Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view
Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view
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DOI：
10.1063/1.4937445
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发表时间：
2016-02-01
影响因子：
1.3
通讯作者：
Morand, Kevin
中科院分区：
文献类型：
--
作者：
Bekaert, Xavier;Morand, Kevin
非相对论时空的“度量”结构由一种单一形式（绝对时钟）组成，其内核被赋予正定度量。与相对论情况相反，度量结构和挠率不能确定唯一的伽利略（即兼容）连接。这种微妙之处与以下事实密切相关：挠率的类时部分与绝对时钟的外导数成正比。当后者不闭合时，自由扭转和公制兼容性是相互排斥的。我们将在一系列论文中探索沿着两条相应替代道路的伽利略连接的概括。在本研究中，我们重点关注兼容连接并研究无扭和扭转情况下的等效问题（即搜索允许唯一确定连接的必要数据）。更准确地说，我们描述了此类连接空间的仿射结构并显示相关的模型向量空间。与相对论情况相反，度量结构并没有为度量兼容连接的空间挑选出一个特权起源。在我们的构造中，列维-奇维塔连接的作用是由一整类特权起源发挥的，即最近在文献中研究的所谓扭转牛顿-嘉当几何。最后，我们讨论牛顿力学与扭转情况的联系的推广。 (C) 2016 AIP 出版有限责任公司。
The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e., compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e., the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case. (C) 2016 AIP Publishing LLC.