Topology of Cauchy Surfaces

A globally hyperbolic spacetime has one spatial topology, even though its spatial geometry can evolve. The precise statement is a smooth splitting theorem: the spacetime is diffeomorphic to a product of the real line with a Cauchy hypersurface, and every smooth spacelike Cauchy hypersurface has the same diffeomorphism type. This result is kinematical. It uses neither Einstein’s equation nor an energy condition.

Required background. Exterior Algebra and Differential Forms supplies pullbacks and orientation language; Hypersurface Normals and Tangential Projectors supplies induced metrics and the distinction between intrinsic and embedded geometry.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes the Lorentzian setting; Affine and Variational Geodesics supplies causal-geodesic language; Causality and Global Structure orients the global assumptions; Timelike Congruences and Spatial Projection helps interpret a foliation geometrically.

Cauchy hypersurfaces and spatial topology

A subset $S$ of a time-oriented spacetime is a Cauchy hypersurface if every inextendible causal curve meets $S$ exactly once. For the differentiable conclusion below, take $S$ to be a smooth embedded spacelike hypersurface. “Exactly once” contains two logically different requirements: acausality prevents a causal curve from meeting $S$ twice, while the intersection requirement prevents it from avoiding $S$ altogether. Achronality alone excludes only timelike connections and would still permit two points joined by a null segment; it is not enough for this version of the definition.

Before stating the theorem, its scope should be explicit.

• The spacetime dimension is any $n\geq2$.
• The spacetime is connected, smooth, time-oriented, and has no manifold boundary.
• Global hyperbolicity is assumed.
• Neither spacetime orientability nor orientability of the Cauchy hypersurface is assumed.
• No asymptotic-flatness, de Sitter, or anti-de Sitter condition is assumed.
• No Ricci-convergence, averaged null energy, Einstein-equation, or matter hypothesis is assumed.

Smooth Cauchy splitting theorem. Under these hypotheses there is a smooth function $T:M\to\mathbb R$ with past-directed timelike gradient such that every level

$$S_t=T^{-1}(t)$$

is a smooth spacelike Cauchy hypersurface. For any one level $S$, there is a diffeomorphism $M\simeq\mathbb R\times S$ in which

$$g=-\beta(T,x)\,dT^2+h_T, \qquad \beta>0,$$

and $h_T$ is a smoothly varying Riemannian metric on $S$. Consequently, any two smooth spacelike Cauchy hypersurfaces in $M$ are diffeomorphic. Bernal and Sánchez give the smooth temporal function and orthogonal metric splitting in Bernal–Sánchez 2005, Theorem 1.1 and proof, pp. 44–49.

The theorem says diffeomorphic, not isometric. The tensors $h_T$ may change without restriction from this result alone. Volumes, curvatures, injectivity radii, and extrinsic curvatures can all evolve while the underlying differentiable manifold remains fixed. Nor does the product form make the spacetime stationary: in general both $h_T$ and $\beta$ depend on $T$.

Why the product follows

The proof has two main stages.

First, global hyperbolicity supplies enough causal compactness to construct a Cauchy time function, and smoothing produces a Cauchy temporal function $T$: its gradient is timelike everywhere and its level sets are Cauchy. Timelikeness of $\nabla T$ is stronger than mere monotonicity. It makes every level regular and spacelike.

Second, follow a future-directed timelike vector field everywhere transverse to the levels. It can be normalized so that its flow parameter is $T$. Each integral curve meets every level $S_t$ exactly once, because the levels are Cauchy. Sending a point $p$ to the pair consisting of $T(p)$ and the point where its flow line meets $S_0$ produces the global diffeomorphism

$$M\longrightarrow\mathbb R\times S_0.$$

Orthogonality follows by choosing the flow along the gradient. Since $dT$ annihilates tangent vectors to $S_t$, no $dT\,dx^i$ cross term remains, giving the displayed metric form.

Now let $C$ be any other smooth spacelike Cauchy hypersurface. Every flow line meets $C$ once, so projection along the flow gives a bijection $C\to S_0$. Spacelikeness makes $C$ transverse to the timelike flow, hence the projection and its inverse are smooth. This is the step that upgrades “all constructed levels look alike” to “all smooth spacelike Cauchy hypersurfaces look alike.” For an arbitrary nonsmooth Cauchy hypersurface, the corresponding general conclusion is homeomorphism rather than an unqualified diffeomorphism.

Topological consequences

Any diffeomorphism invariant of a Cauchy hypersurface is constant throughout globally hyperbolic evolution. In particular, all smooth Cauchy hypersurfaces have isomorphic fundamental and homology groups, the same number of ends, and the same orientability type. A toroidal universe cannot turn into a spherical universe between two regular Cauchy slices.

Compactness is preserved as well. If one Cauchy hypersurface is compact, every Cauchy hypersurface is compact, although the spacetime $\mathbb R\times S$ itself is not compact because of the time factor. If one slice is noncompact, diffeomorphism preserves its end structure: an asymptotically Euclidean end cannot simply disappear on a later Cauchy slice. This statement concerns complete Cauchy hypersurfaces of the same spacetime. Removing a black-hole interior, truncating a slice at a timelike worldtube, or intersecting it with a domain of outer communications produces a manifold-with-boundary or a proper open subset. Such partial spatial sections need not have the topology of the full Cauchy surface, so comparing them does not test the splitting theorem.

The splitting also relates spacetime topology to spatial topology:

$$\pi_k(M)\cong\pi_k(S), \qquad H_k(M;R)\cong H_k(S;R),$$

for every $k$ and any coefficient ring $R$, because $\mathbb R$ is contractible. Moreover, $TM$ splits as a trivial line bundle plus the pullback of $TS$. Thus $M$ is orientable exactly when $S$ is orientable. Time orientability by itself does not imply either one.

Global hyperbolicity does not imply simple connectivity. The flat cylinder

$$(M,g)=(\mathbb R\times S^1,-dt^2+d\phi^2)$$

is globally hyperbolic, and every $t={\rm constant}$ circle has fundamental group $\mathbb Z$. The splitting theorem preserves that nontrivial topology; it does not censor it. Topological censorship needs additional asymptotic and focusing hypotheses, developed on The Topological Censorship Theorem.

Timelike boundaries

The boundaryless hypothesis must not be silently dropped. There is, however, a boundary-capable theorem for a smooth globally hyperbolic spacetime-with-timelike-boundary $\overline M$. Here global hyperbolicity is defined using locally Lipschitz causal curves that may touch or run along $\partial\overline M$. No orientability, field equation, asymptotic condition, or energy condition is required.

In that setting there exists a Cauchy temporal function $\tau$ whose gradient is tangent to the timelike boundary, and

$$\overline M\simeq\mathbb R\times\overline\Sigma, \qquad g=-\Lambda\,d\tau^2+g_\tau,$$

where $\overline\Sigma$ is a Cauchy hypersurface-with-boundary and $\partial\overline M\simeq\mathbb R\times\partial\overline\Sigma$. This is a theorem for the boundary problem itself, not an informal appeal to doubling; see Aké Hau–Flores–Sánchez 2021, Theorem 1.1 and §5.2, pp. 47–49 and 77–79. Null boundaries, corners, degenerate metrics, and arbitrary artificial cutoffs require separate hypotheses and do not follow from this timelike-boundary statement.

Where topology change can hide

If two proposed “initial” and “final” slices have different topology, at least one premise of the Cauchy-splitting argument has failed. The slices may not both be Cauchy; the interpolating region may not be globally hyperbolic; a manifold boundary or singular end may intervene; the metric may become degenerate or nonsmooth; or closed causal curves may appear.

A pair-of-pants cobordism illustrates the point in two spacetime dimensions: one circle evolves into two circles as a statement about an underlying cobordism. It cannot at the same time be a smooth, time-oriented, globally hyperbolic Lorentzian evolution with both boundary circles serving as Cauchy data. For a causally compact Lorentzian interpolating spacetime, absence of closed timelike curves forces a product cobordism Borde 1994, Theorem 1 and proof, pp. 17–18. Allowing a degenerate metric at a critical point changes the mathematical category and evades the theorem rather than contradicting it.

Common pitfalls

A foliation is not automatically a Cauchy foliation. A coordinate family of spacelike slices can cover only a patch or fail to meet some inextendible causal curves. The topology-preservation conclusion uses the Cauchy property globally.

Topology preservation is not geometry preservation. The metrics $h_T$ need not be related by isometries. Cosmological expansion and gravitational radiation are compatible with a fixed diffeomorphism type.

A horizon section is not a Cauchy surface. Horizon cross-sections may merge, crease, or change topology without furnishing two Cauchy surfaces of the full spacetime. Their topology is governed by different global and stability results.

Exercises

Let $(M,g)$ satisfy the smooth boundaryless splitting theorem, and suppose one Cauchy hypersurface $S$ is nonorientable with $\pi_1(S)\neq0$. Show that every smooth spacelike Cauchy hypersurface $C$ is nonorientable, that $M$ is nonorientable, and that $\pi_1(M)\cong\pi_1(C)$.

Solution

The flow construction gives diffeomorphisms $C\simeq S$ and $M\simeq\mathbb R\times S$. Diffeomorphisms preserve orientability and fundamental groups, so $C$ is nonorientable and $\pi_1(C)\cong\pi_1(S)$. Because $\mathbb R$ is contractible, projection $\mathbb R\times S\to S$ is a homotopy equivalence, hence $\pi_1(M)\cong\pi_1(S)\cong\pi_1(C)$. Finally,

$$T(\mathbb R\times S)\cong\underline{\mathbb R}\oplus TS.$$

Adding a trivial oriented line does not remove the obstruction measured by the first Stiefel–Whitney class: $w_1(TM)$ is the pullback of $w_1(TS)$. Since $S$ is nonorientable, $M$ is nonorientable. Time orientability supplies a continuous choice of timelike cone, not a spacetime volume orientation.

References

• Aké Hau, Luis, José L. Flores, and Miguel Sánchez, “Structure of Globally Hyperbolic Spacetimes-with-Timelike-Boundary”, Revista Matemática Iberoamericana 37, no. 1, pp. 45–94, 2021. Open PDF.
• Bernal, Antonio N., and Miguel Sánchez, “Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes”, Communications in Mathematical Physics 257, pp. 43–50, 2005. Open PDF.
• Borde, Arvind, “Topology Change in Classical General Relativity”, preprint, 1994.