Non-Relativistic Regime and Topology: Topological Term in the Einstein Equation

IF 1.2 3区物理与天体物理 Q3 PHYSICS, MULTIDISCIPLINARY

Foundations of Physics Pub Date : 2024-01-19 DOI:10.1007/s10701-023-00749-z

引用次数: 0

Abstract

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of the Einstein equation is only possible in Euclidean topologies, i.e., for which the covering space is \(\mathbb {E}^3\) . We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a universe with a non-Euclidean topology and admitting a non-relativistic limit.

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非相对论时态与拓扑学：爱因斯坦方程中的拓扑术语

摘要我们研究了相对论时空的非相对论（NR）极限与宇宙拓扑的关系。我们首先证明了爱因斯坦方程的非相对论极限只可能存在于欧几里得拓扑结构中，即覆盖空间为\(\mathbb {E}^3\) 。我们将这一结果解释为广义相对论在非欧几里得拓扑中的不一致性，并提出了对广义相对论的修正，允许在任何拓扑中进行极限。为此，除了物理时空连接之外，还引入了第二个参考非动态连接。参考连接的选择与时空拓扑的覆盖空间有关。修正后的爱因斯坦方程不再仅以物理时空里奇张量为特征，而是以物理和参考里奇张量之间的差异为特征。如果想研究一个具有非欧几里得拓扑结构并允许非相对论极限的宇宙，就应该考虑用这一理论来代替广义相对论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Foundations of Physics 物理-物理：综合

CiteScore

2.70

自引率

6.70%

发文量

104

审稿时长

6-12 weeks

期刊介绍： The conceptual foundations of physics have been under constant revision from the outset, and remain so today. Discussion of foundational issues has always been a major source of progress in science, on a par with empirical knowledge and mathematics. Examples include the debates on the nature of space and time involving Newton and later Einstein; on the nature of heat and of energy; on irreversibility and probability due to Boltzmann; on the nature of matter and observation measurement during the early days of quantum theory; on the meaning of renormalisation, and many others. Today, insightful reflection on the conceptual structure utilised in our efforts to understand the physical world is of particular value, given the serious unsolved problems that are likely to demand, once again, modifications of the grammar of our scientific description of the physical world. The quantum properties of gravity, the nature of measurement in quantum mechanics, the primary source of irreversibility, the role of information in physics – all these are examples of questions about which science is still confused and whose solution may well demand more than skilled mathematics and new experiments. Foundations of Physics is a privileged forum for discussing such foundational issues, open to physicists, cosmologists, philosophers and mathematicians. It is devoted to the conceptual bases of the fundamental theories of physics and cosmology, to their logical, methodological, and philosophical premises. The journal welcomes papers on issues such as the foundations of special and general relativity, quantum theory, classical and quantum field theory, quantum gravity, unified theories, thermodynamics, statistical mechanics, cosmology, and similar.

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