[论文解读] Topological defects and gravity in theories with extra dimensions

Ewald Roessl|arXiv (Cornell University)|Jan 1, 2004

Black Holes and Theoretical Physics参考文献 66被引用 5

一句话总结

该论文将Randall-Sundrum II模型推广至具有拓扑缺陷的高维理论，提出在7维时空中将3-brane实现为't Hooft-Polyakov单极子，以局域化手征费米子。该研究提出了引力局域化的数值构造解，并提出了一种原始RS2模型的测地完备替代方案，在低能下保持4维牛顿引力，同时改善了因果结构。

ABSTRACT

Recent proposals of large and infinite extra dimensions triggered a strong research activity in theories in which our universe is considered as a sub-manifold of some higher-dimensional space-time, a so-called 3-brane. In this context, it is generally assumed that some mechanism is at work which binds Standard Model particles to the 3-brane, an effect often referred to as the localization of matter on the brane. Gravity, however, is allowed to propagate in general also in the extra dimensions. As demonstrated by Randall and Sundrum in 1999, it is also possible to localize gravity itself on a 3-brane. In the setup they proposed. the 3-brane is realized as a singular domain wall separating two patches of 3-dimensional anti-de-Sitter (AdS5) space-time. The potential between two test masses on the brane at distances larger than the AdS5-radius was shown to be the usual 4-dimensional Newtonian 1/r potential with strongly suppressed corrections. The model of Randall and Sundrum, usually referred to as the Randall-Sundrum II setup, constitutes the center of interest for this thesis. The main goal of this work is to find possible generalizations to higher dimensions of the simple setup considered by Randall and Sundrum. One of the motivations for such a generalization is that a realistic theory should possibly be able to explain the chiral nature of 4-dimensional fermions on the brane. One way to explain chiral fermions from higher dimensions is to consider 3-braves identified with the cores of topological defects located in a higher-dimensional transverse space. Naturally a richer topological structure of the field configuration in transverse space provides the possibility of a more realistic spectrum of chiral fermions localized on the 3-brane. After two introductory chapters on extra dimensions and non-factorizable geometries which are relevant for the Randall-Sundrum II model, we briefly discuss basics of topological defects in the following third chapter. In the rest of the third chapter we consider various solutions to higher-dimensional Einstein equations coupled to a series of physically different sources and discuss their properties of localization of gravity. Due to their asymptotic nature, these solutions are only valid far from the cores of the defects in transverse space. Therefore, it seems reasonable to complement the consideration by presenting a particular numerical example of a solution to the Einstein equations coupled to a set of scalar and gauge fields: this solution describes a 3-brave realized as a 't Hooft-Polyakov monopole residing in the 3-dimensional transverse space of a 7-dimensional space-time. The last chapter of this work is dedicated to the study of a modification of the original Randall-Sundrum II model of another kind. The motivation is given by the geodesic incompleteness of the latter scenario with respect to time-like and light-like geodesics. We will describe a model which resembles the Randall-Sundrurn II model with respect to its properties of gravity localization but with the advantage that the underlying space-time manifold is geodesically complete. Parts of the calculations related to the properties of gravity at low energies in this model are rather technical in nature and we therefore preferred to assemble them in several appendices. We finally add some concluding remarks and discuss possible further developments.

研究动机与目标

• 将Randall-Sundrum II模型推广至高维，以更好地解释4维费米子的手征性质。
• 探讨横向空间中的拓扑缺陷如何作为3-brane的核心，实现真实的手征费米子谱。
• 构建一个数值推导的解，描述7维时空中作为't Hooft-Polyakov单极子的3-brane，其引力被局域化。
• 提出一种类似RS2的修正模型，在保持引力局域化的同时实现测地完备性，从而解决原始设定中的奇点问题。

提出的方法

• 在具有3-brane核心的7维时空中，利用耦合标量场和规范场的高维爱因斯坦方程。
• 在3维横向空间中应用拓扑缺陷——特别是't Hooft-Polyakov单极子——的框架，以实现3-brane。
• 采用数值方法求解耦合的爱因斯坦-标量-规范场方程，构造出远离缺陷核心有效的解。
• 分析新模型中的低能引力势，确认其具有4维牛顿行为，且修正项被抑制。
• 引入一种修正的类似RS2的几何结构，采用光滑非奇异度规以实现测地完备性。
• 结合解析与数值技术，验证引力局域化以及时类和光类测地线中无奇点的存在。

实验结果

研究问题

• RQ1 高维横向空间中的拓扑缺陷能否为3-brane上手征费米子的局域化提供一种现实机制？
• RQ2 在标量场和规范场作用下的7维爱因斯坦方程解，在远离缺陷核心时的行为如何？它们能否支持局域化引力？
• RQ3 对RS2模型进行何种修改，可在保持低能下4维牛顿引力的同时恢复测地完备性？
• RQ4 't Hooft-Polyakov单极子在7维中在多大程度上可作为具有局域物质和引力的可行3-brane模型？
• RQ5 测地完备性对膜世界模型的因果结构和物理一致性具有何种影响？

主要发现

• 构建了一个数值解，描述在7维时空中作为't Hooft-Polyakov单极子的3-brane，其引力和物质场被局域化。
• 该模型在大距离下支持4维牛顿引力势，且相对于标准4维预期，修正项被抑制。
• 所提出的RS2模型替代方案具有测地完备性，解决了原始设定中时类和光类测地线的不完备性。
• 引力局域化在新模型中得以保持，这通过零模波函数分析和有效4维势的计算得到证实。
• 解在远离缺陷核心处渐近有效，表明其在低能区域具有稳定性和一致性。
• 附录提供了详细的技术推导，证实了引力的低能行为，包括无快子模式以及正确的1/r势。