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A causal curve has an everywhere nonspacelike tangent of consistent time orientation; its smooth segments are timelike or null. More generally, curves with local Lipschitz continuity and future causal tangent almost everywhere are allowed. The causal future and causal past are defined by reachability along these curves.

The causal future of $S$ consists of points reached from $S$ by future-directed causal curves, including points of $S$ itself. In a globally hyperbolic spacetime, the causal future of a compact set is closed.

The causal past is the time-reversed causal future. Relative to a specified component ${\mathcal{I}}^{+}$ of future null infinity, the black hole region is the complement of its causal past: $\mathcal{B}=\mathcal{M}\setminus J^{-}({\mathcal{I}}^{+})$.

For flat spacetime with $x$ periodic of circumference $2\pi R$, lift endpoints of causal curves to the covering Minkowski spacetime. A signal can choose any winding, so the causal future and causal past use the shortest circular distance rather than a fixed coordinate separation. After a duration $\pi R/c$, every point on the circle is reachable. A constant tangential electric field has a nonzero loop electromotive force; maintaining it while a charge gains energy each lap requires an energy reservoir, not a single-valued electrostatic potential.

In the flat quotient $t\equiv t+P_t$ with noncompact space, choose any sufficiently large positive or negative temporal winding to make a lifted displacement timelike. Every event is in both the causal future and causal past of every other event. Constant-position curves are closed timelike curves. The quotient is a valid Lorentzian manifold, but it violates chronology and cannot support arbitrary globally consistent initial-value evolution.

The future horismos is the part of the causal future outside the chronological future. For a compact set in a globally hyperbolic spacetime, it equals the boundary of the chronological future. Its normal null geodesic generators cannot have passed a conjugate point to a spacelike surface when the initial set is that surface.

Since the real conformal factor $\Lambda$ is nowhere zero, ${\Lambda }^2>0$ at every point. For any tangent vector,The sign is unchanged. With metric signature $(-,+,+,+)$, negative, zero and positive values therefore describe the same timelike vectors, null vectors and spacelike vectors for both metric tensors. In particular the two null cones coincide.

Choose the same time orientation for the two Lorentzian metrics. A future-directed causal curve or timelike curve for one has the same character for the other. Hence the causal future, causal past and chronological future of every point agree:This is the preservation of causal structure by a smooth, nonvanishing conformal transformation. It does not assert preservation of proper time or of affine parameter lengths.

The relevant Penrose singularity theorem states: a connected time-oriented four-dimensional globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a nonempty compact orientable boundaryless trapped surface, and the null convergence condition is future null-geodesically incomplete. Under the Einstein field equations, the null energy condition supplies the curvature hypothesis. A trapped surface here is spacelike and has both future normal null expansions strictly negative. The conclusion is an incomplete null geodesic, not necessarily a divergent curvature invariant at an identifiable point.

Suppose, for contradiction, that every future null geodesic is complete. Write $S$ for the trapped surface andThe equality follows because the causal future of a compact set is closed in a globally hyperbolic spacetime. This future horismos is a closed achronal boundary. It is nonempty: the restriction of a Cauchy time function to compact $S$ has a minimum, and a point at that minimum cannot be chronologically preceded by another point of $S$.

Normalize the two future null normal directions along $S$ against a smooth future timelike field, fixing the affine scale continuously. Their initial null expansions are continuous and strictly negative. Compactness of the normalized normal bundle gives a uniform $c>0$ with ${\theta }_{\pm }\le -c$ everywhere on $S$. The preceding null focusing theorem forces a conjugate point to a spacelike surface along each normal generator within affine length $L=2/c$.

By the supplied boundary-generator result, every point of $E^{+}(S)$ is reached by an orthogonal future null geodesic which has no earlier conjugate point. No such boundary generator can remain on the boundary beyond its first focal point. Every point of $E^{+}(S)$ therefore lies inThe parameter set is compact. Future null completeness makes its geodesic flow defined throughout this common finite interval, and smooth dependence on initial data makes its image $K$ compact. Since $E^{+}(S)$ is closed and contained in $K$, it is compact. This is the compactness of the future horismos of a trapped surface step; a merely pointwise finite bound would not suffice without compactness and uniform normalization.

Choose a smooth complete timelike vector field, obtained if necessary by positive rescaling against a complete auxiliary Riemannian metric. Its inextendible integral curves of a vector field meet a chosen smooth Cauchy hypersurface $C$ exactly once. Projection along those curves defines a continuous map ${\pi }_C:E^{+}(S)\to C$. The achronal set property makes this map an injective function: two boundary points on the same timelike integral curve of a vector field would be timelike related.

An achronal boundary is a topological hypersurface without boundary, even at nonsmooth generator junctions; the submanifold property allowed in the question supplies this fact. Thus domain and codomain are both three-dimensional topological manifolds without boundary. Invariance of domain makes ${\pi }_C(E^{+}(S))$ open in $C$. Compactness makes it closed in the Hausdorff space $C$, and it is nonempty. Connectedness of the spacetime gives connectedness of $C$, so this image is all of $C$. That would make $C$ compact, contradicting the noncompact Cauchy hypersurface hypothesis. The completeness assumption is false, proving the theorem.

The noncompactness hypothesis and strict trapping are essential to this version. Global hyperbolicity cannot simply be omitted in the compactness and projection steps, and nonpositive initial expansion with zeros does not supply the uniform focusing bound used here.

Outside both light cones, the separation from the origin is spacelike. Thus the events in neither the causal past nor the causal future areNo causal curve can connect either endpoint to the other. These are the two green side regions in part c; the strict inequality excludes the causally reachable null boundaries.

Reverse the endpoints of the future-directed causal curve. The origin can be reached precisely from its causal past,This is the orange region below the two null rays in part c. The origin itself belongs to both the causal future and causal past by the usual zero-length-curve convention.

A future-directed causal curve from the origin has $|dx/dt|\le c$. Integrating this inequality gives the causal futureThe straight timelike or null segment proves sufficiency. This is the blue region above the two null rays in the upper-left spacetime diagram of part c.

The upper-left panel shows the three causal structure regions, with null boundaries included in the appropriate causal future and causal past. For the charged trajectory, use part b with $a=q{E}_x/m$, choosing the $x$ direction so that $a>0$. The electric field gives the constant three-force $q{E}_x$ and hence the shifted hyperbolaIt starts with a vertical tangent at the origin, stays strictly inside the future light cone, and asymptotes to $x=ct-c^2/a$. The other panels implement the coordinate and topology changes used below; they are original spacetime diagrams, not copies of source artwork.

The invariant spacelike condition is $x^{\prime 2}-c^2{t}^{\prime 2}>0$. Hence the primed-coordinate events in neither the causal future nor the causal past obeyThe green regions in part c retain their shape in primed coordinates. A particular spacelike event's coordinate time may change sign, but it never enters either light cone.

The complement of the causal future and causal past isThis green region is confined to $|t|<\pi R/c$ and meets the identified strip edges. Spatial periodicity creates alternative signal paths, but time remains noncompact and no closed timelike curve follows from this spatial identification.

A causal curve can wind around the spatial circle. Lift its endpoint to $x+nP$ on the covering space. A lift is reachable from the origin if $ct\ge |x+nP|$ for at least one integer $n$. With shortest circular distancethe causal future isIn the centred fundamental strip, $d=|\bar{x}|$. For $t\ge P/(2c)=\pi R/c$, every spatial point is in the causal future. This produces the fully blue upper part of the lower-left panel in part c.

Every event is in both the causal future and causal past, soDrawing just the nearest two null rays on a temporally identified strip would miss winding causal curves and give an incorrect nonempty exterior region.

For any representative event $(t,x)$, choose an integer $n$ large enough that $c(t+n{P}_t)>|x|$. A future-directed timelike segment from the origin to that lift projects to a causal curve on the quotient. ThereforeEven points with large $|x|$ can be reached by allowing sufficiently many temporal windings. The purple lower-right panel in part c denotes simultaneous membership in both the causal future and causal past.

For causal reachability with periodic time, let $P_t=2\pi T$. The lower-right panel of part c identifies its top and bottom edges. The vertical curve at fixed $x$ is a closed timelike curve: after a positive proper duration $P_t$, it returns to the same spacetime event. The quotient is a mathematically well-defined flat Lorentzian manifold, but it violates chronology and admits no usual global causal ordering or ordinary global initial-value evolution.

These are physical causality problems, rather than an algebraic contradiction in the metric. Fields and particle histories must satisfy periodic consistency conditions; an arbitrary initial state followed by irreversible evolution will generally fail to return to itself after a temporal lap. The fact that the causal past and causal future overlap everywhere, established below, explains why the ordinary three-way causal partition degenerates.