球形背景周围轴对称与轴稳态二阶微扰的规范固定和正则性
Gauge fixing and regularity of axially symmetric and axistationary second order perturbations around spherical backgrounds
AI总结：
本文研究球形背景周围稳态轴对称二阶微扰的规范固定与正则性，通过低可微性损失的 Hodge 型分解证明其可化为标准形式并在原点附近保持有界。
摘要：
几何引力理论中的微扰理论是定义在洛伦兹流形（背景时空）上的对称张量的规范理论。规范自由度使微扰理论中的唯一性问题尤为困难，因为在尝试任何唯一性证明之前，需要深入理解规范固定过程。本文是由两篇论文组成的系列中的第一篇，旨在推导广义相对论中刚性旋转恒星在微扰理论二阶意义下的存在性和唯一性结果。一个必要步骤是证明存在适当的规范选择，并理解所得规范张量在某种“标准形式”中的可微性和正则性，尤其是在恒星中心。考虑到更广泛的应用范围，本文在更一般的设定下分析规范固定和正则性问题。特别地，我们处理在球面上将具有有限可微性且性质一直延伸到原点的对称及轴对称张量分解为标量、向量和张量分量的 Hodge 型分解问题，采用一种使可微性损失尽可能低的策略。我们的主要关注点和主要结果是证明：围绕静态球对称背景构型的稳态轴对称二阶微扰，确实可以化为文献中使用的通常“标准形式”，同时仅损失一个可微性阶数，并使所有相关量在原点附近保持有界。
英文摘要：
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 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 loosing only one degree of differentiability and keeping all relevant quantities bounded near the origin.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
