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Scalar Wave Falloff in Asymptotically Anti-de Sitter Backgrounds

S. F. J. Chan, R. B. Mann

TL;DR

This work analyzes how conformally invariant scalar waves decay at late times in asymptotically anti-de Sitter spacetimes, specifically in BTZ and Schwarzschild-AdS backgrounds. Using a combination of analytic Green's-function techniques and numerical finite-difference simulations, it shows exponential decay for BTZ cases (including spinning BTZ) and a more nuanced falloff in Schwarzschild-AdS, where late-time behavior features a mix of returning boundary effects and non-pure power-law tails with a weak exponential component for the maximal peak. The results highlight that AdS asymptotics can qualitatively alter tail behavior and thereby affect mass inflation scenarios near inner horizons. Overall, the paper provides a detailed characterization of scalar-wave falloff in non-flat backgrounds and clarifies how geometry controls late-time dynamics with potential implications for horizon stability in various dimensions.

Abstract

Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti-de Sitter are investigated. We consider both the $(2+1)$-dimensional black hole and $(3+1)$-dimensional Schwarzschild-anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the $(2+1)$ 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.

Scalar Wave Falloff in Asymptotically Anti-de Sitter Backgrounds

TL;DR

This work analyzes how conformally invariant scalar waves decay at late times in asymptotically anti-de Sitter spacetimes, specifically in BTZ and Schwarzschild-AdS backgrounds. Using a combination of analytic Green's-function techniques and numerical finite-difference simulations, it shows exponential decay for BTZ cases (including spinning BTZ) and a more nuanced falloff in Schwarzschild-AdS, where late-time behavior features a mix of returning boundary effects and non-pure power-law tails with a weak exponential component for the maximal peak. The results highlight that AdS asymptotics can qualitatively alter tail behavior and thereby affect mass inflation scenarios near inner horizons. Overall, the paper provides a detailed characterization of scalar-wave falloff in non-flat backgrounds and clarifies how geometry controls late-time dynamics with potential implications for horizon stability in various dimensions.

Abstract

Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti-de Sitter are investigated. We consider both the -dimensional black hole and -dimensional Schwarzschild-anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the 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.

Paper Structure

This paper contains 10 sections, 99 equations, 26 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Wave Equation in $D+1$ Dimensions
  3. Asymptotically Flat Backgrounds
  4. The 3D Black Hole Background
  5. Solution for an Approximate Potential
  6. Higher order correction
  7. Spinning 3D Black Hole Background
  8. Schwarzschild-Anti-de Sitter Background
  9. Conclusion
  10. Solutions in asymptotically flat $(D+1)$-dimensional background

Figures (26)

  • Figure 1: The decay of a scalar wave in Schwarzschild background. Prior to $t=200$ the decay is accompanied by 'ringing' of the quasi-normal modes, after which the falloff rate is that of an inverse power-law.
  • Figure 2: Potential barrier for the Schwarzschild background
  • Figure 3: Exponential decrease on the left side of the barrier in $V(x)$
  • Figure 4: Inverse power decrease on the right side of the barrier of $V(x)$
  • Figure 5: Potential barrier for the Schwarzschild-de Sitter background
  • ...and 21 more figures