Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 75 / 3 / c / Solution

The universal cover of maximally extended AdS is not globally hyperbolic. Its timelike conformal boundary can send signals into the interior after any chosen initial slice, so bulk initial data alone do not determine evolution. In the conformal strip, a null ray can run from $\rho \to \pi /2$ to the center in global time $\pi /2$; it need not meet a sufficiently earlier constant-time slice within the interior spacetime.

The compact-diamond test makes the obstruction explicit. Choose two central events with times $-2$ and $2$ in units $L=1$. Events at $T=0$, $\rho \to \pi /2$ lie between them causally, because their radial coordinate separation is less than2. They form a sequence in the causal diamond with no limit in the AdS interior. The diamond is therefore noncompact. Boundary conditions, such as reflecting conditions on the conformal boundary, can supply a well-posed initial-boundary-value problem; they do not turn the boundary-free interior into a globally hyperbolic spacetime in the usual definition. The periodic-time hyperboloid is also non-globally-hyperbolic, already because it has closed timelike curves.