Why AdS Is Not Globally Hyperbolic

The universal cover of anti-de Sitter spacetime is causal and even admits a smooth temporal coordinate, but it is not globally hyperbolic as a spacetime without boundary. The failure is not a closed causal curve. It is the timelike conformal boundary: causal diamonds can run arbitrarily close to infinity without remaining compact, and causal signals can enter the bulk from boundary times not intercepted by bulk initial data.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal diamonds; Geodesics and Jacobi Fields supplies null curves; Causality and Global Structure supplies global hyperbolicity; Exact Solutions of Einstein’s Equations supplies global AdS; Formulations of General Relativity supplies initial-boundary evolution; Gravitational Actions and Boundary Terms supplies variational boundary data.

The compact-diamond test

Assume the universal covering spacetime in $(d+1)$ dimensions with metric

$$d\widetilde s^2 =\frac{\ell^2}{\cos^2\rho} \left(-dt^2+d\rho^2+\sin^2\rho\,d\Omega_{d-1}^2\right), \qquad 0\leq\rho\lt\frac\pi2.$$

The coordinate $t$ is a temporal function because its gradient is everywhere timelike. Thus the spacetime is stably causal. To test global hyperbolicity, use the equivalent criterion: a causal spacetime is globally hyperbolic only if every diamond $J^+(p)\cap J^-(q)$ is compact. The equivalence and its smooth hypotheses are given in Minguzzi 2019, §§4.2–4.5.

Choose two events at the center,

$$p=(t=-\pi/2,\rho=0), \qquad q=(t=+\pi/2,\rho=0).$$

At $t=0$, every event $x_n$ with radius $\rho_n\lt\pi/2$ lies in the diamond: radial null travel from the center takes time $\rho_n$, and $\pi/2\gt \rho_n$. Take $\rho_n\to\pi/2$. The sequence approaches the conformal boundary but has no convergent subsequence in the physical manifold. Hence

$$J^+(p)\cap J^-(q)$$

is not compact. Under the standard boundaryless definition, AdS is not globally hyperbolic.

This proof separates two rungs of the causal hierarchy. Unwrapping the periodic time removes the closed timelike curves of the AdS hyperboloid and supplies a temporal function, but stable causality does not imply compact diamonds. The missing boundary points are a global compactness failure.

Why no bulk slice is Cauchy

A Cauchy hypersurface must meet every inextendible causal curve exactly once. The natural $t=0$ ball fails. A future-directed null curve can enter from the conformal boundary at a time $t_0\gt 0$, cross the bulk, and return to the boundary without meeting $t=0$. Within the physical spacetime the curve is inextendible in both directions because its ideal endpoints are absent.

Changing the bulk slice does not cure the missing boundary control. The compact-diamond proof already rules out every Cauchy surface by the equivalence theorem. Geometrically, a spacelike bulk slice has an edge at conformal infinity; causal curves can begin to its future on that timelike edge.

This is not geodesic incompleteness. Physical null affine parameter diverges at the conformal boundary. The relevant inextendible causal curves have no physical endpoint in the manifold but can be complete while still avoiding a proposed bulk slice. Cauchy predictability and affine completeness are different properties.

Nor does the failure mean that no deterministic theory can be defined. One can adjoin the timelike conformal boundary and formulate an initial-boundary value problem. The resulting notion is global hyperbolicity with timelike boundary or well-posedness under specified boundary conditions, not the standard theorem for a boundaryless spacetime.

Field propagation exposes the missing data

For a normally hyperbolic field on a globally hyperbolic spacetime, compactly supported Cauchy data determine a unique solution with causal support. AdS lacks the geometric hypothesis behind that theorem. A signal can propagate from the timelike boundary to a bulk event, so bulk data on a slice do not determine whether such incoming radiation was present.

The simplest analogue is the wave equation on a half-space. Let $x\gt 0$ and prescribe zero initial data at $t=0$. Without a boundary condition at $x=0$, both the zero solution and a solution produced by later incoming boundary data satisfy the same bulk initial data. Boundary conditions close the evolution problem by specifying how flux behaves at the edge.

For AdS fields, the allowable choices depend on field type, effective mass, dimension, and energy norm. Ishibashi and Wald classify self-adjoint extensions for scalar, electromagnetic, and gravitational perturbations in Ishibashi and Wald 2004, §2 and §§3.1–3.3, pp. 2985–3012. Their result is not “always choose Dirichlet”; in some mass ranges multiple positive self-adjoint extensions exist.

The standard Green-operator theorem for normally hyperbolic equations and the role of global hyperbolicity are developed in Bär, Ginoux, and Pfäffle 2007, Chapter 3, §§3.2–3.5, pp. 73–102. Applying it to AdS without boundary data would omit a required hypothesis.

Boundary conditions are physical structure

A boundary condition determines more than uniqueness. It selects a domain for the evolution generator, controls symplectic or energy flux, and restricts the asymptotic symmetry group. Reflective conditions return radiation to the bulk; transparent conditions generally require coupling to external degrees of freedom rather than a closed AdS system; mixed conditions can encode boundary interactions.

For gravity, the conformal boundary metric and subleading normalizable data are constrained by the Einstein equations. Fixing a boundary conformal class, allowing it to vary, or adding boundary dynamics leads to different phase spaces and charges. A variational principle must be compatible with the same choice.

Thus the causal statement is precise but limited: bulk initial data alone are insufficient. It does not choose the correct boundary condition, establish nonlinear stability, or imply information loss. Those are dynamical and physical questions layered on the timelike-boundary geometry.

Counterexamples to overgeneralization

Minkowski spacetime also has finite-speed signals, but its null infinity is reached only at infinite coordinate time in standard Cauchy evolution, and its causal diamonds are compact. De Sitter spacetime has a spacelike conformal boundary; future infinity behaves as an asymptotic final surface rather than a timelike source of incoming data. The causal character, not the mere presence of a conformal boundary, drives the AdS failure.

A finite radial cavity in Minkowski space resembles AdS only after a timelike wall and its boundary conditions have been added. The cavity is not a counterexample to Minkowski global hyperbolicity; it is a different manifold-with-boundary problem.

Exercise

Using the global AdS conformal metric, show directly that the sequence $x_n=(0,\rho_n,\Omega_0)$ with $\rho_n\to\pi/2$ lies in $J^+(p)\cap J^-(q)$ for the center events chosen above and has no convergent subsequence in the physical spacetime.

Solution

Radial null curves obey $\lvert\Delta t\rvert=\lvert\Delta\rho\rvert$. From $p$ to $x_n$, $\Delta t=\pi/2$ while $\Delta\rho=\rho_n\lt\pi/2$, so a timelike radial curve exists; likewise from $x_n$ to $q$. Thus every $x_n$ lies in the diamond.

Any limit in the coordinate completion has $\rho=\pi/2$, but that surface is excluded from the physical manifold. No subsequence converges in AdS. The diamond is noncompact, so the spacetime fails global hyperbolicity despite the temporal function $t$ and absence of closed causal curves.

Boundary Conditions and AdS Evolution next compares the admissible ways to complete the dynamical problem.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, Chapter 3, §§3.2–3.5, pp. 73–102, European Mathematical Society, 2007.
• Ishibashi, Akihiro, and Robert M. Wald, “Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime”, Classical and Quantum Gravity 21, §2 and §§3.1–3.3, pp. 2981–3014, 2004.
• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§4.2–4.5, 2019.