Covering Spaces and Causal Topology

A covering spacetime converts homotopy into destination data. A causal curve that winds nontrivially downstairs can lift to a curve joining different copies of the same asymptotic region upstairs. Geometry can then rule out communication between those copies. This is the characteristic method behind topological censorship: topology labels sheets, causality selects a curve, and null focusing forbids that curve.

Required background. Exterior Algebra and Differential Forms supplies pullbacks and orientation language; Hypersurface Normals and Tangential Projectors supplies induced geometry and normal expansions.

Helpful background. Pseudo-Riemannian Metrics and Signature supplies causal cones and conformal structure; Affine and Variational Geodesics supplies null geodesics and focal points; Causality and Global Structure supplies the global causal language; Timelike Congruences and Spatial Projection prepares the focusing step.

Lorentzian geometry on a covering space

Begin with a connected smooth time-oriented Lorentzian $n$-manifold $(M,g)$, $n\geq2$, without boundary, and a connected covering map

$$\pi:\widetilde M\longrightarrow M.$$

Smooth manifolds are locally path connected and semilocally simply connected, so their connected covers correspond to subgroups of $\pi_1(M)$; the universal cover corresponds to the trivial subgroup Hatcher 2002, §1.3, pp. 56–72. Give $\widetilde M$ the pullback metric $\widetilde g=\pi^*g$ and the lifted time orientation. Then $\pi$ is a local isometry.

Several consequences are immediate but important.

• Timelike, null, and spacelike tangent vectors retain their causal type under $d\pi$.
• A geodesic lifts uniquely once its initial lift is chosen, and affine parameters are unchanged.
• Curvature tensors pull back. Thus pointwise null convergence and the value of an averaged curvature integral along a lifted null geodesic agree with their downstairs values.
• Local expansion, shear, conjugate points, and focal points lift because their Jacobi equations are local.
• Deck transformations are isometries of $(\widetilde M,\widetilde g)$.

No orientability, asymptotics, field equation, or energy condition is needed for these lifting facts. If $M$ is globally hyperbolic, then every connected cover is globally hyperbolic as well. To see this, lift a Cauchy hypersurface $S$. An inextendible causal curve upstairs projects to an inextendible causal curve downstairs—an extension downstairs would lift uniquely and extend the original curve—and hence meets $S$ once. The lifted curve therefore meets $\pi^{-1}(S)$ once. This gives a Cauchy hypersurface upstairs.

The converse direction is not automatic: a quotient by isometries can create closed causal curves. Local preservation of light cones does not preserve every global causal property under quotienting.

Homotopy becomes an endpoint sheet

Fix a reference curve $\gamma_0$ between points $p$ and $q$ in $M$, and choose a lift $\widetilde p$ of $p$. The endpoint of the lifted curve is determined by the fixed-endpoint homotopy class of the path. Two paths from $p$ to $q$ are homotopic relative to their endpoints exactly when their lifts beginning at $\widetilde p$ have the same endpoint in the universal cover.

For topological censorship, $p$ and $q$ lie on, or are limiting points of, an asymptotic region. Choose a simply connected neighborhood $U$ of infinity. Its inverse image is a disjoint union

$$\pi^{-1}(U)=\coprod_{\alpha}U_\alpha,$$

with each $U_\alpha$ mapped diffeomorphically onto $U$. Under the additional conformal-completion hypotheses of a censorship theorem, each copy carries its own lifted component $\mathscr I_\alpha$. A curve homotopic to the reference curve near infinity lifts from $\mathscr I^-_0$ to $\mathscr I^+_0$. A curve in a different relative homotopy class ends on $\mathscr I^+_\alpha$ with $\alpha\ne0$.

That last statement requires more than merely writing down the universal cover. One must know that the asymptotic neighborhood is evenly covered and that the lifted physical spacetime has the asserted conformal boundary components. These are theorem hypotheses, not automatic properties of an arbitrary conformal completion.

The focusing contradiction upstairs

Suppose a lifted causal curve communicates between two distinct asymptotic copies. Near the initial copy choose a large compact codimension-two cut $T$. Its two future null normal families have different meanings: one points outward toward the same infinity, while the other points inward toward the physical region. Asymptotic control makes the inward expansion negative.

Global causal compactness is then used to replace an arbitrary connecting curve by a generator of an achronal boundary such as $\dot J^+(T)$. The generator must begin orthogonally at $T$. Asymptotic separation excludes the outward family, so it belongs to the inward, initially converging family. Null convergence, or an appropriately stated averaged condition along the relevant complete half-geodesic, creates a focal point. Beyond a focal point the geodesic cannot remain on $\dot J^+(T)$, which is the contradiction.

This architecture appears in the original Friedman–Schleich–Witt covering argument Friedman–Schleich–Witt 1993, Theorem 1 and proof, pp. 1486–1489. The existence of the required boundary generator and its avoidance of spatial infinity need additional care; the next page states a later theorem with explicit assumptions rather than treating that historical proof as complete.

A boundary-capable version

There is a particularly clean formulation when infinity is timelike. Let $M$ be the interior of a smooth spacetime-with-boundary $M'=M\cup\mathscr I$, with $g'=\Omega^2g$ on $M$, $\Omega=0$ and $d\Omega\ne0$ at the timelike boundary. Choose one connected component $\mathscr I$ and define its domain of outer communications $D=I^+(\mathscr I)\cap I^-(\mathscr I)$.

The following hypotheses are all part of the result:

• $D'=D\cup\mathscr I$ is globally hyperbolic as a spacetime-with-timelike-boundary, so causal curves may run along the boundary;
• $\mathscr I$ admits a compact spacelike cut;
• for every $p$ sufficiently near $\mathscr I$ and every future-complete affinely parametrized null geodesic $\eta:[0,\infty)\to D$ beginning at $p$,

$$\int_0^\infty R_{ab}\dot\eta^a\dot\eta^b\,ds\geq0;$$

• the metric and completion are smooth enough for causal-boundary and focusing arguments.

No global orientability assumption is needed for this censorship statement, and the spacetime dimension is arbitrary above the two-dimensional degenerate case. Under these assumptions, every causal curve in $D'$ with endpoints on $\mathscr I$ is fixed-endpoint homotopic to a curve in $\mathscr I$. This is the principle of topological censorship proved in Galloway–Schleich–Witt–Woolgar 1999, Theorems 2.1–2.2 and proofs, pp. 7–11.

This source treats the manifold boundary directly. No doubling argument is needed. The half-geodesic integral is stronger in directionality than an averaged condition stated only on complete two-sided geodesics, and it is exactly what the proof uses.

The subgroup cover and the algebraic conclusion

For a boundary component that is not simply connected, the universal cover is not the most efficient choice. Inclusion $i:\mathscr I\hookrightarrow D'$ induces

$$i_*:\pi_1(\mathscr I,p)\longrightarrow\pi_1(D',p).$$

Take the connected cover of $D'$ corresponding to the subgroup $G=i_*\pi_1(\mathscr I,p)$. The selected lift $\widetilde{\mathscr I}$ is a copy of $\mathscr I$, and every loop in the cover is homotopic to a loop in that copy. If a causal curve downstairs with endpoints on $\mathscr I$ lifted to a curve ending on a different boundary component, the noncommunication result obtained by focusing would be violated. Hence both lifted endpoints lie on $\widetilde{\mathscr I}$, the lifted curve deforms into it, and projection gives the desired homotopy downstairs.

An immediate algebraic consequence is that $i_*$ is surjective Galloway–Schleich–Witt–Woolgar 1999, Proposition 3.1 and Corollary 3.3, pp. 11–13. Therefore:

• if $\mathscr I$ is simply connected, then $D'$ and $D$ are simply connected;
• if $\mathscr I$ has nontrivial fundamental group, the theorem permits exterior loops inherited from infinity but forbids new loop classes inaccessible at infinity;
• the conclusion concerns the chosen domain of outer communications, not the entire spacetime behind its horizons.

Limits and checks

The flat cylinder $\mathbb R\times S^1$ is globally hyperbolic and has a nontrivial universal cover, but it has none of the required asymptotic boundary structure. It is a check that global hyperbolicity alone cannot produce the focusing contradiction.

Likewise, a nontrivial topology hidden behind an event horizon need not yield any causal curve between different lifted copies of the same exterior infinity. The covering method constrains what causal probes from the specified boundary can traverse; it does not erase topology from the full Cauchy surface.

Finally, a pointwise energy condition and an averaged one cannot be swapped without revisiting completeness and the focusing lemma. The integral must be imposed on the geodesics and in the direction actually produced by the global argument.

Exercises

Assume the boundary-capable principle above and suppose $\mathscr I$ is simply connected. Prove that every loop in $D'$ is null-homotopic. Then explain why this does not prove that the full physical spacetime $M$ is simply connected.

Solution

The principle implies that inclusion induces a surjection

$$i_*:\pi_1(\mathscr I)\twoheadrightarrow\pi_1(D').$$

Because $\mathscr I$ is simply connected, its fundamental group is trivial. A surjective homomorphism from the trivial group can have only the trivial group as its target, so $\pi_1(D')=0$. Equivalently, in the subgroup-cover construction, the relevant cover is the universal cover; focusing prevents more than one lifted boundary component from communicating, forcing that cover to be one sheet.

But $D'$ is the completed domain of outer communications associated with one boundary component. Regions behind future or past event horizons are not contained in $D'$. A loop in the full spacetime $M$ may pass through such a region and represent topology that no two-ended causal probe from $\mathscr I$ can sample. The result is simple connectivity of the exterior domain, not of all $M$.

References

• Friedman, John L., Kristin Schleich, and Donald M. Witt, “Topological Censorship”, Physical Review Letters 71, pp. 1486–1489, 1993; erratum, volume 75, p. 1872, 1995. Open PDF.
• Galloway, Gregory J., Kristin Schleich, Donald M. Witt, and Eric Woolgar, “Topological Censorship and Higher Genus Black Holes”, Physical Review D 60, 104039, 1999. Open PDF.
• Hatcher, Allen, Algebraic Topology, Cambridge University Press, Cambridge, 2002.