de Sitter and Anti-de Sitter Charts

de Sitter and anti-de Sitter spacetime have equal-magnitude constant curvature with opposite signs. Their local formulas look parallel, but their global causal structures do not: a de Sitter static chart ends at observer horizons, whereas the universal cover of anti-de Sitter has a timelike conformal boundary and no global static-coordinate horizon.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification; Constant Curvature and Maximal Symmetry derives the invariant geometry; Exact Solutions of Einstein’s Equations supplies the chart-domain and invariant-check method.

Curvature and embedding conventions

In spacetime dimension $d\ge3$, write

$$K=\begin{cases}+\ell^{-2},&\text{de Sitter},\\-\ell^{-2},&\text{anti-de Sitter},\end{cases} \qquad \ell\gt0.$$

Both families obey

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc}), \quad R_{ab}=(d-1)K g_{ab}, \quad R=d(d-1)K,$$

so the vacuum Einstein equation fixes

$$\Lambda=\frac{(d-1)(d-2)}{2}K.$$

de Sitter is the one-sheeted quadric $-X_0^2+\sum_{i=1}^{d}X_i^2=\ell^2$ in $\mathbb R^{1,d}$. Anti-de Sitter is the quadric $-X_{-1}^2-X_0^2+\sum_{i=1}^{d-1}X_i^2=-\ell^2$ in $\mathbb R^{2,d-1}$. Pulling back the ambient metric fixes both the chart metric and the sign of $K$. Reall 2013, §§13.1–13.2, printed pp. 149–157, PDF develops the constant-curvature and de Sitter construction; Bičák 2000, §1.3, printed pp. 8–10, PDF compares the constant-curvature backgrounds and their identifications.

de Sitter chart card

Let $d\Omega_n^2$ denote the unit $n$-sphere metric.

Chart	Metric and range	Coverage and boundary
Global	$ds^2=-dt^2+\ell^2\cosh^2(t/\ell)d\Omega_{d-1}^2$, $t\in\mathbb R$	Covers the complete hyperboloid, apart from ordinary angular overlaps. Constant-$t$ slices are $S^{d-1}$.
Static	$ds^2=-(1-r^2/\ell^2)dt_s^2+dr^2/(1-r^2/\ell^2)+r^2d\Omega_{d-2}^2$, $t_s\in\mathbb R$, $0\le r\lt\ell$	Covers one observer’s causal diamond, apart from ordinary angular overlaps. Its coordinate boundary contains future and past cosmological Killing horizons and their bifurcation sphere; it is not a curvature boundary.
Planar	$ds^2=-dt_p^2+e^{2t_p/\ell}d\mathbf x^2$, $t_p\in\mathbb R$, $\mathbf x\in\mathbb R^{d-1}$	Covers one half of the hyperboloid. Intrinsically flat spatial slices do not make the spacetime flat.

Anninos 2012, revised 2025, §1.1, printed pp. 4–6 gives the embedding and the global, planar, static, and hyperbolic patch package in four dimensions.

The embedding maps make those coverage claims testable. With unit sphere vectors $\mathbf n$, the global map is

$$X_0=\ell\sinh(t/\ell), \qquad (X_1,\ldots,X_d)=\ell\cosh(t/\ell)\mathbf n, \qquad \mathbf n\in S^{d-1}.$$

One static diamond uses

$$\begin{aligned} X_0&=\sqrt{\ell^2-r^2}\sinh(t_s/\ell),\\ X_d&=\sqrt{\ell^2-r^2}\cosh(t_s/\ell),\\ (X_1,\ldots,X_{d-1})&=r\mathbf n, \qquad \mathbf n\in S^{d-2}. \end{aligned}$$

It is real and one-to-one only where $X_d\gt\lvert X_0\rvert$. At fixed finite $t_s$, the limit $r\to\ell$ reaches only the bifurcation $S^{d-2}$. Generic future or past horizon points require a correlated limit $r\to\ell$, $t_s\to\pm\infty$ with $\sqrt{\ell^2-r^2}\,e^{\lvert t_s\rvert/\ell}$ finite. The norm of $\partial_{t_s}$ then vanishes while $R$ and $R_{abcd}R^{abcd}=2d(d-1)/\ell^4$ stay constant. The failure is in the map, not the spacetime; other static diamonds require other sign choices.

For the planar chart, put $u=t_p/\ell$ and $\mathbf x^2=\sum_{i=1}^{d-1}(x^i)^2$. Then

$$\begin{aligned} X_0&=\ell\sinh u+\frac{e^u\mathbf x^2}{2\ell},\\ X_i&=e^u x^i,\qquad i=1,\ldots,d-1,\\ X_d&=\ell\cosh u-\frac{e^u\mathbf x^2}{2\ell}. \end{aligned}$$

Thus $X_0+X_d=\ell e^u\gt0$: even though $t_p$ spans all real values, this chart covers only one half of the hyperboloid. On its overlap with any other chart,

$$e^u=\frac{X_0+X_d}{\ell}, \qquad x^i=\frac{\ell X_i}{X_0+X_d},$$

and the transformation ceases to exist at the patch boundary $X_0+X_d=0$. The exact de Sitter atlas derives the overlaps and causal interpretation.

Anti-de Sitter chart card

The embedded anti-de Sitter quadric admits the angular coordinate $\tau\sim\tau+2\pi$ in its timelike two-plane. Those identified orbits are closed timelike curves. The standard spacetime used in gravity is therefore the universal cover, on which $\tau\in\mathbb R$. Unwrapping time changes global topology but no local curvature tensor.

Chart	Metric and range	Coverage and boundary
Global cover	$ds^2=\ell^2[-\cosh^2\rho\,d\tau^2+d\rho^2+\sinh^2\rho\,d\Omega_{d-2}^2]$, $\rho\ge0$, $\tau\in\mathbb R$	Covers the universal cover. The center $\rho=0$ is regular.
Conformal global	$ds^2=\ell^2\cos^{-2}\chi[-d\tau^2+d\chi^2+\sin^2\chi\,d\Omega_{d-2}^2]$, $0\le\chi\lt\pi/2$	$\chi=\pi/2$ is timelike conformal infinity after rescaling; it is not part of the physical manifold.
Poincaré	$ds^2=\ell^2z^{-2}(dz^2+\eta_{\alpha\beta}dx^\alpha dx^\beta)$, $z\gt0$, $x^\alpha\in\mathbb R^{1,d-2}$, $\alpha=0,\ldots,d-2$, $\eta_{\alpha\beta}=\operatorname{diag}(-,+,\ldots,+)$	Covers one wedge. $z=0$ approaches its conformal boundary; $z\to\infty$ is a patch horizon, not a curvature singularity.

Aharony et al. 2000, §2.2.1, printed pp. 40–43 derives the global cover, conformal boundary, and Poincaré patch. Their inverse radial coordinate $u$ is related to this card’s convention by $z=1/u$.

On the AdS quadric, the global embedding map is

$$\begin{aligned} X_{-1}&=\ell\cosh\rho\cos\tau,\\ X_0&=\ell\cosh\rho\sin\tau,\\ (X_1,\ldots,X_{d-1})&=\ell\sinh\rho\,\mathbf n, \qquad \mathbf n\in S^{d-2}. \end{aligned}$$

It is periodic in $\tau$ on the embedded quadric; the universal cover lifts $\tau$ to the real line and is not one-to-one in this ambient map. The conformal global coordinate is the overlap transformation

$$\tan\chi=\sinh\rho, \qquad 0\leq\chi\lt\frac{\pi}{2}.$$

For the Poincaré map, write $x^2=\eta_{\alpha\beta}x^\alpha x^\beta$. Then

$$\begin{aligned} X_{-1}&=\frac{\ell^2+z^2+x^2}{2z}, &X_0&=\frac{\ell x^0}{z},\\ X_i&=\frac{\ell x^i}{z},\quad i=1,\ldots,d-2, &X_{d-1}&=\frac{\ell^2-z^2-x^2}{2z}. \end{aligned}$$

It obeys $X_{-1}+X_{d-1}=\ell^2/z\gt0$, which is precisely the Poincaré wedge. Equating it with the global map gives, on that overlap,

$$\begin{aligned} \tau&=\operatorname{atan2}\!\left(2\ell x^0, \ell^2+z^2+x^2\right),\\ \sinh\rho\,n_i&=\frac{x^i}{z},\qquad i=1,\ldots,d-2,\\ \sinh\rho\,n_{d-1}&=\frac{\ell^2-z^2-x^2}{2\ell z}. \end{aligned}$$

The branch of $\tau$ is lifted continuously on the universal cover; this is an overlap map, not a global identification of the two coordinate domains.

The global radial coordinate $r=\ell\sinh\rho$ gives

$$ds^2=-\left(1+\frac{r^2}{\ell^2}\right)dt^2 +\frac{dr^2}{1+r^2/\ell^2}+r^2d\Omega_{d-2}^2, \qquad t=\ell\tau.$$

Because $1+r^2/\ell^2$ has no real zero, this global static generator has no Killing horizon. Radial null curves nevertheless reach conformal infinity in finite $t$ while requiring infinite affine parameter. A boundary condition for fields is therefore additional problem data, not something fixed by the local Einstein equation. The exact anti-de Sitter atlas develops the covering-space and boundary analysis. General causal completion and boundary construction belong to Conformal Completion, Causal Boundaries, and Penrose Diagrams.

Independent checks and limits

• Substitute each embedding map into its quadric and pull back the ambient metric. This checks coefficients, signs, and the actual patch inequality.
• Contract the constant-curvature tensor to recover $R_{ab}$, $R$, and the stated cosmological constant. No chart boundary should change these invariants.
• At fixed local distances and times, $\ell\to\infty$ sends $K$, $\Lambda$, and the curvature invariants to zero and recovers a Minkowski neighborhood.
• A covering map is a local isometry: the AdS quadric and its universal cover have identical local curvature, but only the quadric identifies timelike orbits periodically.

Common pitfalls

A horizon is not a curvature singularity. The de Sitter static horizon and the AdS Poincaré horizon mark failures of particular charts. Their curvature is the same finite constant as everywhere else in the patch.

A conformal boundary is not an ordinary wall. It is added only after a singular conformal rescaling. Whether a field reflects there is a boundary condition, not a property encoded by $R_{abcd}$ alone.

“AdS” must name its covering choice. The embedded quadric and universal cover solve the same local field equation but have different causal curves. Holographic interpretation is developed separately and lies outside this chart card.

References

• Aharony, Ofer, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz. “Large N Field Theories, String Theory and Gravity.” Physics Reports 323 (2000): 183–386. DOI. Open manuscript.
• Anninos, Dionysios. “De Sitter Musings.” International Journal of Modern Physics A 27 (2012): 1230013; revised 2025. DOI. Open manuscript.
• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 1–126. Springer, 2000. arXiv:gr-qc/0004016.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.