In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to
Einsteins equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einsteins equations obtained in the generalized wave map gauge formalism combined with Gerochs symmetry reduction, and focus on cosmological spacetimes with spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7].
We present a new spectral scheme for analysing functions of half-integer spin-weight on the $2$-sphere and demonstrate the stability and convergence properties of our implementation. The dynamical evolution of the Dirac equation on a manifold with sp
atial topology of $mathbb{S}^2$ via pseudo-spectral method is also demonstrated.
In a recent paper (Beyer and Hennig, 2012 [9]), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub-NUT solutions. Here we derive a three-parametric family of exact solutions within this class,
which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a true spike in analogy to previously known Gowdy symmetric solutions with spatial T3-topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain false spikes.
Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through use of integer spin weighted spherical harmonics we present a method which allows for the implementation of arbitrary tensorial ev
olution equations. Our method combines two numerical techniques that were originally developed with different applications in mind. The first is Huffenberger and Wandelts spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Lubans method, to convert numerically divergent linear recursions into stable nonlinear recursions, to the calculation of reduced Wigner d-functions. We give a detailed discussion of the theory and numerical implementation of our algorithm. The properties of our method are investigated by solving the scalar and vectorial advection equation on the sphere, as well as the 2+1 Maxwell equations on a deformed sphere.
The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.
