Existence and Uniqueness of the Maximal Globally Hyperbolic Development

Constraint-satisfying vacuum initial data determine a spacetime in two stages. A wave-gauge argument first gives a local solution with continuous dependence; constraint propagation and geometric uniqueness then turn that coordinate solution into an invariant development. Patching all such developments produces a unique maximal globally hyperbolic development, but only in the class for which the initial hypersurface remains Cauchy.

Helpful background. Causality and Global Structure supplies Cauchy hypersurfaces and global hyperbolicity; 3+1 Formulations of General Relativity supplies the constraint equations, gauge data, and initial-value decomposition.

Vacuum data and the local Einstein development

Let $\Sigma$ be a connected smooth three-manifold without boundary, let $h$ be a Riemannian metric, and let $K$ be a symmetric two-tensor. The sufficient Sobolev class used for the global theorem on this page is

$$h\in H^s_{\mathrm{loc}}(\Sigma), \qquad K\in H^{s-1}_{\mathrm{loc}}(\Sigma), \qquad s\in\mathbb N, \qquad s\gt\frac52 \quad(\text{hence }s\geq3),$$

with the usual uniform or asymptotic hypotheses when a chosen global Sobolev formulation requires them. Using $K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$, the vacuum constraints are

$$R(h)+(\operatorname{tr}_hK)^2-K_{ij}K^{ij}=0, \qquad D^jK_{ij}-D_i(\operatorname{tr}_hK)=0.$$

Use Chruściel’s $H^s_{\mathrm{space,loc}}$ spacetime class throughout the global statement. In each local coordinate cylinder $I\times U$, with $t$ the local time coordinate, its time-slice regularity requires

$$\partial_t^k g_{\mu\nu} \in C^0_{\mathrm{loc}}\bigl(I;H^{s-k}_{\mathrm{loc}}(U)\bigr), \qquad 0\leq k\leq s.$$

The compatible embeddings and isometries are understood in the corresponding $H^s_{\mathrm{space,loc}}$ structure. Every development, embedding, and uniqueness claim below is restricted to this same category. Smooth initial data recover the smooth theorem. Chruściel proves this integer-index result in Chruściel 2013, Thm. 1.1, printed pp. 1–2, and Def. 2.1, printed pp. 3–4, Open PDF. The paper identifies extension to real $s\gt5/2$ as expected but not established by that theorem, so the broader real-index local theory is not used to enlarge the global claim here.

After compatible lapse, shift, and gauge data are chosen, Wave-Gauge Local Existence reduces the equations to a quasilinear wave system. Energy estimates give a short-time metric, uniqueness in the fixed gauge, and continuous dependence on $(h,K)$; smoother data give smoother solutions. This is the analytic layer described in Friedrich and Rendall 2000, §§ 2.4 and 5.1–5.3, arXiv v1 PDF pp. 21–24 and 75–81 and Ringström 2009, Chs. 13–15, pp. 147–175.

The reduced equations are not yet the full vacuum equations. If $C_\mu$ is the wave-gauge constraint, the contracted Bianchi identity gives the homogeneous subsidiary system

$$\Box_g C_\mu+R_\mu{}^\nu C_\nu=0.$$

Compatible gauge initialization gives $C|_\Sigma=0$, while the reduced equations together with the Hamiltonian and momentum constraints give $\nabla_nC|_\Sigma=0$. Uniqueness for this wave equation therefore gives $C=0$ throughout the local domain of dependence, so the reduced metric satisfies $R_{\mu\nu}=0$. Constraint and Gauge Propagation develops this step.

Fixed-gauge uniqueness becomes geometric uniqueness by placing two solutions in a common wave-map gauge. The resulting overlap isometry fixes the initial data, and compatible overlap maps can be patched; see Geometric Uniqueness and Patching. This passage from coordinate uniqueness to geometry is essential: arbitrary coordinate representatives need not agree.

The maximal-development theorem

A globally hyperbolic development of $(\Sigma,h,K)$ is a triple $(M,g,\iota)$ in which $(M,g)$ is a connected, time-oriented vacuum spacetime, $\iota:\Sigma\hookrightarrow M$ induces $h$ and $K$, and $\iota(\Sigma)$ is a Cauchy hypersurface—every inextendible causal curve meets it exactly once.

The Choquet-Bruhat–Geroch theorem provides a development

$$(M_{\max},g_{\max},\iota_{\max})$$

with the following universal property: for every globally hyperbolic development $(M,g,\iota)$ of the same data, there is an initial-data-preserving isometric embedding in the direction

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

If $(M'_{\max},g'_{\max},\iota'_{\max})$ has the same property, there is an isometry $F:M_{\max}\to M'_{\max}$ satisfying $F^*g'_{\max}=g_{\max}$ and $F\circ\iota_{\max}=\iota'_{\max}$. Thus uniqueness is precisely uniqueness up to a data-fixing isometry. The original smooth construction and theorem occupy Choquet-Bruhat and Geroch 1969, pp. 329–335; the stated Sobolev-class result is Chruściel 2013, Thm. 1.1, printed pp. 1–2, Open PDF; and its relation to local geometric uniqueness is also developed in Ringström 2009, Chs. 13–16, pp. 147–183. The Maximal Globally Hyperbolic Development gives the full construction.

Limits of maximality

“Maximal” means maximal only among globally hyperbolic developments of these data in the stated solution class. The theorem does not imply geodesic completeness, curvature boundedness, or absolute inextendibility as a Lorentzian spacetime. $M_{\max}$ may admit an extension across a Cauchy horizon, provided $\iota(\Sigma)$ is no longer Cauchy there; such a non-globally-hyperbolic extension does not contradict the universal property. Nor does the theorem by itself establish curvature blow-up, create a singular point, or locate an event horizon.

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.
• Chruściel, Piotr T. “On Maximal Globally Hyperbolic Vacuum Space-Times.” arXiv:1112.5779v2 [gr-qc] (2013). Open PDF.
• 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/0002074v1.
• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.