Scalar wave falloff in asymptotically anti–de Sitter backgrounds

Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti–de Sitter spacetimes are investigated. We consider both the $\mathrm{}(2+1)\mathrm{}$-dimensional black hole and $\mathrm{}(3+1)\mathrm{}$-dimensional Schwarzschild–anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the $\mathrm{}(2+1)\mathrm{}$-dimensional black hole background. However, the falloff pattern of the conformal scalar waves in the Schwarzschild–anti-de Sitter background is generally neither exponential nor an inverse power rate, although the approximate falloff of the maximal peak is weakly exponential. We discuss the implications of these results for mass inflation.