Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 55 / 4 / 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 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.