= Solution
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 by
$$
X^0=\sinh t,\qquad X^1=\cosh t\cos\chi,\qquad X^2=\cosh t\sin\chi.
$$
Thus \b[$t\in\mathbb R$ and $\chi$ is periodic modulo $2\pi$], for example $0\leq\chi<2\pi$. 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 $(++-)$, with
$$
X^0=\cosh r\cos t,\qquad X^1=\cosh r\sin t,\qquad X^2=\sinh r.
$$
On this quadric \b[$r\in\mathbb R$ 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 \b[$r,t\in\mathbb R$], with time no longer identified. The signed radial coordinate has two spatial ends; restricting to $r\geq0$ 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>, introduce
$$
\eta=\arctan(\sinh t),\qquad d\eta=\operatorname{sech}t\,dt,\qquad
\cosh t=\sec\eta.
$$
Then
$$
\boxed{ds^2=\sec^2\eta\,(d\eta^2-d\chi^2),\qquad-\pi/2<\eta<\pi/2,\quad\chi\sim\chi+2\pi.}
$$
The <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>, use
$$
\psi=\arctan(\sinh r),\qquad dr=\sec\psi\,d\psi,\qquad\cosh r=\sec\psi,
$$
so on the <universal cover>
$$
\boxed{ds^2=\sec^2\psi\,(dt^2-d\psi^2),\qquad-\pi/2<\psi<\pi/2,\quad t\in\mathbb R.}
$$
Its 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>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-55-conformal-structure.png]
{title=Conformal cylinder and observer horizons of two-dimensional <de Sitter> <spacetime>, compared with the timelike boundaries and <geodesics> of the <AdS> <universal cover>}
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 $p=\cosh^2t\,\dot\chi$, and normalization gives
$$
\dot t^2-\frac{p^2}{\cosh^2t}=\kappa.
$$
Writing $y=\sinh t$ reduces this to $\dot y^2=p^2+\kappa(1+y^2)$. For a future-directed <timelike geodesic>,
$$
y=\sqrt{1+p^2}\sinh(\tau-\tau_0),\qquad
\chi(t)=\chi_0+\arcsin\left(\frac{p\tanh t}{\sqrt{1+p^2}}\right).
$$
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>, $p\ne0$ 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>, $|p|\geq1$ 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 $2\pi\operatorname{sgn}p$. 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 $E=\cosh^2r\,\dot t$. Their norm equation gives
$$
\dot r^2=\frac{E^2}{\cosh^2r}-\kappa,\qquad
\left(\frac{d}{ds}\sinh r\right)^2=E^2-\kappa(1+\sinh^2r).
$$
A future <timelike geodesic> has $E\geq1$ and
$$
\boxed{\sinh r=\sqrt{E^2-1}\sin(\tau-\tau_0).}
$$
Its turning points are $r=\pm\operatorname{arcosh}E$. <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 when
$$
d(\chi,0)<\pi/2-\eta.
$$
It 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 $\cosh^2r>0$ everywhere. Thus \b[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 $|d\chi/dt|\leq\operatorname{sech}t$; 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>. \b[<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 https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html[David Tong's general relativity lectures].
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