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Abstract

We present a new formulation of Einstein's equations for an axisymmetric
spacetime with vanishing twist in vacuum. We propose a fully constrained scheme
and use spherical polar coordinates. A general problem for this choice is the
occurrence of coordinate singularities on the axis of symmetry and at the
origin. Spherical harmonics are manifestly regular on the axis and hence take
care of that issue automatically. In addition a spectral approach has
computational advantages when the equations are implemented. Therefore we
spectrally decompose all the variables in the appropriate harmonics. A central
point in the formulation is the gauge choice. One of our results is that the
commonly used maximal-isothermal gauge turns out to be incompatible with tensor
harmonic expansions, and we introduce a new gauge that is better suited. We
also address the regularisation of the coordinate singularity at the origin.