Wave computation on the Poincaré dodecahedral space

Abstract

Abstract. We compute the waves propagating on a compact 3-manifold of constant positive curvature with a non trivial topology: the Poincaré dodecahedral space that is a plausible model of multi-connected universe. We transform the Cauchy problem to a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We adopt a variational approach using a space of finite elements that is invariant under the action of the binary icosahedral group. The computation of the transient waves is validated with their spectral analysis by computing a lot of eigenvalues of the Laplace-Beltrami operator.

Waves on accelerating dodecahedral universes

“…The construction is based on the quaternionic calculus. It is detailled in [5]. Here we just present the main features.…”

“…This appendix is devoted to a brief description of F and F v . For more details on the construction of these domains we refer to [5].…”

“…This model has received a lot of interest and generated numerous works (see e.g. [1,5,8,20,25,30,36]). These spacetimes are Lorentzian manifolds (R t × K, g) that are globally hyperbolic and multi-connected.…”

“…To reach this goal we adopt the variational method associated to (I.8) with suitable finite elements satisfying constraint (I.9). In the case of the stationary universes we used in [5] finite elements of P 1 type. To be able to treat the accelerating universes, we need more precision and we use elements of second order.…”

Analytic Calculation of Quasi-Normal Modes

“…The asymptotic form of QNMs for large n is ω n T H = (2n + 1)πi + ln 3 (14) independent of the angular momentum quantum number ℓ. This form was first derived numerically [3][4][5][6][7] and subsequently confirmed analytically [8]. The large imaginary part of the frequency (ℑω n ) makes the numerical analysis cumbersome but is easy to understand because the spacing of frequencies is 2πiT H which is the same as the spacing of poles of a thermal Green function on the Schwarzschild black hole background.…”