hyperbolic. An upper-bound argument then produces a maximal element. A final common-development argument proves that this element contains every other development, not merely every member of one chosen chain.

Coordinates cannot be used as the equivalence relation. Two developments may use unrelated gauges; their points are identified only where local geometric uniqueness supplies an isometry compatible with the initial embeddings. Geometric Uniqueness and Patching examines the Hausdorff and overlap issue in detail.

What maximality does and does not mean

It is maximal among globally hyperbolic developments. Any extension in which $\iota(\Sigma)$ remains Cauchy would itself be a globally hyperbolic development and must already embed into $M_{\max}$.

It need not be geodesically complete. Geodesics may end after finite affine parameter at a genuine singularity or at an extendible boundary.

It can admit non-globally-hyperbolic extensions. Exact charged or rotating black-hole models illustrate how a Cauchy horizon can bound the maximal globally hyperbolic region while a smooth extension exists beyond it. The initial surface is no longer Cauchy in that extension. Questions about whether generic data admit extensions depend on the regularity class and belong to cosmic censorship.

It is not a largest coordinate chart. Coordinate singularities can occur strictly inside a perfectly regular maximal development. The construction is invariant under diffeomorphism.

It is tied to a matter model and solution class. Changing the constitutive equations, regularity, boundary conditions, or weak-solution notion changes the category in which maximality is asserted.

These qualifications are not technical footnotes. They distinguish deterministic evolution from the separate claims of completeness and inextendibility.

Check your understanding

Take all of Minkowski spacetime and the slab $|t|<1$, both with the $t=0$ slice as initial hypersurface. Which is maximal as a globally hyperbolic development of the flat data?

Solution

The slab is globally hyperbolic but embeds properly into full Minkowski spacetime while fixing the $t=0$ data, so it is not maximal. Full Minkowski spacetime has the universal property for these data and is the maximal globally hyperbolic development. The example shows that local inextendibility of a chosen coordinate interval is irrelevant.

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.
• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record; Chs. 13–16, pp. 147–183.
• Wald, Robert M. General Relativity. University of Chicago Press, 1984. Publisher record.