球面周围轴对称和静止二阶微扰的规范固定和规则性

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Advances in Theoretical and Mathematical Physics Pub Date : 2020-07-24 DOI:10.4310/atmp.2022.v26.n6.a8
M. Mars, B. Reina, R. Vera
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引用次数: 3

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,需要深入了解规范固定的过程。这是一系列两篇论文的第一篇,旨在推导广义相对论中微扰理论中二阶刚性旋转恒星的存在唯一性结果。一个必要的步骤是证明一个合适的规范选择的存在性,并理解一些“规范形式”的规范张量的可微性和正则性,特别是在恒星的中心。考虑到更广泛的应用,本文在更一般的情况下分析了固定和规则问题。特别地,我们解决了hodge型分解为具有有限可微性的对称和轴对称张量球体上的标量,矢量和张量分量的问题,利用了一种策略,其中可微性的损失尽可能低。我们的主要兴趣和主要结果,是证明围绕静态和球对称背景构型的静态和轴对称二阶扰动确实可以用文献中使用的通常的“规范形式”来表示,同时只失去一次可微性并保持所有相关量在原点附近有界。
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Gauge fixing and regularity of axially symmetric and axistationary second order perturbations around spherical backgrounds
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.
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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