Anti-de Sitter Spacetime

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Passing to the universal cover is essential because the standard anti-de Sitter hyperboloid has a periodic timelike coordinate and therefore closed timelike curves. Unwrapping that circle leaves the local metric and constant negative curvature unchanged while providing a nonperiodic global time. The resulting universal covering spacetime is what is normally meant by anti-de Sitter spacetime.

Required background. The Riemann Curvature Tensor supplies the constant-curvature checks; The Einstein Equation fixes the negative cosmological constant.

Helpful background. Killing Fields and Isometry Groups supplies the covering and quotient interpretation of the time-translation isometry.

Hyperboloid and its closed timelike circles

In $\mathbb R^{2,3}$ with

ds_5^2=-dX_{-1}^2-dX_0^2+dX_1^2+dX_2^2+dX_3^2,

the four-dimensional anti-de Sitter hyperboloid of radius $L>0$ is

-X_{-1}^2-X_0^2+X_1^2+X_2^2+X_3^2=-L^2.

Set

X_{-1}=L\cosh\rho\cos\tau, \quad X_0=L\cosh\rho\sin\tau, \quad (X_1,X_2,X_3)=L\sinh\rho\,\mathbf n_2.

The induced metric is

ds^2=L^2\left[-\cosh^2\rho d\tau^2+d\rho^2+\sinh^2\rho d\Omega_2^2\right].

The curvature sign follows without a long coordinate computation. The position vector is normal to the quadric, and its squared norm is $-L^2$ in the ambient metric. Consequently the second fundamental form is proportional to $g/L$ . Because the ambient space is flat and the normal is timelike, the Gauss equation gives

R_{abcd}=-\frac{1}{L^2}(g_{ac}g_{bd}-g_{ad}g_{bc}).

This derivation checks both the magnitude and the minus sign: changing the quadric to one with a spacelike normal produces the positive-curvature de Sitter sign. It also proves immediately that every point has the same local curvature data.

On the embedded hyperboloid, $\tau\sim\tau+2\pi$ . At $\rho=0$ , the curve with varying $\tau$ has $ds^2=-L^2d\tau^2$ and closes after $2\pi$ : it is timelike and closed. The universal cover replaces the circle by $\tau\in\mathbb R$ . Bičák states the topology $S^1\times\mathbb R^3$ of the hyperboloid and the standard unwrapping convention in Bičák 2000, §1.3, printed p. 8, Open PDF.

This covering operation is a global change. It creates no local curvature difference, so no scalar or Cartan invariant can diagnose whether time was unwrapped; the identification must be specified to define the spacetime.

The cover also clarifies terminology. Sometimes “AdS” denotes the embedded quadric and sometimes its cover. Any statement about causal curves must say which one is intended. The quadric’s closed timelike curves are forced by the time identification, not by negative curvature alone. On the universal cover those curves open, while further discrete quotients may introduce new global identifications.

Global radial form

Define $r=L\sinh\rho$ and $t=L\tau$ on the cover. Then

ds^2=-\left(1+\frac{r^2}{L^2}\right)dt^2 +\frac{dr^2}{1+r^2/L^2}+r^2d\Omega_2^2,

with $t\in\mathbb R$ and $r\ge0$ . The center $r=0$ is regular after the sphere collapses in the usual way. There is no zero of $1+r^2/L^2$ , hence no static-coordinate Killing horizon.

The curvature is

R_{\mu\nu\rho\sigma} =-\frac1{L^2}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}), \quad R_{\mu\nu}=-\frac3{L^2}g_{\mu\nu}, \quad R=-\frac{12}{L^2}.

Thus four-dimensional vacuum Einstein gravity requires $\Lambda=-3/L^2$ . The Weyl tensor vanishes, so AdS is Petrov O.

The rotation in the timelike two-plane and the three rotations of the spatial three-plane, together with the six generators mixing a timelike and a spacelike direction, restrict to ten independent Killing fields. This saturates the four-dimensional upper bound $4(4+1)/2=10$ . Their algebra is $\mathfrak{so}(2,3)$ on both the quadric and its universal cover; only the global group realization changes when the timelike circle is unwrapped. Constant negative curvature, maximal local symmetry, and the absence of a global time identification are therefore separate pieces of information.

The absence of a static root does not mean that AdS has no important boundary. Proper radial distance on a constant-time slice grows without bound as $r$ tends to infinity, while null rays reach conformal infinity in finite global coordinate time. These statements refer to different metrics: infinite physical distance is compatible with a finite location after conformal rescaling.

Conformal boundary and Poincaré patch

Introduce $\chi$ by $\tan\chi=\sinh\rho$ , with $0\le\chi<\pi/2$ . The global metric becomes

ds^2=\frac{L^2}{\cos^2\chi} \left[-d\tau^2+d\chi^2+\sin^2\chi d\Omega_2^2\right].

After removing the conformal factor, $\chi=\pi/2$ is a timelike cylinder with topology $\mathbb R\times S^2$ on the universal cover. Radial null curves satisfy $d\tau=\pm d\chi$ and reach this boundary in finite global coordinate time. This fact makes boundary conditions indispensable for hyperbolic evolution, but their general theory belongs to Causality and Global Structure.

Affine parameter gives an independent check of what “reach” means. For a radial null geodesic in the static form, the conserved energy is

E=\left(1+\frac{r^2}{L^2}\right)\frac{dt}{d\lambda}.

The null condition then gives $dr/d\lambda=\pm E$ . Thus $r\to\infty$ requires infinite affine parameter, even though

\Delta t=\int_{r_0}^{\infty}\frac{dr}{1+r^2/L^2}

is finite. The physical null geodesic is complete toward that end; the finite global-time endpoint belongs only to the conformal completion. This calculation reconciles the apparently conflicting distance and causal statements without treating the boundary as a curvature defect.

The Poincaré coordinates

ds^2=\frac{L^2}{z^2} \left(-dt_P^2+dx^2+dy^2+dz^2\right), \qquad z>0,

follow from the embedding map

\begin{aligned} X_{-1}&=\frac{L^2+z^2+x^2+y^2-t_P^2}{2z}, &X_3&=\frac{L^2-z^2-x^2-y^2+t_P^2}{2z},\\ X_0&=\frac{Lt_P}{z}, &X_1&=\frac{Lx}{z}, \qquad X_2=\frac{Ly}{z}. \end{aligned}

These expressions satisfy the AdS quadric and obey

X_{-1}+X_3=\frac{L^2}{z}>0.

They therefore cover only that wedge of the embedded AdS hyperboloid. On the universal cover, choose one connected lift of the wedge: the embedding coordinates alone do not distinguish lifts separated by $\tau\mapsto\tau+2\pi$ in dimensionless global time. The surface $z=0$ approaches the conformal boundary; $z\to\infty$ is a Poincaré horizon of this patch, not a curvature singularity. The invariant curvature remains constant at both limits.

There is a quick intrinsic check that the Poincaré metric has the same radius. Write $g=\Omega^2\eta$ with $\Omega=L/z$ . The conformal-curvature formula gives $R=-12/L^2$ , and the traceless Ricci and Weyl tensors vanish. Hence the patch is locally AdS rather than merely a conformally flat metric with a similar boundary. Nevertheless this calculation cannot establish global coverage: the embedding-coordinate inequalities defining the patch exclude part of the hyperboloid, just as Rindler coordinates exclude most of Minkowski space.

To check the timelike character of the global conformal boundary, examine the rescaled metric at fixed $\chi=\pi/2$ . Its induced form contains a negative time direction and positive sphere directions. Consequently boundary data can influence the bulk after finite coordinate time. This is a geometric domain fact; choosing reflecting, transparent, or other field boundary conditions is extra input not fixed by the local Einstein equation.

Checks and scope

Pulling back the ambient metric checks the global form. Contracting the constant-curvature tensor checks $R_{\mu\nu}=-3g_{\mu\nu}/L^2$ . Letting $L\to\infty$ at fixed local distances recovers Minkowski space. None of these local checks distinguishes the hyperboloid from its cover, which is why the domain and identification statement is scientifically essential.

This page does not develop holography, AdS quantum boundary conditions, or global causal theorems.

It is also important not to call conformal infinity a curvature singularity or an ordinary timelike wall. Along a constant- $\tau$ radial curve, the physical proper distance to $\chi=\pi/2$ is

L\int^{\pi/2}\frac{d\chi}{\cos\chi}=\infty,

whereas a radial null curve crosses the finite coordinate interval according to $|\Delta\tau|=|\Delta\chi|$ . The conformally completed cylinder records causal endpoints, but it is not part of the physical manifold. Whether fields reflect there is a boundary-value choice, not a consequence of the constant-curvature tensor alone.

As an independent local limit, choose a neighborhood of $r=0$ with $r$ and $t$ fixed and send $L$ to infinity. Both the metric coefficients and curvature approach their Minkowski values. By contrast, taking $z$ to zero at fixed Poincaré coordinates approaches the conformal boundary while physical curvature remains negative and constant. These are different limits and should not be conflated.

Check your understanding

Do the embedded AdS hyperboloid and its universal cover have different Einstein tensors or local Killing algebras?

Solution

No. A covering projection is a local diffeomorphism and local isometry, so the Riemann, Ricci, and Einstein tensors pull back unchanged. Both have the same ten-dimensional local Killing algebra and solve the same vacuum equation with $\Lambda=-3/L^2$ . They differ globally: the hyperboloid identifies $\tau$ modulo $2\pi$ and contains closed timelike circles, whereas the universal cover has $\tau\in\mathbb R$ . Local field equations cannot select between them.

References

Bičák, Jiří. “Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics.” Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540 (2000), 1–126. arXiv:gr-qc/0004016, §1.3, printed pp. 8–10.