Cobordism and Topology Change

A cobordism can connect spaces of different topology, but a cobordism is not yet a physically admissible spacetime evolution. Once the interpolating manifold carries a smooth nondegenerate time-oriented Lorentz metric, regular spacelike initial and final boundaries, and sufficient causal control, topology change forces a failure of causality or of another stated hypothesis. The obstruction is sharper than the slogan that “topology change needs a singularity.”

Required background. Exterior Algebra and Differential Forms supplies boundary orientation and characteristic classes; Hypersurface Normals and Tangential Projectors supplies spacelike boundary geometry.

Helpful background. Pseudo-Riemannian Metrics and Signature distinguishes nondegenerate Lorentz metrics from degenerate interpolations; Affine and Variational Geodesics supplies completeness and genericity; Causality and Global Structure supplies Cauchy horizons and chronology; Timelike Congruences and Spatial Projection supplies focusing.

Cobordisms and Lorentzian interpolations

Let $S_0$ and $S_1$ be smooth closed $(n-1)$-manifolds. A compact smooth $n$-manifold $W$ is a cobordism from $S_0$ to $S_1$ if

$$\partial W=S_0\sqcup S_1.$$

For an oriented cobordism the boundary convention is $\partial W=(-S_0)\sqcup S_1$. Cobordism asks whether an interpolating manifold exists. It says nothing about a metric, causal curves, field equations, or whether the boundaries can serve as initial and final data.

A Lorentzian cobordism additionally carries a smooth nondegenerate metric of signature $(-,+,\ldots,+)$ for which $S_0$ and $S_1$ are spacelike. Time orientability supplies a nowhere-zero timelike vector field pointing into $W$ at $S_0$ and out of $W$ at $S_1$. This vector-field requirement already gives selection rules. In the compact closed-boundary setting, an even-dimensional $W$ requires $\chi(W)=0$; for odd-dimensional $W$, the corresponding inward/outward field requires $\chi(S_0)=\chi(S_1)$. These conditions concern existence of a Lorentzian interpolation, not its causal acceptability Borde 1994, §III.C, pp. 12–16.

For example, many pairs of closed three-manifolds are cobordant, and a four-dimensional cobordism can often be modified to meet the Euler-characteristic condition. That kinematical freedom does not contradict topology preservation in globally hyperbolic evolution: the resulting Lorentz metric may contain closed timelike curves, fail to admit Cauchy slices, or be causally open to a missing region.

The causal product theorem

The boundary assumptions matter, so state them explicitly. Let $W^n$, $n\geq2$, be a connected smooth time-oriented interpolating spacetime. Its boundary consists of an initial spacelike surface $S_0$, a final spacelike surface $S_1$, and optionally a compact timelike tube $T$. When present, $T$ meets $S_i$ in $\partial S_i$ and is diffeomorphic to $\partial S_0\times[0,1]$, with its timelike flow joining the two ends. No point of $W$ lies chronologically before $S_0$ or after $S_1$.

Assume causal compactness: causal influence cannot escape through a hole or an uncontrolled end; concretely, the closure of the region chronologically related to each point is compact in $W$. Assume also chronology, meaning that $W$ contains no closed timelike curve. No orientability of $W$ or $S_i$, global hyperbolicity, asymptotic structure, Einstein equation, ANEC, or convergence condition is required.

Causal product theorem. Under those hypotheses,

$$W\simeq S_0\times[0,1], \qquad S_0\simeq S_1.$$

This is Borde’s causally compact form of the Geroch topology-change obstruction Borde 1994, Theorem 1 and proof, pp. 17–18. Geroch’s original compact and externally Lorentzian versions appear in Geroch 1967, pp. 782–786.

The theorem includes genuine manifold boundaries rather than pretending they are absent. It does not apply to an arbitrary timelike boundary: the tube must have the stated product structure and causal compactness must include it. If the boundary has corners beyond $S_i\cap T$, null pieces, branching pieces, or missing points, a separate result is needed.

Causal compactness is neither ordinary compactness nor global hyperbolicity. An actually compact interpolating spacetime automatically supplies the required compact containment, but Borde’s condition also covers some noncompact situations whose causal futures and pasts cannot leak to infinity. Conversely, a noncompact manifold may fail causal compactness even when the proposed topology change occurs in a visually bounded part of a diagram. Global hyperbolicity would be stronger in a different direction: it forbids closed causal curves and provides a Cauchy surface. The causal product theorem deliberately assumes only chronology plus the compactness property needed for its flow argument.

The optional tube also has a precise physical interpretation. It lets one isolate a finite region while requiring the exterior boundary to carry no topology-changing surprise: its generators join corresponding points of $\partial S_0$ and $\partial S_1$. If a handle crosses that tube, if one tube component branches, or if data are injected through a deleted part of the boundary, the problem is no longer the stated interpolation. The conclusion must not be used after erasing that boundary information.

Why chronology forces a product

Choose a smooth future-directed timelike vector field $V$, tangent to $T$ when the timelike tube is present. Its integral curves are candidate histories connecting $S_0$ to $S_1$. Curves already on $T$ have the required endpoints by assumption.

Suppose an interior integral curve fails to reach $S_1$. Its future tail cannot end at a regular interior point because $V$ is smooth and nowhere zero. Causal compactness confines that tail to a compact region, so it has an accumulation point. Smooth dependence of integral curves then produces a recurrence: a later segment enters the chronological past of an earlier segment, closing a timelike loop. Chronology forbids this. The time-reversed argument shows that every integral curve also reaches $S_0$.

Normalize the flow parameter to be $0$ on $S_0$ and $1$ on $S_1$. Smooth dependence on initial data turns the flow map into a diffeomorphism $S_0\times[0,1]\to W$. The proof is not a curvature argument. Its engine is compact imprisonment plus uniqueness of timelike flow lines.

A globally hyperbolic evolution is an important special case. A Cauchy temporal function foliates the spacetime by diffeomorphic levels, as shown on Topology of Cauchy Surfaces. Thus two differently shaped hypersurfaces can bound a cobordism without representing topology change; they represent topology change only if they are not diffeomorphic, and then they cannot both be Cauchy surfaces of the same globally hyperbolic spacetime.

A dynamical strengthening

Allowing closed timelike curves does not automatically make topology change compatible with ordinary Einstein dynamics. Borde proves a stronger result for a smooth time-oriented causally compact interpolating spacetime of dimension $n\geq3$. Besides the boundary assumptions above, require:

1. every null geodesic having no endpoint on $S_0$ or $S_1$ obeys the null generic condition somewhere,

$$k_{[a}R_{b]cd[e}k_{f]}k^ck^d\ne0;$$

2. on every past-complete affinely parametrized null geodesic $\gamma(u)$ and for every chosen $u_0$, the past half-integral convergence condition holds: for each $\delta>0$ there is $b>0$ such that, for every $u_1<u_0$, some interval $I\subset(-\infty,u_1)$ satisfies $\lvert I\rvert\geq b$ and

$$\int_u^{u_0}R_{ab}k^ak^b\,du'\geq-\delta \qquad\text{for every }u\in I.$$

Pointwise null convergence is a sufficient, stronger condition. No asymptotic condition or global orientability is assumed. Under these hypotheses $W$ is again a product and both boundary surfaces are connected Borde 1994, Theorem 3 and proof, pp. 20–28.

The proof assumes topology change, obtains a nonempty Cauchy horizon, and uses causal compactness to find a compactly imprisoned horizon generator. Such a generator is complete in the needed direction even though geodesic completeness was never assumed. Genericity and the half-integral convergence condition force focusing inconsistent with an achronal horizon generator. Hence ordinary incomplete geodesics do not by themselves rescue the topology change under these assumptions.

Degeneracy, singularities, and escape routes

The two-dimensional pair of pants connects one circle to two. It is a valid cobordism. A smooth nondegenerate time-oriented Lorentz metric can be placed on suitable topology-changing examples, but causal compactness then makes closed timelike curves unavoidable. If instead one demands a global time function whose levels pass through the crotch, its differential must vanish at a Morse critical point; the associated metric construction becomes degenerate there. A degenerate bilinear form is not a Lorentz metric, so this changes the hypothesis rather than evading its conclusion.

Morse theory makes the same obstruction visible without drawing the pair of pants. Let $t:W\to[0,1]$ be a proper smooth function with $S_i=t^{-1}(i)$. If $dt$ never vanishes, the regular-level theorem and the flow of a vector field satisfying $V(t)=1$ identify all levels and give a product. Attaching a handle changes a level’s topology only when $t$ passes a critical point. But a temporal function on a Lorentzian spacetime has timelike, hence nonzero, gradient everywhere. A critical point therefore signals that $t$ is not temporal there; attempts to build the metric directly from that gradient commonly make the metric degenerate. This is a precise incompatibility between regular Lorentzian time evolution and Morse handle attachment, not a claim that the underlying cobordism cannot exist.

There is also a difference between topology of a complete spatial slice and topology of a selected region within it. A black-hole exterior can acquire new boundary components when trapped regions are excised, and an event-horizon cut can pass through a crease set. Neither construction gives two regular Cauchy hypersurfaces of the full interpolating spacetime. The product theorem constrains the latter.

Other escape routes are equally specific.

• Lose causal compactness: a handle can enter from infinity or through a deleted region. The compact-imprisonment step then fails.
• Lose chronology: closed timelike curves allow the timelike flow to recur instead of joining the two boundaries.
• Lose regularity or manifold structure: branching spaces, singular identifications, or metrics undefined at a critical set are not smooth Lorentzian cobordisms.
• Use non-Cauchy slices: partial spatial sections can have different topology while the full Cauchy topology remains fixed.
• Change theory: a Euclidean path integral sums Riemannian cobordisms and is not a classical Lorentzian evolution theorem.

“A singularity permits topology change” is therefore too broad. A singular boundary may prevent the two slices from belonging to one regular globally hyperbolic spacetime, but Borde’s dynamical theorem shows that geodesic incompleteness alone is not always enough.

Exercises

Suppose $W$ is compact, time oriented, has spacelike boundary $S_0\sqcup S_1$, and contains no closed timelike curves. Assume $S_0$ and $S_1$ are not diffeomorphic. Which hypothesis of the causal product theorem must fail? Why does merely declaring one null geodesic incomplete not repair the contradiction when the dynamical strengthening also applies?

Solution

Compactness makes causal compactness automatic in the intended closed-universe setting, and the problem gives time orientation and chronology. If all interpolation and boundary hypotheses held, the theorem would give $W\simeq S_0\times[0,1]$ and hence $S_0\simeq S_1$, contrary to the premise. Therefore no smooth nondegenerate Lorentz metric on this cobordism can satisfy all those properties. In practice chronology, spacelikeness of a boundary, time orientability, or metric nondegeneracy must fail.

If the null generic and half-integral convergence hypotheses of the dynamical theorem also hold, its relevant Cauchy-horizon generator becomes complete because it is imprisoned in a compact set. The theorem never assumes that every geodesic is complete. Making some unrelated null geodesic incomplete therefore does not remove the complete generator used in the focusing contradiction and does not authorize topology change.

References

• Borde, Arvind, “Topology Change in Classical General Relativity”, preprint, 1994.
• Geroch, Robert P., “Topology in General Relativity”, Journal of Mathematical Physics 8, pp. 782–786, 1967.