Spacetime Extensions and Maximality

A spacetime extension is a larger Lorentzian spacetime containing the original one as a proper isometric open submanifold. This simple definition separates a removable coordinate patch from a maximal geometry, but “maximal” has several meanings. Maximal analytic extension, maximal extension in a fixed differentiability class, and maximal globally hyperbolic development of initial data answer different questions and need not coincide.

Required background. Geodesics and Jacobi Fields supplies curve continuation; Curvature supplies extension obstructions. Helpful background. Lorentzian Metrics and Causal Cones supplies nondegenerate Lorentzian metrics; Causality and Global Structure supplies domains and horizons; Exact Solutions of Einstein’s Equations supplies coordinate and analytic extensions.

Proper isometric embeddings

Let $(M,g)$ and $(M',g')$ be connected time-oriented Lorentzian manifolds of the same dimension and declared regularity. An extension of $(M,g)$ is an isometric embedding

$$\iota:M\longrightarrow M', \qquad \iota^*g'=g,$$

whose image is a proper subset of $M'$. Equal dimension and nondegeneracy make the image open. “Proper” here means not onto; it does not mean a proper map in the topological sense.

The extension must preserve the full metric, not merely its coordinate formula on a shared range. It also includes the topology and smooth structure of $M'$. A coordinate substitution that makes components finite suggests an extension but does not establish one until the transformed chart, transition maps, and nondegenerate metric are defined across the old edge.

An extension may be required to preserve time orientation, field content, or an Einstein–matter solution. Pure Lorentzian extendibility only asks for a metric. If the physical problem includes matter fields, the fields and equations may fail to extend even when the metric does. The category must therefore be stated.

The spacetime is $C^k$-inextendible when no proper extension exists with a $C^k$ Lorentzian metric and compatible embedding. Since higher regularity is more restrictive,

$$C^0\text{-inextendible}\ \Longrightarrow\ C^1\text{-inextendible} \ \Longrightarrow\ C^2\text{-inextendible} \ \Longrightarrow\ C^\infty\text{-inextendible},$$

but not conversely. This hierarchy becomes central to strong cosmic censorship.

Coordinate extension and geometric extension

Minkowski space in Rindler coordinates covers one wedge. The coordinate transformation degenerates at the wedge’s null boundary, but the full Minkowski metric is already a smooth extension. The Rindler horizon is an observer horizon, not an obstruction to the geometry.

The Schwarzschild exterior gives a more consequential example. Static coordinates have a singular coefficient at areal radius $r=2GM$, while curvature invariants remain finite there. Ingoing Eddington–Finkelstein coordinates extend across the future horizon; Kruskal coordinates produce an analytic extension across future and past horizons. This demonstrates that the exterior chart was not maximal.

Yet the phrase “the maximal Schwarzschild spacetime” still needs qualification. The Kruskal solution is the standard connected, simply connected, maximal analytic vacuum extension of the exterior. Discrete quotients can change global topology, and a collapse spacetime with matter can contain the same exterior region without the white-hole sector or second asymptotic end. The exterior data alone do not select a unique global physical history.

A finite chart boundary is therefore classified by a three-stage check: attempt a regular coordinate change, construct the larger manifold and metric if possible, and state the category in which no further extension is claimed. Curvature divergence can obstruct a $C^2$ extension, but finite scalar invariants alone do not guarantee any extension.

Maximal extensions and their nonuniqueness

A spacetime is maximal in a chosen category if it has no proper extension in that category. A maximal extension of a given spacetime is an extension that is itself inextendible there. Zorn-type arguments can produce maximal elements under suitable set-theoretic control, but maximal need not mean unique.

Nonuniqueness can arise when more than one inequivalent region may be attached beyond a boundary. Taub spacetime has inequivalent extensions across its Cauchy horizon; neither is selected solely by the original globally hyperbolic region. Branching cannot be represented by one Hausdorff manifold containing all continuations simultaneously. This is a global failure of deterministic continuation even though each branch can be smooth.

Analytic maximality is still narrower. An analytic metric may have a unique analytic continuation locally under favorable conditions, but smooth extensions need not be analytic. Conversely, demanding analyticity can hide lower-regularity extensions. Physical initial data are generally smooth or Sobolev rather than analytic, so analytic maximality is not the default notion in evolution problems.

Maximality does not imply completeness: an inextendible spacetime can contain finite-affine-parameter geodesics. There is, however, an important implication in the other direction. A smooth timelike geodesically complete spacetime is $C^2$-inextendible. A regular extension would provide a timelike geodesic crossing its boundary at finite proper time, contradicting completeness of its portion in the original spacetime. With global hyperbolicity added, timelike completeness even implies $C^0$-inextendibility Galloway–Ling–Sbierski 2018, Introduction and Theorem 1. Thus completeness and inextendibility are not equivalent, but neither are they logically unrelated. The extension regularity determines which implication one may use.

Maximal globally hyperbolic development

For admissible Einstein initial data, local well-posedness yields a globally hyperbolic development containing the data as a Cauchy surface. Choquet-Bruhat and Geroch prove that there is a maximal globally hyperbolic development (MGHD), unique up to an isometry fixing the embedded initial data Choquet-Bruhat and Geroch 1969, pp. 329–335.

The adjective “maximal” here is restricted to globally hyperbolic developments of those data. The MGHD may admit a larger Lorentzian extension across a Cauchy horizon, but the extension is no longer a globally hyperbolic development with the original Cauchy surface. Reissner–Nordström and Kerr interiors provide exact-model illustrations. Thus

$$\text{maximal globally hyperbolic} \quad\not\Rightarrow\quad \text{inextendible as a spacetime}.$$

Strong cosmic censorship is motivated by this gap: for generic admissible data, should the MGHD be inextendible in a specified regularity class? Changing $C^0$ to $C^2$, the matter model, cosmological constant, or topology changes the conjecture.

The MGHD is also not “the largest subset” under ordinary inclusion, because developments live on different manifolds. Its universal property is formulated through isometric embeddings preserving the initial-data embedding. This category-level statement is what makes uniqueness meaningful.

Exercise

Let $M=\{(t,x)\in\mathbb R^{1,1}:t\lt0\}$ with the flat metric. Is it maximal as (a) a smooth Lorentzian spacetime, (b) a globally hyperbolic development of data on $t=-1$ if only $t\lt0$ is admitted, and (c) an analytic spacetime?

Solution

It is not maximal in either the smooth or analytic Lorentzian category: inclusion into full Minkowski spacetime is a proper isometric extension of both types. It is also not the maximal globally hyperbolic development of the data on $t=-1$, because full Minkowski space is a globally hyperbolic development containing the same induced data and $t=-1$ remains a Cauchy surface there. Merely declaring the truncated domain admissible does not make it maximal; maximality is tested against all extensions in the chosen category.

Regularity Classes and Inextendibility next explains why an extension question is incomplete until the differentiability of its metric is fixed.

References

• Choquet-Bruhat, Yvonne, and Robert Geroch, “Global Aspects of the Cauchy Problem in General Relativity”, Communications in Mathematical Physics 14, pp. 329–335, 1969.
• Galloway, Gregory J., Eric Ling, and Jan Sbierski, “Timelike Completeness as an Obstruction to C⁰-Extensions”, Communications in Mathematical Physics 359, pp. 937–949, 2018. Open PDF.
• Steinbauer, Roland, “The Singularity Theorems of General Relativity and Their Low Regularity Extensions”, Jahresbericht der Deutschen Mathematiker-Vereinigung 125, §2.3, pp. 8–9, 2023.