I am referencing to a passage on wikipedia's page about AdS space:

Because the conformal infinity of AdS is timelike, specifying the initial data on a spacelike hypersurface would not determine the future evolution uniquely (i.e. deterministically) unless there are boundary conditions associated with the conformal infinity.

(from https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Coordinate_patches, 02.12.2023)

Maybe I lack the required knowledge on PDE's, but unfortunately I do not fully understand why the reachability of AdS' conformal boundary requires prescription of boundary condition:

$\quad$ Considers e.g. a Penrose diagram of AdS space, as seen in the figure below. Within that figure, I highlighted two surfaces of constant $t$ (red). In order to calculate the correct evolution of e.g. a wave, one must consider the past lightcones (blue) of the points of evaluation, as these tell you what region of spacetime may causally influence your points of interest when time-stepping from the first surface $t = t_1$ to $t = t_2$. The way I see it, the boundary on surface $t = t_2$, is only influenced by the data on $t = t_1$, the evolution on the boundary may therefore be calculated without prescription of any boundary conditions. So why do we need to prescribe boundary conditions if they may as well be calculated by evolving the initial data?

enter image description here

(original image: https://de.wikipedia.org/wiki/Anti-de-Sitter-Raum#/media/Datei:AntiDeSitterPenrose.png, 02.12.2023)


1 Answer 1


So why do we need to prescribe boundary conditions if they may as well be calculated by evolving the initial data?

Because there could be multiple solutions of the wave equation (corresponding to different boundary conditions) with the same set of initial data. This is what “non-globally-hyperbolic spacetime” means.

Consider a massless scalar field with conformal coupling propagating on universal covering of anti de Sitter spacetime (abbreviated as CAdS, universal covering is needed to avoid closed timelike curves). The key observation, following [$1$], is to note that CAdS spacetime can be conformally mapped onto one half of Einstein static universe (ESU), where the “spatial slices” of AdS are mapped onto a half-spheres of spatial slices of ESU. So a solution of a wave equation on ESU restricted to one half of the universe would provide a solution for our field on CAdS. But the initial data on CAdS do not provide a full set of initial data needed to specify a solution on ESU! And if we consider solutions with initial data on ESU with support restricted to the “other half-sphere”, then such solutions would give rise to solutions on CAdS where zero initial data with time give rise to nontrivial fields. Due to linearity of wave equation such pathological solutions could be added to the evolution of any initial data, making the dynamics non-unique.

Imposing boundary conditions on the fields in CAdS spacetime would eliminate this ambiguity making the dynamics deterministic. In terms of our ESU map such boundary conditions allow us to unambiguously extend initial data from half-sphere onto the whole sphere. For example, we can simply reflect the values of field onto the second half-sphere or we can reflect with the change of sign. Thus we would obtain two types of boundary conditions: Neumann and Dirichlet. Other types of boundary conditions are possible corresponding to the choice of self-adjoint extensions of a certain differential operator [$2$].

  1. Avis, S. J., Isham, C. J., & Storey, D. (1978). Quantum field theory in anti-de Sitter spacetime. Physical Review D, 18(10), 3565, DOI:10.1103/PhysRevD.18.3565.

  2. Ishibashi, A., & Wald, R. M. (2004). Dynamics in non-globally-hyperbolic static spacetimes: III. Anti-de Sitter spacetime. Classical and Quantum Gravity, 21(12), 2981, DOI:10.1088/0264-9381/21/12/012, arXiv:hep-th/0402184.

  • $\begingroup$ Thank you for your comment. Unfortunately, I still have questions: 1.) In [1], it is explained why CAdS is not globally hyperbolic, i.e. due to some "loss of information to spatial infinity". Is the boundary of CAdS "permeable"? What is meant by "information is lost" - is the reflective effect of the boundary only correctly modeled when having a certain B.C. on it? 2.) Wikipedia states that if a spacetime has a Cauchy surface, then it is globlly hyperbolic. However, CAdS does have such surfaces, doesn't it? And why would a timelike boundary contradict the existence of Cauchy surfaces? $\endgroup$
    – Octavius
    Dec 3, 2023 at 17:46
  • $\begingroup$ loss of information means that a massless particle can leave the CAdS space (unless boundary conditions prevent that). From quantum perspective this means that evolution can become non-unitary. CAdS does not have Cauchy surfaces: specifying $\phi$ and $\dot\phi$ at a hypersurface $t=t_0$ does not guarantee a unique evolution, precisely because of a timelike boundary, through which a wave can not only leave but also enter the spacetime without any prior indicators. $\endgroup$
    – A.V.S.
    Dec 4, 2023 at 8:44
  • $\begingroup$ Sorry to keep bothering you, I want to clear up my confusion. According to WALDS General Relativity, a Cauchy surface is a closed achronal set such that every inextensible curve intersects it. Consider now a timelike curve in CAdS that starts (or ends) at the conformal boundary and remains timelike all over. Since the boundary is its starting point (end point, respectively), it is not inextensible and therefore irrelevant to finding a Cauchy surface, right? So how does the boundary even contribute to CAdS being non-globally hyperbolic. $\endgroup$
    – Octavius
    Dec 4, 2023 at 14:17
  • $\begingroup$ Conformal boundary is not a part of a manifold, so if a curve starts at boundary, it can be inextendible. $\endgroup$
    – A.V.S.
    Dec 4, 2023 at 15:19
  • $\begingroup$ Thank you again! $\endgroup$
    – Octavius
    Dec 4, 2023 at 15:29

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