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Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

A complete inertial timelike geodesic in Minkowski spacetime has $\mathbf{x}=\mathbf{b}+\mathbf{v}t$ with $|\mathbf{v}|<1$. Under $p=\arctan (t-r)$, $q=\arctan (t+r)$, both $p,q$ tend to the same endpoint, giving past timelike infinity and future timelike infinity. A complete null geodesic has one finite limiting null coordinate at each end and reaches past null infinity and future null infinity. Complete spacelike geodesics have $r>|t|$ asymptotically and both ends approach spacelike infinity. The endpoints have finite compactified coordinates but occur at infinite physical affine parameter. A conformal transformation preserves null paths as unparametrized geodesics; it does not generally preserve timelike or spacelike geodesic paths.

A geodesic on the embedded two-dimensional de Sitter spacetime is the intersection with a two-plane through the ambient origin. For initial point $P$ and tangent $V$ satisfying $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, its affine parametrization is $P\cosh s+V\sinh s$ when timelike, $P+sV$ when null and $P\cos s+V\sin s$ when spacelike. These solve $X^{\prime \prime }=\epsilon X$. The spacelike geodesics are closed; timelike geodesics and null geodesics extend to both conformal infinities with infinite affine length.

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.

In ordinary spherical coordinates, $t\in \mathbb{R}$, $r\in [0,\infty )$, $\theta \in [0,\pi ]$ and $\varphi \in [0,2\pi )$ with periodic identification. The regular coordinate chart excludes the origin and polar axes; the endpoint ranges include the usual spherical identifications. With null coordinates $u=t-r$ and $v=t+r$, both are real and their joint domain is $v\ge u$. The inverse relations giveThe arctangent coordinates obey $-\pi /2<p\le q<\pi /2$. Since $du={\sec }^{2}pdp$ and $dv={\sec }^{2}qdq$, the rescaled metric isNow $T=p+q$, $R=q-p$, so $4dpdq=d{T}^2-d{R}^2$. Thus the Minkowski conformal compactification hasThe full coordinate ranges are linked, not independent: $T\in (-\pi ,\pi )$ and $R\in [0,\pi )$ with $|T|+R<\pi$. Inverse coordinates are $p=(T-R)/2$, $q=(T+R)/2$, $t=[\tan q+\tan p]/2$, $r=[\tan q-\tan p]/2$. The conformal factor isThe unphysical metric extends as the unit-radius Einstein static universe on $\mathbb{R}\times S^3$, where $0\le R\le \pi$ with the endpoint two-spheres collapsed. Physical Minkowski spacetime occupies the triangular radial domain $R\ge 0$, $|T|+R<\pi$; the edge $R=0$ is the ordinary spatial center, not an infinity or physical boundary.

In the closure, the infinite-coordinate limits have locationsThese are future timelike infinity, past timelike infinity, spacelike infinity, future null infinity and past null infinity. For example ${\mathcal{I}}^{+}$ has $v\to +\infty$ with finite $u$, while ${\mathcal{I}}^{-}$ has $u\to -\infty$ with finite $v$. A diagram point with $0<R<\pi$ stands for a two-sphere, and each null boundary is topologically $\mathbb{R}\times S^2$; their three corners are points after the angular collapse. On the null boundaries $\Omega =0$ and $d\Omega \ne 0$, whereas the three corners have a degenerate zero gradient. Thus a compactifying metric can be smooth there even though those infinity points are not ordinary smooth null-boundary points.

The geodesic endpoints in Minkowski conformal compactification can be established directly. A physical inertial timelike geodesic has $t=t_0+E\tau$, $\mathbf{x}={\mathbf{x}}_0+\mathbf{p}\tau$, with $E>0$ and $E^2-|\mathbf{p}{|}^2=1$. At future infinity $u,v\to +\infty$, so $p,q\to \pi /2$ and the curve ends at $i^{+}$. At past infinity both tend to $-\infty$, giving $i^{-}$. Every complete inertial timelike path has these endpoints, regardless of velocity.

For a future-directed null geodesic, $t=t_0+E\lambda$, $\mathbf{x}={\mathbf{x}}_0+E\mathbf{n}\lambda$ with $|\mathbf{n}|=1$. As $\lambda \to +\infty$, $v\to +\infty$ but $u$ has a finite limit; the endpoint lies on ${\mathcal{I}}^{+}$. As $\lambda \to -\infty$, $u\to -\infty$ but $v$ has a finite limit, giving an endpoint on ${\mathcal{I}}^{-}$. A conformal transformation preserves these null paths as unparametrized geodesics. The null affine parameter under a conformal rescaling obeys $d\tilde{\lambda }/d\lambda ={\Omega }^2$; here ${\Omega }^2=O({\lambda }^{-2})$ at the null endpoints, so the unphysical affine parameter reaches them in finite time, although the physical affine parameter is infinite.

A physical spacelike geodesic has $t=t_0+a\lambda$, $\mathbf{x}={\mathbf{x}}_0+\mathbf{b}\lambda$, with $|\mathbf{b}|>|a|$. At either end $r>|t|$ asymptotically, $u\to -\infty$ and $v\to +\infty$, so both ends approach $i^0$. Timelike and spacelike physical geodesic paths are not generally geodesics of the rescaled metric. Their physical proper time or length is infinite, while $\Omega =O(|\lambda {|}^{-2})$ gives finite unphysical proper time or length near the endpoints. Infinity at finite diagram coordinates does not imply incompleteness of physical Minkowski geodesics.

The original Penrose diagram suppresses the angular spheres. The displayed physical paths illustrate their distinct endpoint types; the two sloping boundaries are null infinity, while the left edge is the regular center.

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.

The physical spherical coordinates cover $t\in \mathbb{R}$, $r\in [0,\infty )$, $\theta \in [0,\pi ]$, and $\varphi \in [0,2\pi )$ with periodic identification. The center and the polar axes are the usual coordinate singularities, rather than physical curvature singularities; a regular polar chart restricts $r>0$ and $0<\theta <\pi$. For the null coordinates,Thus the physical metric tensor isThe conformal transformation followed by $p=\arctan u$, $q=\arctan v$ gives $du=(1+u^2)dp$, $dv=(1+v^2)dq$, andConsequently $d{s}^2=4dpdq-{\sin }^2(q-p)d{\Sigma }^2$. Finally $T=p+q$, $R=q-p$ converts $4dpdq$ to $d{T}^2-d{R}^2$, yieldingThe ranges must retain their joint constraints:Equivalently, $-\pi <T-R<\pi$ and $-\pi <T+R<\pi$, with $R\ge 0$. The angles retain their original ranges and identifications. Extending $R$ to negative values would double-count the radial polar geometry. The conformal factor isThe unphysical metric tensor extends onto the Einstein static universe, while physical Minkowski spacetime occupies the triangular region in its radial Penrose diagram. The edge $R=0$ is the regular center and belongs to the physical region away from its endpoints.

The conformal boundary consists of the following infinities. Future null infinity has $q=\pi /2$, hence $T+R=\pi$ with $0<R<\pi$, and corresponds to finite $u$ with $v\to +\infty$. Past null infinity has $p=-\pi /2$, hence $T-R=-\pi$, and corresponds to finite $v$ with $u\to -\infty$. Future timelike infinity is $(T,R)=(\pi ,0)$, past timelike infinity is $(-\pi ,0)$, and spacelike infinity is $(0,\pi )$. Although angles label points on null infinity, their two-spheres collapse at these three endpoint locations in the compactified picture.

For the geodesic endpoints in Minkowski conformal compactification, one can check the limits directly. A timelike inertial trajectory is $\mathbf{x}=\mathbf{b}+\mathbf{w}t$ with $|\mathbf{w}|<1$. At $t\to +\infty$ both $t-r$ and $t+r$ tend to $+\infty$, so $p,q\to \pi /2$ and the trajectory ends at future timelike infinity. At $t\to -\infty$ both tend to $-\infty$, giving past timelike infinity. This includes a stationary inertial worldline with bounded $r$.

For a null inertial trajectory, $|\mathbf{w}|=1$. As $t\to +\infty$, $r=t+\mathbf{w}\cdot \mathbf{b}+o(1)$, so $u$ has a finite limit while $v\to +\infty$; its future endpoint lies on future null infinity. As $t\to -\infty$, $r=-t-\mathbf{w}\cdot \mathbf{b}+o(1)$, giving a finite limiting $v$ and $u\to -\infty$, hence a past endpoint on past null infinity. Radial null geodesic paths have $T\pm R$ constant. A ray passing through the center changes its angular direction; in the angle-suppressed radial diagram it reflects at $R=0$ rather than terminating there.

A spacelike geodesic is also a straight line, now parametrized so that $|d\mathbf{x}/d\lambda |>|dt/d\lambda |$. At either end, $r>|t|$ asymptotically, $u\to -\infty$, $v\to +\infty$, and the endpoint is spacelike infinity. A radial spacelike line at $t=0$ has two opposite-angle halves which project onto the same horizontal interval in the diagram.

The conformal compactification endpoints occur at finite $T$ but at infinite physical proper time or affine parameter; Minkowski spacetime is geodesically complete. The conformal preservation of null geodesic paths does not preserve an affine parameter: if $\lambda$ is physically affine, an unphysical affine parameter satisfies $d\tilde{\lambda }/d\lambda ={\Omega }^2$ up to a constant. Along a complete null ray this can have a finite integral at infinity. Timelike and spacelike physical geodesic paths are not generally geodesics of the rescaled metric tensor, so their drawn curves must not be interpreted as freely falling unphysical observers.

Finally, every physical event can send an outgoing null ray to future null infinity, so $J^{-}({\mathcal{I}}^{+})$ is all of Minkowski spacetime. Its black-hole region and intrinsic event horizon are empty. The null edges in the diagram are infinity, not horizons concealing part of the physical manifold. Complete inertial observers likewise have no observer event horizon. Accelerated observers restricted to a Rindler wedge can have a Rindler horizon, but that is observer-dependent and does not contradict the horizon-free Minkowski spacetime. The conformal diagram compactifies infinity; it does not introduce a physical horizon.

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.

The original spherical coordinates have $t\in \mathbb{R}$, $0\le r<\infty$, $0\le \theta \le \pi$ and $0\le \varphi <2\pi$, with $\varphi$ periodic. The angular coordinates degenerate at the polar axis and all angles identify at $r=0$; a regular spherical chart uses $r>0$, $0<\theta <\pi$ and suitable overlapping angular patches.

The retarded time $u=t-r$ and advanced coordinate $v=t+r$ both range over $\mathbb{R}$, with the joint restriction $u\le v$. Their inverse relations are $t=(u+v)/2$ and $r=(v-u)/2$, soSet $p=\arctan u$, $q=\arctan v$. ThenThe conformal factor is $\Omega =2\cos p\cos q>0$. Since $(\tan q-\tan p)\cos p\cos q=\sin (q-p)$, multiplication by ${\Omega }^2$ givesFinally $T=p+q$ and $R=q-p$ give $4dpdq=d{T}^2-d{R}^2$. HenceThis is a region of the Einstein static universe, with positive time and negative spatial signature. The exact joint coordinate domain isor equivalently $R\ge 0$ and $|T|+R<\pi$. The regular radial chart has $R>0$; $R=0$ is the ordinary centre. The angular ranges are unchanged. It is important not to treat the stated individual ranges of $T,R$ as independent. The inverse map and conformal factor provide a useful check:

The resulting Minkowski conformal compactification has a triangular radial Penrose diagram. The two sloping boundary edges areThese are future null infinity and past null infinity. The three vertices are future timelike infinity $i^{+}=(\pi ,0)$, past timelike infinity $i^{-}=(-\pi ,0)$ and spacelike infinity $i^0=(0,\pi )$. They are ideal boundary points, not events of physical Minkowski spacetime.

For a complete inertial timelike geodesic, $\mathbf{x}=\mathbf{b}+\mathbf{v}t$ with $|\mathbf{v}|<1$. As $t\to +\infty$, both $t-r$ and $t+r$ tend to $+\infty$, so $p,q\to \pi /2$ and the curve ends at $i^{+}$. As $t\to -\infty$, both tend to $-\infty$, giving $i^{-}$. These endpoints occur at infinite physical proper time.

For a complete null geodesic, write $\mathbf{x}=\mathbf{b}+\mathbf{n}t$, $|\mathbf{n}|=1$. To the future, $r=t+\mathbf{n}\cdot \mathbf{b}+O(1/t)$, so $u\to -\mathbf{n}\cdot \mathbf{b}$ is finite and $v\to +\infty$. Thus $q\to \pi /2$ with $p$ finite: the endpoint lies on ${\mathcal{I}}^{+}$. To the past, $v\to -\mathbf{n}\cdot \mathbf{b}$ remains finite while $u\to -\infty$, giving ${\mathcal{I}}^{-}$. Physical affine parameter is infinite at both ends. Along a radial null path, $dT=\pm dR$, so the diagram represents its null directions at $45$ degrees.

For a complete spacelike geodesic, take $t=t_0+a\lambda$, $\mathbf{x}=\mathbf{b}+\mathbf{w}\lambda$ with $|\mathbf{w}|>|a|$. At either end $r>|t|$ asymptotically, so $u\to -\infty$, $v\to +\infty$ and $(T,R)\to (0,\pi )$. Both ends therefore reach $i^0$, again at infinite physical affine parameter. These are the geodesic endpoints in Minkowski conformal compactification.

Only conformal preservation of null geodesic paths is guaranteed. The pictured timelike and spacelike curves are images of physical geodesics, not generally geodesics of the unphysical metric. For a null path, an unphysical affine parameter satisfies $d\tilde{\lambda }/d\lambda \propto {\Omega }^2$. Since $\Omega =O(1/|\lambda |)$ near a generic null-infinity endpoint, the unphysical parameter can converge while the physical parameter diverges. At $i^{\pm }$ and $i^0$, the usual smooth-boundary condition $d\Omega \ne 0$ degenerates; these vertices should not be treated as regular points of null infinity.

Complete Minkowski spacetime has no black-hole event horizon. Every physical event can send an outgoing null signal to ${\mathcal{I}}^{+}$, so the black-hole region $\mathcal{M}\setminus J^{-}({\mathcal{I}}^{+})$ is empty. There is likewise no particle or future event horizon for an eternal inertial observer: any finite event can eventually signal that observer. Null infinity is a conformal boundary, not a physical horizon. Uniformly accelerated observers can have Rindler horizons, but that is an observer-dependent restriction and is not a horizon of the full inertial Minkowski spacetime.

For the full unit-radius two-dimensional de Sitter spacetime, the coordinate ranges areOne can use $0\le \chi <2\pi$ with endpoints identified. If the identification is omitted, the same line element describes the universal covering spacetime instead. The periodicity follows from the hyperboloid embeddingThe induced ambient Minkowski metric is precisely the given Lorentzian metric.

Using an affine parameter $s$, the geodesic Lagrangian is $\frac{1}{2}({\dot{t}}^2-{\cosh }^{2}t{\dot{\chi }}^2)$. The nonzero Christoffel symbols areThus the geodesic equations and their first integrals areHere $\epsilon =1$ for unit-speed timelike geodesics, $0$ for null geodesics and $-1$ for unit-speed spacelike geodesics. The conserved $J$ is the geodesic conserved quantity from a Killing vector for the angular symmetry.

All geodesics of two-dimensional de Sitter spacetime can be written especially simply in the embedding. If $P$ is the initial point and $V$ the initial tangent, with $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, the geodesic has normal ambient acceleration $X^{\prime \prime }=\epsilon X$. ThereforeThese curves stay in the two-plane spanned by $P$ and $V$, preserve both constraints and have no tangential acceleration. They consequently solve the geodesic equation and exhaust all initial directions.

An isometry can move the chosen point to $P=(0,1,0)$, namely $(t,\chi )=(0,0)$. For future-directed timelike geodesics take $V=(\cosh \alpha ,0,\sinh \alpha )$, obtainingThe continuous angular branch has $|\chi |<\pi /2$, and every real $\alpha$ gives a distinct timelike direction. The two null geodesics have $V=(1,0,\pm 1)$ andFor spacelike geodesics take $V=(\sinh \alpha ,0,\cosh \alpha )$, givingThey are closed with period $2\pi$ in affine arclength and $|t|\le |\alpha |$. In particular, $\alpha =0$ is the equatorial geodesic $t=0$. These three families, with reversals of parametrization allowed, are the requested complete set through the chosen point.

The conformal time substitution givesHenceMultiplying by ${\cos }^{2}T$ gives the regular conformal cylinder of two-dimensional de Sitter spacetime. Adding $T=\pm \pi /2$ completes the conformal compactification; both boundary circles are spacelike. Null geodesics are straight lines $\chi =\pm T$ through the origin. The other families have the useful diagram equationsThe image sketches representative directions from each family; the formulas allow all intermediate directions.

For the complete observer $\chi =0$, let $d(\chi ,0)\in [0,\pi ]$ be the shortest angular distance on the circle. Since light travels one unit of angular distance per unit conformal time, an event at $(T,\chi )$ can signal to the observer at some finite future time precisely whenEquality defines its future observer event horizon. Likewise an event can receive a signal emitted by the observer precisely whenand equality defines the past horizon. In the lift $-\pi \le \chi \le \pi$, their branches are $\chi =\pm (\pi /2-T)$ and $\chi =\pm (T+\pi /2)$. The common communication region is the diamond $|\chi |+|T|<\pi /2$, the observer's static patch of de Sitter spacetime. These are the observer horizons in two-dimensional de Sitter spacetime. Every complete timelike observer has analogous horizons after a de Sitter isometry; they depend on the observer, rather than enclosing an intrinsic black-hole region.

Despite these horizons, each constant-$T$ circle is a Cauchy hypersurface of the full spacetime: any inextendible causal curve crosses it exactly once. The global spacetime is therefore globally hyperbolic and has no Cauchy horizons. The conformal boundaries lie at infinite proper time for complete timelike geodesics and infinite affine distance for complete null geodesics, so their finite height in the diagram is not a physical incompleteness.

The radial part of the Minkowski metric becomes $-dudv$, and $r=(v-u)/2$. With $u=\tan p$, $v=\tan q$ and the positive conformal factor $\Omega =2\cos p\cos q$, multiplication by ${\Omega }^2$ givesConsequently the Minkowski conformal compactification isThe spatial metric $d{R}^2+{\sin }^{2}Rd{\Sigma }^2$ is the round unit three-sphere. The extended unphysical metric is therefore that of the Einstein static universe, $\mathbb{R}\times S^3$ with unit spatial radius. Physical Minkowski spacetime occupies only the diamond-shaped causal patch of this cylinder, or a triangle after restricting to nonnegative spherical radius; it is not the whole cylinder.

The complete coordinate ranges and joint restrictions areThe polar angles have the usual coordinate degeneracies at their poles and at $r=0$. In the extended Einstein static universe one may let $T\in \mathbb{R}$ and $0\le R\le \pi$, but these are not the ranges of the original physical patch. In its interior $\Omega >0$. The line $R=0$ is the regular centre $r=0$, not a component of infinity.

The conformal completion has future timelike infinity $i^{+}=(\pi ,0)$, past timelike infinity $i^{-}=(-\pi ,0)$, and spacelike infinity $i^0=(0,\pi )$. The open sloping boundaries arenamely future null infinity and past null infinity. Each ordinary interior point of the radial diagram represents a two-sphere, and each point on either open null boundary represents a two-sphere of null directions.

A timelike geodesic is an inertial worldline $\mathbf{x}=\mathbf{b}+\mathbf{\beta }t$ with $|\mathbf{\beta }|<1$. As $t\to +\infty$, both $t-r$ and $t+r$ tend to $+\infty$, giving $p,q\to \pi /2$ and endpoint $i^{+}$. At past infinity both tend to $-\infty$, giving $i^{-}$. These endpoints are reached at infinite physical proper time, although at finite conformal coordinate time $T$. A null geodesic has $r\sim |t|$. In its outgoing future, $u$ approaches a finite retarded time while $v\to +\infty$, giving an endpoint on ${\mathcal{I}}^{+}$; in its incoming past, $v$ has a finite limit and $u\to -\infty$, giving an endpoint on ${\mathcal{I}}^{-}$. A spacelike geodesic is a straight line with spatial tangent larger than its temporal tangent, so $u\to -\infty$, $v\to +\infty$ at either end and both endpoints are $i^0$ in the compactified radial description. All these physical geodesics are complete. Only the unparametrized null geodesics are necessarily preserved as geodesics of the unphysical metric; timelike and spacelike images need not be its geodesics.