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The causal hierarchy orders restrictions on the causal structure of a spacetime, from exclusion of closed timelike curves through stable causality and global hyperbolicity.

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.

On the equatorial plane of the Kerr metric, the displayed component is negative for sufficiently small negative $r$. A curve of constant $t,r,\theta =\pi /2$, with $\varphi$ increasing through $2\pi$, is therefore a regular closed timelike curve. It avoids the ring at $r=0$. Such circles occur on the negative-radius sheet of the exact analytic extension, not in the exterior region accessible from positive-radius infinity.

First use the original null coordinates $u=t+m/\lambda$, $v=t-m/\lambda$. They obey $u-v=2m/\lambda$, and the radial metric tensor becomesApply the given substitutions $u=\tan (U/2)$ and $v=-\cot (V/2)$. The identitiesgive the analytic extension of a spacetimeThe poles of the individual tangent and cotangent substitutions are now ordinary coordinate lines wherever $\cos ((U-V)/2)\ne 0$. Set $T=(U+V)/2$ and $X=(U-V)/2$. Allowing these coordinate functions to continue across every such line givesThis is the universal cover of two-dimensional anti-de Sitter spacetime of radius $m$, times a two-sphere of radius $m$: the simply connected Bertotti-Robinson spacetime. In defining this maximal analytic extension, $T$ ranges over the whole real line without periodic identification. A periodic identification would instead create closed timelike curves and would be an additional global choice.

For clarity, the original patch is embedded by the branches $-\pi <U<\pi$ and $0<V<2\pi$. The inequality $u>v$ is equivalent on these branches to $|U-V|<\pi$, so the original patch occupiesThe first spacetime extension in part (a) occupies the larger band $0<T-X<2\pi$, with $|X|<\pi /2$. Its finite-$v$ Killing horizon is the interior line $T+X=\pi$. The edges $T-X=0,2\pi$ of that band are further regular Poincaré horizons; continuing through them produces the full conformal strip of two-dimensional anti-de Sitter spacetime.

The original coordinate $\lambda$ has a smooth continuation throughout the strip:Consequently the complete set of zero-level Killing horizons isThese are two alternating families of parallel null lines; different families meet only at the excluded conformal boundaries, never inside the spacetime. The continuation of the original Killing vector field isThe field never vanishes in the interior, including on the Killing horizons.

To obtain the Penrose diagram, suppress the two-sphere and multiply by the conformal factor ${\cos }^{2}X/m^2$. The resulting radial metric tensor is $-d{T}^2+d{X}^2$, so the diagram is an infinitely tall strip of width $\pi$, with null lines at 45 degrees and timelike edges $X=\pm \pi /2$. Generically,At the special edge points where the two families of Killing horizons end, the displayed formula is a $0/0$ limit and $\lambda$ has no unique limiting value. Thus the infinity labels on the diagram refer to the open edge segments, not those vertices. Each interior point of the diagram represents a two-sphere of fixed radius $m$; neither edge is a centre where that two-sphere shrinks.

Finally, these timelike edges are conformal boundaries, not omitted regular points of a further spacetime extension. One way to establish maximality is geodesic completeness. The two-dimensional anti-de Sitter spacetime factor embeds locally as $Z\cdot Z=-m^2$ in a flat space with metric signature $(-,-,+)$. Along an affinely parametrized geodesic with constant $ϵ=\dot{Z}\cdot \dot{Z}$, the ambient equation isIts oscillatory, linear or hyperbolic solutions for negative, zero or positive $ϵ$ exist for all real affine parameters and lift to the universal cover. The two-sphere is also geodesically complete, and geodesics of the product split into geodesics of its factors. Hence $\tilde{g}$ is geodesically complete. A proper regular spacetime extension would furnish a geodesic reaching its boundary in finite affine parameter, contradicting that completeness. This proves that the displayed simply connected analytic spacetime is maximal.

The local metric tensors do not determine the global coordinate identifications. The usual complete two-dimensional de Sitter spacetime is the hyperboloid $(X^0{)}^2-(X^1{)}^2-(X^2{)}^2=-1$ in ambient signature $(+--)$, withPulling back the ambient metric tensor gives $d{t}^2-{\cosh }^{2}td{\chi }^2$, withUnwrapping $\chi$ gives the universal cover of two-dimensional de Sitter spacetime instead. The usual embedded two-dimensional anti-de Sitter spacetime has $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in signature $(++-)$ andHere $r\in \mathbb{R}$ and $t$ is initially periodic modulo $2\pi$. The time circles are closed timelike curves. Its physically standard universal cover removes that identification:The two signs of $r$ describe two spatial ends; imposing $r\ge 0$ would retain only half of this complete model.

Both spacetimes have constant curvature and are geodesically complete. In the paper's convention their Ricci scalars are respectively $+2$ and $-2$. Their embeddings show that geodesics are intersections with planes through the ambient origin. They nevertheless have very different causal structures.

For the geodesics of two-dimensional de Sitter spacetime, use an affine parameter $\lambda$, normalization $\kappa ={\dot{t}}^2-{\cosh }^{2}t{\dot{\chi }}^2\in \{1,0,-1\}$ and conserved momentum $L={\cosh }^{2}t\dot{\chi }$. ThenTimelike geodesics extend to both infinite proper times; $\chi =\text{constant}$ is a simple example. Null geodesics obey $\sinh t=\pm |L|(\lambda -{\lambda }_0)$, so their affine parameter is also infinite at either temporal end. Spacelike geodesics have $|L|\ge 1$, obey $\sinh t=\sqrt{L^2-1}\sin (\lambda -{\lambda }_0)$ and are closed curves on the embedded cylinder; $t=0$ is its simplest spacelike geodesic circle. Unwrapping the spatial circle removes closure but not completeness.

For the geodesics of two-dimensional anti-de Sitter spacetime, the timelike Killing vector ${\partial }_t$ gives $E={\cosh }^{2}r\dot{t}$. The normalization is $\kappa ={\cosh }^{2}r{\dot{t}}^2-{\dot{r}}^2$, henceChoose a future-directed timelike geodesic, so $E\ge 1$ andFor $E>1$, it oscillates radially with proper period $2\pi$; $E=1$ gives the central geodesic $r=0$. In both cases, global time advances by $2\pi$ over a proper time interval $2\pi$. The curve is closed on the original hyperboloid, while its lift on the universal cover is nonclosed and future-directed. Null geodesics have $\sinh r=\pm E(\lambda -{\lambda }_0)$ and reach either spatial end only at infinite affine parameter. Spacelike geodesics satisfy $\sinh r=\sqrt{E^2+1}\sinh (\lambda -{\lambda }_0)$ and also have infinite proper length toward either end. Thus the finite coordinate time discussed below does not imply physical geodesic incompleteness.

The conformal cylinder of two-dimensional de Sitter spacetime follows by setting $\eta =\arctan (\sinh t)$:Its past and future conformal boundaries are spacelike circles. Null geodesics have $d\chi /d\eta =\pm 1$ and can travel only a finite angular distance over the entire infinite proper-time history. For the complete observer $\chi =0$, an event can send a signal to that observer precisely when its shortest angular distance $d(\chi ,0)$ is less than $\pi /2-\eta$. The observer horizons in two-dimensional de Sitter spacetime are therefore the null curves $d(\chi ,0)=\pi /2-\eta$; its past signal horizon has $d(\chi ,0)=\eta +\pi /2$. These are observer horizons, without curvature singularities. Their intersection bounds a static patch with metric tensorThe horizons at $\rho =\pm 1$ are regular Killing horizons beyond which this static chart fails, while the global coordinates remain smooth.

For the conformal strip of two-dimensional anti-de Sitter spacetime, set $\psi =\arctan (\sinh r)$. On its universal cover,The two conformal boundaries are timelike. A null ray from the center approaches a boundary after coordinate time $\pi /2$, though its affine parameter diverges. Every interior event can signal to the complete central static observer in finite global time, so that observer has no event horizon. A Poincaré horizon concerns a restricted coordinate patch, and accelerated observers can have different causal horizons.

Finally, the global de Sitter cylinder and its spatial cover are globally hyperbolic: constant-time slices are Cauchy hypersurfaces. The anti-de Sitter universal cover is not globally hyperbolic, because timelike infinity admits incoming signals within finite global time. A field's evolution therefore requires boundary conditions at its two conformal boundaries in addition to initial data. The opposite placement of the conformal boundaries explains why de Sitter observers have cosmological horizons despite global hyperbolicity, while the central anti-de Sitter observer has no event horizon despite the failure of global hyperbolicity.

Take the causally unwrapped universal cover of Anti-de Sitter spacetime, with curvature radius $L$. In global coordinates,The ordinary radial Penrose diagram is a half-strip: the left edge $\rho =0$ is a regular center and the right edge $\rho =\pi /2$ is a timelike conformal boundary. For displaying signed spatial motion, the figure doubles the radial diameter to $-\pi /2<\rho <\pi /2$. Its two ends represent antipodal spatial directions of the higher-dimensional boundary; they are not a claim that the full AdS boundary has two disconnected components.

The vertical edges are spatial infinity at every finite global time, forming the timelike boundary $\mathbb{R}\times S^{d-2}$. The arrows mark the future and past timelike ends $T\to \pm \infty$, conventionally $i^{+}$ and $i^{-}$. There are no separate null-infinity edges like Minkowski ${\mathcal{I}}^{\pm }$: radial null rays satisfy $dT=\pm d\rho$ and reach the timelike boundary in finite global time. Their physical affine parameter is nevertheless infinite there. The drawing is a finite excerpt of an infinitely tall strip, not an identification of its top and bottom. On the original hyperboloid global time is periodic and there are closed timelike curves; unwrapping it is essential to the usual physical maximally extended AdS spacetime.

Both metrics have curvature radius one and signature $(+-)$. Their global topology must be specified as well as their local metric. The usual two-dimensional de Sitter spacetime is the hyperboloid $(X^0{)}^2-(X^1{)}^2-(X^2{)}^2=-1$ in three-dimensional Minkowski spacetime, parametrized byThus $t\in \mathbb{R}$ and $\chi$ is periodic modulo $2\pi$, for example $0\le \chi <2\pi$. Spatial slices are circles. Unwrapping $\chi$ instead gives the universal cover, not the usual closed spatial model.

The ordinary two-dimensional anti-de Sitter spacetime hyperboloid is $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in ambient signature $(++-)$, withOn this quadric $r\in \mathbb{R}$ and $t$ is periodic modulo $2\pi$. The closed time circles are timelike, so it contains closed timelike curves. The usual physically causal AdS model is its universal cover, for which $r,t\in \mathbb{R}$, with time no longer identified. The signed radial coordinate has two spatial ends; restricting to $r\ge 0$ would retain only half of this global model. The metric by itself cannot distinguish periodic time from the cover. The scalar curvature is $+2$ for de Sitter and $-2$ for AdS in the Ricci convention of Q2. Both are maximally symmetric and have no curvature singularity; these toy two-dimensional metrics illustrate causal geometry rather than the ordinary four-dimensional matter field equations.

For the conformal cylinder of two-dimensional de Sitter spacetime, introduceThenThe conformal boundaries $\eta =\pm \pi /2$ are spacelike circles, representing past and future infinity. Null geodesics are straight lines $\chi ={\chi }_0\pm (\eta -{\eta }_0)$, with the periodic spatial identification. The finite total conformal-time interval explains the cosmological horizons: light cannot complete a full trip around the circle during the entire history.

For the conformal strip of two-dimensional anti-de Sitter spacetime, useso on the universal coverIts two conformal boundaries at $\psi =\pm \pi /2$ are timelike. Null rays are $t\pm \psi =\text{constant}$, and the strip extends for arbitrarily long coordinate time. The distances in these conformal diagrams are not physical lengths: a conformal boundary may be reached in finite coordinate time while lying infinitely far away in physical affine parameter.

The sides of the de Sitter diagram are identified; its top and bottom are spacelike infinities. The AdS diagram shows a finite window of an infinite time strip, with no identification of top and bottom. Null rays have slope one in both diagrams. The blue curves are timelike geodesics, while the colored de Sitter lines bound the central observer's causal domains.

The geodesics of two-dimensional de Sitter spacetime can be derived directly. Let an overdot denote an affine parameter normalized so that the tangent norm is $\kappa =1,0,-1$ for timelike, null and spacelike geodesics. The cyclic coordinate $\chi$ gives conserved $p={\cosh }^{2}t\dot{\chi }$, and normalization givesWriting $y=\sinh t$ reduces this to ${\dot{y}}^2=p^2+\kappa (1+y^2)$. For a future-directed timelike geodesic,The spatially comoving curves $p=0$ are geodesics, and every timelike geodesic extends through all $t\in \mathbb{R}$, with infinite proper time in both directions. Initially comoving nearby observers have physical separation proportional to $\cosh t$, displaying de Sitter's tidal defocusing. Relative separation can initially decrease for other initial velocities; not every pair is monotonically separating.

For a nontrivial null geodesic, $p\ne 0$ and $\sinh t=\pm |p|\lambda +\text{constant}$. Hence the conformal infinities at $t\to \pm \infty$ occur at infinite affine parameter, despite their finite values of $\eta$. For a spacelike geodesic, $|p|\ge 1$ and $y=\sqrt{p^2-1}\sin (s-s_0)$. Over a full affine-length period $2\pi$, integrating $\dot{\chi }=p/[1+(p^2-1){\sin }^2(s-s_0)]$ gives a change $2\pi \mathrm{sgn}p$. Thus the spacelike geodesics close on the usual cylinder. The special case $|p|=1$ is the equatorial circle $t=0$. These closed spacelike curves do not threaten causality.

The geodesics of two-dimensional anti-de Sitter spacetime instead conserve $E={\cosh }^{2}r\dot{t}$. Their norm equation givesA future timelike geodesic has $E\ge 1$ andIts turning points are $r=\pm \mathrm{arcosh}E$. Timelike geodesics oscillate rather than escaping to the spatial boundary. For $E=1$, the central curve $r=0$ is geodesic. Integrating $\dot{t}=E/[1+(E^2-1){\sin }^2(\tau -{\tau }_0)]$ shows that a full proper-time period $2\pi$ advances coordinate time by $2\pi$. They close on the periodically identified hyperboloid but remain complete nonclosed curves on the universal cover. Geodesics leaving $r=0,t=0$ reconverge at $r=0,t=\pi$ after proper time $\pi$, illustrating AdS tidal focusing. A static observer at nonzero fixed $r$ is accelerated, not one of these oscillating free trajectories.

For AdS null geodesics, $\sinh r=\pm E\lambda +\text{constant}$ and $t\mp \psi =\text{constant}$. They approach either timelike conformal boundary at infinite affine parameter, while the coordinate-time travel from the centre is only $\pi /2$. A spacelike geodesic has $\sinh r=\sqrt{E^2+1}\sinh (s-s_0)$ and connects the two spatial ends with infinite affine length. Both global models are geodesically complete. Reflecting a signal at the AdS conformal boundary is a choice of boundary condition for fields; it is not an automatic continuation of a geodesic at a finite affine endpoint.

For the complete de Sitter observer at $\chi =0$, let $d(\chi ,0)\in [0,\pi ]$ be the shortest angular distance. An event at $(\eta ,\chi )$ can send a signal reaching the observer at a finite future time exactly whenIt can receive a signal sent by the observer at a finite past time exactly when $d(\chi ,0)<\eta +\pi /2$. Equality gives the future and past observer horizons in two-dimensional de Sitter spacetime; their null branches are respectively $\chi =\pm (\pi /2-\eta )$ and $\chi =\pm (\eta +\pi /2)$ on the displayed cylinder. Their intersection is the central static patch, $d(\chi ,0)+|\eta |<\pi /2$. All complete timelike geodesic observers have analogous horizons by de Sitter symmetry. These are observer-dependent cosmological horizons, not singularities of the global coordinates or Cauchy horizons.

In contrast, the complete central AdS observer can exchange signals with every event at finite $r$: the conformal distance $|\psi |$ is finite and the observer exists for all $t\in \mathbb{R}$. Its static Killing vector has squared norm ${\cosh }^{2}r>0$ everywhere. Thus global AdS on its cover has no cosmological event horizon for this observer, although other restricted patches or accelerated observers can have horizons.

The causal predictability distinction is also important. Global de Sitter is globally hyperbolic: constant-$t$ circles are Cauchy surfaces. A causal curve is monotone in $t$ and satisfies $|d\chi /dt|\le \mathrm{sech}t$; if it ended at finite $t$, compactness of the spatial circle would allow an endpoint and extension. Hence an inextendible causal curve crosses every such slice once. AdS on the cover has no closed timelike curves, but it is not globally hyperbolic. For two central events at $t=-T$ and $t=T$, with $T>\pi /2$, their causal diamond contains events at $t=0$ and arbitrarily large $r$, so it is not compact. Initial data alone do not determine field evolution without conditions at the timelike conformal boundaries. de Sitter has spacelike conformal infinities and cosmological observer horizons; covered AdS has timelike conformal infinities and requires boundary data. The distinction between the AdS quadric and its universal cover is also explained in David Tong's general relativity lectures.

For the printed spacetime, yes. The surfaces $T=\text{constant}$, equivalently $t=\text{constant}$, are Cauchy surfaces in Minkowski spacetime. Every inextendible causal curve intersects each once; if its monotone time coordinate had a finite endpoint, the bound $|dx/dT|\le 1$ would give a finite spatial endpoint too, allowing continuation. Causal diamonds are closed and bounded in $(T,x)$ and hence compact. Therefore the printed spacetime is globally hyperbolic.

Initial data on a Cauchy slice determine unique causal evolution for the usual well-posed hyperbolic field equations, without boundary data at spatial infinity at finite time. The time reparametrization preserves that property.

For the qualified AdS interpretation, no. On the universal cover, choose two central events with enough time separation for a null ray to approach a strip edge and return. Their causal diamond contains a sequence with $\rho \to \pi /2$ but no limit in the physical spacetime, so it is noncompact. A constant-$t$ surface fails to meet every inextendible causal curve arriving from timelike conformal infinity. Thus even the cover is not globally hyperbolic. Field evolution needs appropriate boundary conditions, such as a reflecting condition compatible with the field theory, as well as initial data. If AdS time itself is periodically identified, closed timelike curves give an additional and more immediate obstruction. These are consequences of the alternative spatial-lapse geometry, not of the printed flat metric tensor.

Take both curvature radii to be one. Coordinate ranges depend on the global identification, which a line element alone cannot fix. For the ordinary two-dimensional de Sitter spacetime,This is the unit hyperboloid $X_0^2-X_1^2-X_2^2=-1$ in ambient signature $(+--)$, parametrized by $X_0=\sinh t$, $X_1=\cosh t\cos \chi$, $X_2=\cosh t\sin \chi$. Unwrapping $\chi$ instead gives its universal cover.

For two-dimensional anti-de Sitter spacetime, the embedding $X_0^2+X_1^2-X_2^2=1$ in ambient signature $(++-)$ is parametrized by $X_0=\cosh r\cos t$, $X_1=\cosh r\sin t$, $X_2=\sinh r$. On this hyperboloid $r\in \mathbb{R}$ and $t$ is periodic modulo $2\pi$, creating closed timelike circles. The physically usual universal cover removes that periodicity:Both signs of $r$ are needed for the full two-dimensional spatial line. Taking $r\ge 0$ without an additional construction would omit one of its two ends. The following causal comparison uses the AdS universal cover.

Both models are maximally symmetric Lorentzian manifolds, with constant sectional curvature and no curvature singularities. In the paper's convention their Ricci scalars are respectively $+2$ and $-2$. The de Sitter spatial scale factor $a(t)=\cosh t$ contracts to a nonzero minimum and then expands; $t=0$ is not a big-bang singularity. The AdS metric is static, and ${\partial }_t$ is timelike everywhere because $g_{tt}={\cosh }^{2}r>0$.

The embeddings give a concise classification of all affinely parametrized geodesics. Geodesic acceleration in the ambient space is normal to the hyperboloid, hence proportional to $X$. Differentiating the constraint and setting $X^{\prime 2}=\kappa$, with $\kappa =1,0,-1$ for timelike, null and spacelike curves, givesEvery solution stays in the two-plane spanned by its initial point $P$ and tangent $V$, so every geodesic is a plane section through the ambient origin. For de Sitter, these are $P\cosh s+V\sinh s$ for timelike curves, $P+sV$ for null curves, and $P\cos s+V\sin s$ for spacelike curves. Thus timelike geodesics and null geodesics are complete and nonclosed, while spacelike geodesics are closed circles on the ordinary hyperboloid. These are the geodesics of two-dimensional de Sitter spacetime.

For AdS the roles of the circular and hyperbolic solutions are reversed: timelike curves are $P\cos s+V\sin s$, null curves are $P+sV$, and spacelike curves are $P\cosh s+V\sinh s$. Timelike curves close after proper time $2\pi$ on the hyperboloid, but their lifts on the universal cover continue to ever later time rather than forming closed curves. All three kinds extend for arbitrary affine parameter. This is geodesics of two-dimensional anti-de Sitter spacetime, not an assertion that geodesics reach the conformal boundary in finite physical affine length.

The coordinate first integrals make the behavior concrete. In de Sitter, spatial rotational symmetry gives a conserved $P_{\chi }={\cosh }^{2}td\chi /ds$, and normalization givesConstant-$\chi$ observers are timelike geodesics. Nontrivial null geodesics have affine parameter proportional to $\sinh t$, which diverges at both $t\to \pm \infty$. In AdS the conserved static energy is $E={\cosh }^{2}rdt/ds$, withFor timelike curves, $E\ge 1$ andThey oscillate through the center and never reach spatial infinity; $r=0$ is the $E=1$ geodesic. For null curves, $\lambda =\pm \sinh r/E+\text{constant}$, so both ends are at infinite affine parameter. Spacelike AdS geodesics likewise extend to the two spatial ends.

For the conformal structure, introduce de Sitter conformal timeMultiplying by ${\cos }^2\eta$ gives the conformal cylinder of two-dimensional de Sitter spacetime. The past and future conformal infinities are spacelike circles at $\eta =\mp \pi /2$. Null curves are $\chi =\pm \eta +\text{constant}$ modulo $2\pi$. Although the conformal-time interval is finite, their physical affine parameters and timelike observers' proper times are infinite at its ends. Constant-$\eta$ circles are Cauchy hypersurfaces; global de Sitter is globally hyperbolic.

For AdS putThe conformal strip of two-dimensional anti-de Sitter spacetime has unbounded time and timelike conformal boundaries at $\psi =\pm \pi /2$. Null curves have $t\pm \psi =\text{constant}$. A signal from any finite $r$ reaches the center in coordinate time $|\psi |<\pi /2$, and a null curve crosses the entire conformal strip in time $\pi$. This finite coordinate travel time coexists with infinite physical affine length to the boundary. The covering spacetime has no closed timelike curves, but is not globally hyperbolic: causal curves can arrive from timelike infinity without meeting a proposed initial slice. Field evolution consequently requires boundary conditions as well as initial data. Unwrapping time removes closed timelike curves, not the timelike boundary.

The contrast in observer event horizons follows directly from these null curves. For the complete de Sitter observer $\chi =0$, let $d(\chi ,0)\in [0,\pi ]$ be shortest angular distance. An event can send a signal to this observer before its future endpoint precisely whenEquality is its future observer event horizon. It can receive a signal emitted by the observer after its past endpoint precisely when $d(\chi ,0)<\eta +\pi /2$, whose equality is the past horizon. In the observer's fundamental domain these horizons are null linesTheir intersection encloses the observer's static patch of de Sitter spacetime, the diamond $d(\chi ,0)<\pi /2-|\eta |$. They are observer-dependent cosmological horizons, not curvature singularities or Cauchy horizons. Every inertial observer has the corresponding horizons by de Sitter symmetry.

For the complete AdS static observer at $r=0$, every event at finite $r$ can send a signal to the observer, and can receive one, because time is unbounded and the coordinate distance $|\psi |$ is finite. Thus there is no corresponding global static-observer event horizon. The lapse never vanishes in these coordinates. Restricted accelerated-observer patches can have observer horizons, but those are distinct from the global static model. On the original periodic-time AdS hyperboloid, the more serious issue is closed timelike curves.

The figure identifies the de Sitter spatial edges and displays only a finite time window of the AdS covering strip, whose time continues indefinitely. Solid red curves are null geodesics; the oscillating AdS curve is a timelike geodesic. The shaded de Sitter diamond is the static observer's two-way communication region, not the entire spacetime.

The concise comparison is de Sitter has spacelike conformal infinities and cosmological observer horizons; covering AdS has timelike conformal boundaries and no horizon for an eternal global static observer. Both are constant-curvature and geodesically complete, but their geodesic recurrence, global topology, causal boundaries and initial-value properties differ.

On the equatorial plane, the Kerr metric coefficient simplifies toPut $r=-\epsilon$ on the negative-radius sheet. Then $g_{\varphi \varphi }={\epsilon }^2+a^2-2M{a}^2/\epsilon <0$ for all sufficiently small $\epsilon >0$. The curveis a closed timelike curve, since its tangent is ${\partial }_{\varphi }$ and the angular coordinate is periodic. It never meets the ring at $r=0$. These are the closed timelike circles in negative-radius Kerr spacetime; they need not be geodesics. Along them $\tilde{t}$ is also constant, because its difference from $t$ depends only on $r$.

For $0<M<a$, the ring is a naked singularity and there is no event horizon shielding the extension's closed timelike curves. The weak cosmic censorship conjecture predicts that such superextreme Kerr geometries are not formed by generic collapse of regular isolated initial data satisfying the appropriate energy conditions. This is a conjectural physical restriction, not a proof from the exact metric alone.

For $M>a>0$, the closed timelike curves lie deep inside the ideal analytic extension, beyond the inner Cauchy horizon, so weak cosmic censorship does not itself exclude them. However, that inner horizon is unstable to perturbations, with mass inflation expected in realistic collapse. The strong cosmic censorship conjecture is the relevant obstruction to a regular continuation into the ideal chronology-violating region. Thus the exact Kerr extension exhibits such curves mathematically, but neither regular collapse nor the uniqueness theorem establishes that they occur in Nature.

The periodic axial orbits have norm $m^2=g_{\phi \phi }$. On the equator,Thus a neighborhood on the negative sheet has $m^2<0$. A fixed-$(t,r,\theta )$ curve running once around periodic $\phi$ is a closed timelike curve: its timelike tangent returns to the same event.

No horizon in the over-rotating solution prevents passage from positive-$r$ infinity through the nonsingular disk to this region. Timelike periodic axial orbits provide the time-machine mechanism. This property of the analytic solution does not establish that such a naked geometry can be manufactured stably from generic regular collapse, as envisaged by the weak cosmic censorship conjecture.

The relevant implications in the causal hierarchy areThe first follows from strict cone widening: failure of strong causality would give causal curves leaving and re-entering arbitrarily small neighborhoods, and the wider metric turns the limiting recurrence into a closed timelike curve. For the second, if distinct $p,q$ had $I^{+}(p)=I^{+}(q)$, arbitrarily short timelike segments near either point and this equality would produce a causal curve leaving and re-entering a sufficiently small causally convex neighborhood. ThereforeThe stated null geodesic completeness is stronger than needed: stable causality already gives the conclusion.

Let $t^a$ exhibit the stable causality of $g$, so $g-t_a{t}_b$ has no closed timelike curves. SetThe vector $r^a$ remains timelike for $g^{\prime }$. Widening the cones of $g^{\prime }$ once more using $r$ giveswhich is causal by assumption. Hence

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.

Choose an integer $n$ sufficiently negative that $c(t+n{P}_t)<-|x|$. That lift lies in the ordinary causal past of the origin, and its projected causal curve reaches the origin. HenceFuture-directed curves can connect the origin and an arbitrary event in either order. This property includes the origin's nontrivial closed timelike curves.

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.

A spacetime is stably causal when its light cones can be widened slightly without creating a closed timelike curve. Equivalently, it admits a continuous time function strictly increasing along every future-directed causal curve.

The unit hyperboloid $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in ambient signature $(++-)$ has $X^0=\cosh r\cos t$, $X^1=\cosh r\sin t$, $X^2=\sinh r$. It gives the displayed metric with $r\in \mathbb{R}$ and periodic $t$ modulo $2\pi$. Its time circles are timelike, so the universal cover, with $t\in \mathbb{R}$, removes these closed timelike curves. The metric alone does not prescribe the global identification. Both versions have constant negative curvature and two spatial ends.