Wave Computation on the Hyperbolic Double Doughnut

Abstract

We compute the waves propagating on the compact surface of constant negative curvature and genus 2 that is a toy model in quantum chaos theory and cosmic topology. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. Despite the ergodicity of the dynamics that is quantum weak mixing, the computation is very accurate. A spectral analysis of the transient waves allows to compute the spectrum and the eigenfunctions of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping satisfying the assumption of geometric control.Mathematics subject classification: 65M, 65N, 58J, 37D.

Stability of Reissner-Nordström black hole in de Sitter background under charged scalar perturbation

“…This supported the argument in Ref. [22] and the necessity of a bound state in the equivalent Schrödinger problem [23]. We have further explored the physical nature about why the instability is triggered by the l = 0 mode in the four-dimensional RN-dS black hole.…”

“…They found that some potentials with a negative gap still do not imply instability. The criterion to determine whether a system is stable or not against linear perturbation is whether the timedomain profile for the evolution of the perturbation is decaying or not, or, in more general terms, the potential has to permit the existence of bound states [23]. Thus, in order to study the stability of the RN-dS black hole against charged scalar perturbation, we have to examine the evolution of the perturbation.…”

Waves on accelerating dodecahedral universes

“…= 0, for various t * . This is the initial data presented in the first picture of figure (4). First we check the accuracy of the numerical localization of a future horizon.…”

“…We remark that the initial perturbation of the metric leads to a temperature fluctuation that shows spherical correlations along six pairs of antipodal matched circles (the famous circles-in-the-sky) that are a signature of a complex topology with a positive curvature. In contrast, a complex topology and a negative curvature lead to chaotic temperature fluctuations [4]. We now present a more complex solution associated to an initial data u(t * , .)…”

“…An approximated formula for low spherical harmonics is also given by the ordinary Sachs-Wolfe formula, (I. 4) δT (θ, ϕ) T 1 3 Ψ(η ls , χ = η 0 − η ls , θ, ϕ).…”