Nonlinear (in)stability of Anti–de Sitter under different boundary conditions

Establish that the Anti–de Sitter spacetime solving the Einstein vacuum equations with negative cosmological constant is nonlinearly unstable under reflective boundary conditions at conformal infinity and asymptotically stable under optimally dissipative boundary conditions at conformal infinity.

Background

The paper reviews linear decay properties for the conformal wave equation on Anti–de Sitter and Schwarzschild–Anti–de Sitter backgrounds under various boundary conditions, and discusses implications for gravitational perturbations. Under reflective (e.g., Dirichlet) conditions, AdS admits time-periodic solutions and shows no decay for the scalar model, while under dissipative conditions, arbitrarily fast inverse polynomial decay of energy (with derivative loss) has been established in pure AdS.

Building on these linear analyses and on further studies of gravitational perturbations, the cited literature has proposed a broader dynamical picture for the full Einstein vacuum equations with a negative cosmological constant. The conjecture highlighted in this paper articulates that Anti–de Sitter should be nonlinearly unstable when energy is perfectly reflected at the boundary, but asymptotically stable when the boundary is optimally dissipative.

Confirming or refuting this conjecture would significantly advance the understanding of AdS dynamics and boundary effects in the nonlinear regime of general relativity.

References

The above results plus further analyses of gravitational perturbations presented in , lead the authors to conjecture that Anti-de Sitter spacetime is non-linearly unstable for reflective and asymptotically stable for optimally dissipative boundary conditions.

— Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions  (2604.02084 - Tullini, 2 Apr 2026) in Subsection "Further discussion" (Introduction)

There is considerable debate on which boundary conditions render the Cauchy problem well-posed in gravity. There are arguments against DBC and for CBC, but also CBC is still under debate .

— Brane effective actions and their island rule from $T\overline T$ flows  (2608.30533 - Callebaut et al., 31 Aug 2026) in Section 'Discussion and outlook', paragraph 'Higher dimensions and well-posedness of the boundary conditions'