arXiv 2601.200032026-01-27gr-qchep-th

哥德尔对称背景与显式时空对称破缺

Gödel-symmetric backgrounds and explicit spacetime symmetry breaking

Universidad de Concepción（康塞普西翁大学）；Universidad del Bío-Bío（比奥比奥大学）；Universidade Federal de Mato Grosso（马托格罗索联邦大学）

César Riquelme, Carlos M. Reyes, A. F. Santos

备注：: 16 pages

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AI总结：

本文研究了哥德尔度量与最小引力标准模型扩展中，利用背景场的对称性破缺方法，展示了其与约束方程的一致性及因果性破坏的分析。

摘要：

在引力理论中，使用非动力学背景场显式破坏微分同胚对称性可能导致运动方程与底层伪黎曼几何之间的一致性问题。这些理论产生了一个约束方程，该方程源自修改的爱因斯坦方程和收缩 Bianchi恒等式，其没有简单的解，并引发了对扩展到广义相对论的背景结构一致性的质疑。相比之下，自发对称破缺中，当场获得真空期望值时，这些问题得以避免。然而，某些方法能够一致地实现显式微分同胚破缺：i) 引入几何限制，这些限制可能动态地恢复微分同胚不变性；ii) 使用 Stückelberg 形式主义，其中辅助标量场可以携带与自发破缺对称性相关的自由度；更近期的是 iii) 利用给定引力配置在背景存在下的对称性。最终，这要求背景场必须沿着 Killing 向量 Lie-拖曳。在本文中，我们进一步发展并应用了后者的方法，重点研究涉及哥德尔度量和最小引力标准模型扩展的具体例子。我们证明所得到的动力学完全符合约束方程，并产生新的诺特定理。此外，我们通过分析临界半径来研究因果性破坏，发现该依赖性取决于背景场，这种依赖性允许在时空内出现因果和非因果区域。

英文摘要：

Explicit breaking of diffeomorphism symmetry with nondynamical background fields in gravitational theories can lead to inconsistencies between the equations of motion and the underlying pseudo-Riemannian geometry. These theories produce a constraint equation that follows from the modified Einstein's equations and the contracted Bianchi identity which has no simple solution and raises questions about the consistency of backgrounds structures in extensions to general relativity. In contrast, spontaneous symmetry breaking where fields acquire a vacuum expectation value are shown to avoid these problems. Nevertheless, there are some approaches that consistently implement explicit diffeomorphism breaking by: i) introducing geometrical restrictions which may dynamically restore diffeomorphism invariance, ii) using the Stückelberg formalism where auxiliary scalar fields can carry degrees of freedom associated to spontaneously broken symmetries, and more recently by iii) exploiting the isometries of a given gravitational configuration in the presence of a background; which ultimately requires the background fields to be Lie-dragged along the Killing vectors. In this work, we further develop and apply the latter approach focusing on a concrete example involving the Gödel metric and the minimal gravitational Standard-Model Extension. We show that the resulting dynamics is fully consistent with the constraint equation and produce new Noether's identities. Furthermore we investigate causality violations by analyzing the critical radius, which is found to depend explicitly on the background fields. This dependence allows for the emergence of both causal and non-causal regions in spacetime.

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