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arXiv 1203.3563gr-qchep-th
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任意维的极小张量和纯电或纯磁时空

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

Sigbjørn Hervik, Marcello Ortaggio, Lode Wylleman

备注：
43 pages. v2: new proposition 4.10; some text reshuffled (former sec. 2 is now an appendix); references added; some footnotes cancelled, others incorporated into the main text; some typos fixed and a few more minor changes made
发表信息：
Class. Quantum Grav. 30 (2013) 165014
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摘要：

我们考虑时间反演变换以获得任意维n的洛伦兹空间上的任何张量的双重正交分裂。应用于时空的外尔张量，这导致了相对于观察者的电和磁部分的定义（即，一个单位类时向量场u），在任何n中。我们研究的情况下，其中一个部分消失，特别是，即，纯电（PE）或磁（PM）时空。我们推广了一些结果从四维到高维，并讨论了高维的新特征。我们证明了唯一允许的Weyl类型是G，I_i和D，并讨论了u与WAND的可能关系，给出了描述PE/PM时空的不变量条件，如Bel-Debever准则或标量不变量的约束，并将PE/PM部分与u的运动学量联系起来;我们提出的条件下，直积时空（和某些扭曲）是PE/PM，这使我们能够构建明确的例子。特别是，它还表明，所有静态时空必然是PE，而稳态时空（例如，旋转的黑洞）通常既不是PE也不是PM。充足的类PE时空存在，但PM解决方案是难以捉摸的，我们证明了PM爱因斯坦时空的D型不存在，任何n。最后，我们得到相应的结果的电/磁部分的黎曼张量。这也导致了PM时空在更高维度的第一个例子。我们还注意到，PE/PM外尔张量提供了最小张量的例子，我们最近证明的对齐定理的连接。这反过来又为基于零对齐的外尔张量分类提供了新的思路，提供了进一步的不变特征，将G/I/D型与II/III/N型区分开来。

英文摘要：

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (i.e., a unit timelike vector field u), in any n. We study the cases where one of these parts vanishes in particular, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. We prove that the only permitted Weyl types are G, I_i and D, and discuss the possible relation of u with the WANDs; we provide invariant conditions that characterize PE/PM spacetimes, such as Bel-Debever criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (e.g., spinning black holes) are in general neither PE nor PM. Ample classes of PE spacetimes exist, but PM solutions are elusive, and we prove that PM Einstein spacetimes of type D do not exist, for any n. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor. This also leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem. This in turn sheds new light on classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the types G/I/D from the types II/III/N.

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