The purpose of this work is to introduce a new analytical and numerical approach to the treatment of the initial value problem for the vacuum Einstein field equations on spacetimes with spatial topologies S3 or S1 × S2 and symmetry groups U(1) or U(1)×U(1). The general idea consists of taking advantage of the action of the symmetry group U(1) to rewrite those spacetimes as a principal fiber bundle, which is trivial for S1 × S2 but not for S3. Thus, the initial value problem in four dimensions can be reduced to a three-dimensional initial value problem for a certain manifold with spatial topology S2. Furthermore, we avoid coordinate representations that suffer from coordinate singularities for S2 by expressing all the fields in terms of the spin-weighted spherical harmonics.
We use the generalized wave map formalism to reduce the vacuum Einstein field equations on a manifold with three spatial dimensions to a system of quasilinear wave equations in terms of generalized gauge source functions with well-defined spin-weights. As a result, thanks to the fully tensorial character of these equations, the system of evolution equations can be solved numerically using a 2 + 1-pseudo-spectral approach based on a spin-weighted spherical harmonic transform. In this work, however, we apply our infrastructure to the study of Gowdy symmetric spacetimes, where thanks to the symmetry group U(1) × U(1), the system of hyperbolic equations obtained from the vacuum Einstein field equations can be reduced to a 1+1-system of partial differential equations. Therefore, we introduce an axial symmetric spin-weighted transform that provides an efficient treatment of axially symmetric functions in S2 by reducing the complexity of the general transform.
To analyse the consistency, accuracy, and feasibility of our numerical infrastructure, we reproduce an inhomogeneous cosmological solution of the vacuum Einstein field quations with spatial topology S3 . In addition, we consider two applications of our infrastructure. In the first one, we numerically explore the behaviour of Gowdy S1 × S2 spacetimes using our infrastructure. In particular, we study the behaviour of some geometrical quantities to investigate the behaviour of those spacetimes when approach a future singularity. As a second application, we conduct a systematic investigation on the non-linear instability of the Nariai spacetime and the asymptotic behaviour of its perturbations
