Perturbation theory in geometric theories of gravitation is a gauge theory of symmetric tensors defined on a Lorentzian manifold (the background spacetime). The gauge freedom makes uniqueness problems in perturbation theory particularly hard as one needs to understand in depth the process of gauge fixing before attempting any uniqueness proof. This is the first paper of a series of two aimed at deriving an existence and uniqueness result for rigidly rotating stars to second order in perturbation theory in General Relativity. A necessary step is to show the existence of a suitable choice of gauge and to understand the differentiability and regularity properties of the resulting gauge tensors in some “canonical form”, particularly at the centre of the star. With a wider range of applications in mind, in this paper we analyse the gauge fixing and regularity problem in a more general setting. In particular we tackle the problem of the Hodge-type decomposition into scalar, vector and tensor components on spheres of symmetric and axially symmetric tensors with finite differentiability down to the origin, exploiting a strategy in which the loss of differentiability is as low as possible. Our primary interest, and main result, is to show that stationary and axially symmetric second order perturbations around static and spherically symmetric background configurations can indeed be rendered in the usual “canonical form” used in the literature while losing only one degree of differentiability and keeping all relevant quantities bounded near the origin.

中文翻译：

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球面背景下轴对称和轴静止二阶扰动的规范固定和规律性

几何引力理论中的微扰理论是在洛伦兹流形（背景时空）上定义的对称张量的规范理论。规范自由使得微扰理论中的唯一性问题变得特别困难，因为在尝试任何唯一性证明之前需要深入理解规范固定的过程。这是一系列论文中的第一篇，旨在导出广义相对论微扰理论中刚性旋转恒星二阶的存在性和唯一性结果。必要的一步是证明存在合​​适的规范选择，并了解所得规范张量在某种“规范形式”中的可微性和规律性特性，特别是在恒星中心。考虑到更广泛的应用，在本文中，我们在更一般的环境中分析了规范固定和规律性问题。特别是，我们利用一种可微性损失尽可能低的策略，解决了在对称和轴对称张量球上将 Hodge 型分解为标量、向量和张量分量的问题，并具有有限可微性直至原点。我们的主要兴趣和主要结果是表明，围绕静态和球对称背景配置的静态和轴对称二阶扰动确实可以以文献中使用的通常“规范形式”呈现，同时丢失 对称和轴对称张量球上的向量和张量分量，具有有限可微分至原点，采用可微分损失尽可能低的策略。我们的主要兴趣和主要结果是表明，围绕静态和球对称背景配置的静态和轴对称二阶扰动确实可以以文献中使用的通常“规范形式”呈现，同时丢失 对称和轴对称张量球上的向量和张量分量，具有有限可微分至原点，采用可微分损失尽可能低的策略。我们的主要兴趣和主要结果是表明，围绕静态和球对称背景配置的静态和轴对称二阶扰动确实可以以文献中使用的通常“规范形式”呈现，同时丢失只有一级可微性，并将所有相关量保持在原点附近。