# Puncture black hole initial data: a single domain Galerkin-Collocation method for trumpet and wormhole data sets

## TL;DR

This paper introduces a unified Galerkin-Collocation method in spherical coordinates for efficiently computing initial data for single and binary black holes in trumpet and wormhole geometries, including spinning cases.

## Contribution

A novel single domain Galerkin-Collocation approach that unifies the treatment of trumpet and wormhole black hole initial data in spherical coordinates.

## Key findings

- Effective combination of Galerkin and Collocation methods.
- Unified expression for conformal factor in different geometries.
- Demonstrated deformation of the limit surface due to spin.

## Abstract

We present a single domain Galerkin-Collocation method to calculate puncture initial data sets for single and binary, either in the trumpet or wormhole geometries. The combination of aspects belonging to the Galerkin and the Collocation methods together with the adoption of spherical coordinates in all cases show to be very effective. We have proposed a unified expression for the conformal factor to describe trumpet and spinning black holes. In particular, for the spinning trumpet black holes, we have exhibited the deformation of the limit surface due to the spin from a sphere to an oblate spheroid. We have also revisited the energy content in the trumpet and wormhole puncture data sets. The algorithm can be extended to describe binary black holes.