Hyperboloidal slices are spacelike slices that reach future null infinity. Their asymptotic behaviour is different from Cauchy slices, which are traditionally used in numerical relativity simulations. This work uses free evolution of the formally-singular conformally compactified Einstein equations in spherical symmetry. One way to construct gauge conditions suitable for this approach relies on building the gauge source functions from a time-independent background spacetime metric. This background reference metric is set using the height function approach to provide the correct asymptotics of hyperboloidal slices of Minkowski spacetime. The present objective is to study the effect of different choices of height function on hyperboloidal evolutions via the reference metrics used in the gauge conditions. A total of 10 reference metrics for Minkowski are explored, identifying some of their desired features. They include 3 hyperboloidal layer constructions, evolved with the non-linear Einstein equations for the first time. Focus is put on long-term numerical stability of the evolutions, including small initial gauge perturbations. The results will be relevant for future (puncture-type) hyperboloidal evolutions, 3D simulations and the development of coinciding Cauchy and hyperboloidal data, among other applications.

双曲面切片是达到未来零无穷大的类似空间的切片。它们的渐近行为不同于传统上用于数值相对论模拟的柯西切片。这项工作使用了球对称性中形式奇异共形压缩爱因斯坦方程的自由进化。构建适合此方法的规范条件的一种方法依赖于从与时间无关的背景时空度量构建规范源函数。此背景参考指标是使用高度函数方法设置的，以提供 Minkowski 时空双曲面切片的正确渐近。目前的目标是通过仪表条件中使用的参考指标研究不同高度函数选择对双曲面演化的影响。总共探讨了 Minkowski 的 10 个参考指标，确定了它们所需的一些特征。它们包括 3 个双曲面层结构，首次与非线性爱因斯坦方程一起发展。重点放在演化的长期数值稳定性上，包括小的初始规范扰动。结果将与未来的（穿刺型）双曲面演化、3D 模拟以及重合柯西和双曲面数据的开发以及其他应用相关。